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Existence of groundstates for a class of nonlinear Choquard equations

Existence of groundstates for a class of nonlinear Choquard equations TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 9, September 2015, Pages 6557–6579 S 0002-9947(2014)06289-2 Article electronically published on December 18, 2014 EXISTENCE OF GROUNDSTATES FOR A CLASS OF NONLINEAR CHOQUARD EQUATIONS VITALY MOROZ AND JEAN VAN SCHAFTINGEN 1 N Abstract. We prove the existence of a nontrivial solution u ∈ H (R )to the nonlinear Choquard equation −Δu + u = I ∗ F (u) F (u)in R , where I is a Riesz potential, under almost necessary conditions on the non- linearity F in the spirit of Berestycki and Lions. This solution is a groundstate and has additional local regularity properties; if moreover F is even and mono- tone on (0, ∞), then u is of constant sign and radially symmetric. Contents 1. Introduction 6557 2. Construction of a solution 6562 3. Regularity of solutions and Pohoˇ zaev identity 6566 4. From solutions to groundstates 6571 5. Qualitative properties of groundstates 6573 6. Alternative proof of the existence 6576 References 6577 1. Introduction We consider the problem (P ) −Δu + u = I ∗ F (u) f (u)in R , where N ≥ 3, α ∈ (0,N ), I : R → R is the Riesz potential defined for every x ∈ R \{0} by N −α Γ( ) I (x)= , N/2 α N −α Γ( )π 2 |x| Received by the editors March 14, 2013 and, in revised form, September 22, 2013. 2010 Mathematics Subject Classification. Primary 35J61; Secondary 35B33, 35B38, 35B65, 35Q55, 45K05. Key words and phrases. Stationary Choquard equation, stationary nonlinear Schr¨ odinger– Newton equation, stationary Hartree equation, Riesz potential, nonlocal semilinear elliptic prob- lem, Pohoˇ zaev identity, existence, variational method, groundstate, mountain pass, symmetry, polarization. The second author was supported by the Grant n. 2.4550.10 “Etude qualitative des solutions d’´ equations aux d´ eriv´ ees partielles elliptiques” of the Fonds de la Recherche Fondatementale Col- lective (F´ ed´ eration Wallonie–Bruxelles). c 2014 American Mathematical Society Reverts to public domain 28 years from publication License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6558 VITALY MOROZ AND JEAN VAN SCHAFTINGEN F ∈ C (R; R)and f := F . Solutions of (P ) are formally critical points of the functional defined by 1 1 2 2 I(u)= |∇u| + |u| − I ∗ F (u) F (u). 2 2 N N R R We are interested in the existence and some qualitative properties of solutions to (P ). Problem (P ) is a semilinear elliptic equation with a nonlocal nonlinearity. For N =3, α =2 and F (s)= it covers in particular the Choquard–Pekar equation 2 3 (1.1) −Δu + u =(I ∗|u| )u in R , introduced at least in 1954, in a work by S. I. Pekar describing the quantum me- chanics of a polaron at rest [32]. In 1976 P. Choquard used (1.1) to describe an electron trapped in its own hole, in a certain approximation to Hartree–Fock the- ory of one component plasma [22]. In 1996 R. Penrose proposed (1.1) as a model of self-gravitating matter [30]. In this context equation (1.1) is usually called the non- linear Schr¨ odinger–Newton equation. Note that if u solves (1.1), then the function it ψ defined by ψ(t, x)= e u(x) is a solitary wave of the focusing time-dependent Hartree equation 2 N iψ +Δψ = −(I ∗|ψ| )ψ in R × R . t 2 + In this context (1.1) is also known as the stationary nonlinear Hartree equation. The existence of solutions for stationary equation (1.1) was proved by variational methods by E. H. Lieb, P.-L. Lions and G. Menzala [22, 24, 28] and also by ordinary differential equations techniques [11, 30, 38]. In the more general case of equation 1 p (P)with F (s)= |s| ,problem (P ) is known to have a solution if and only if N +α N +α <p < ([27, p. 457], [31, Theorem 1]; see also [15, Lemma 2.7]). N N −2 The existence results for (P ) up until now were only available when the nonlin- earity F is homogeneous. This situation contrasts with the striking existence result for the corresponding local problem (1.2) −Δu + u = g(u)in R , which can be considered as a limiting problem of (P)when α → 0, with g = Ff . H. Berestycki and P.-L. Lions [6, Theorem 1] have proved that (1.2) has a nontrivial solution if nonlinearity g ∈ C (R; R) satisfies the assumptions 2N N−2 (g ) there exists C> 0 such that for every s ∈ R, sg(s) ≤ C |s| + |s| , G(s) 1 G(s) (g ) lim < and lim sup ≤ 0, 2N s→0 |s| 2 N−2 |s|→∞ |s| (g ) there exists s ∈ R \{0} such that G(s ) > , 3 0 0 where G(s)= g(σ)dσ (and if g = Ff,then G = ). They also proved that 0 2 ∞ N if u ∈ L (R ) is a finite energy solution of (1.2), then u satisfies the Pohoˇ zaev loc identity [6, Proposition 1] N − 2 N 2 2 (1.3) |∇u| + |u| = N G(u). 2 2 N N N R R R This, in particular, implies that assumptions (g ), (g )and (g ) are “almost nec- 1 2 3 essary” for the existence of nontrivial finite energy solutions of (1.2). Indeed, the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6559 necessity of (g ) follows directly from (1.3). For (g )and (g ), if f (s)= s with 3 1 2 N +2 s ∈ (1, ), then (1.3) immediately implies that (1.2) does not have any bounded N −2 finite-energy nontrivial solution. In this spirit, we prove the existence of solutions to Choquard equation (P ), assuming that nonlinearity f ∈ C (R; R) satisfies the growth assumption: N +α N +α N N−2 (f )there exists C> 0 such that for every s ∈ R, |sf (s)|≤ C |s| + |s| , its antiderivative F : s ∈ R → f (σ)dσ is subcritical : F (s) F (s) (f ) lim = 0 and lim =0, N +α N +α s→0 |s|→∞ N N−2 |s| |s| and nontrivial : (f)thereexists s ∈ R \{0} such that F (s ) =0. 3 0 0 It is standard to check using Hardy–Littlewood–Sobolev inequality that if f ∈ C (R; R) satisfies growth assumption (f ), then I defines on the Sobolev space 1 N H (R ) a continuously differentiable functional and critical points of I are weak solutions of equation (P ). In what follows, solutions of (P ) are always understood in the weak sense. 1 N We say u ∈ H (R ) \{0} is a groundstate of (P)if u is a solution of (P)and 1 N (1.4) I(u)= c := inf I(v): v ∈ H (R ) \{0} is a solution of (P ) . Our main result in this paper is the following. Theorem 1 (Existence of a groundstate). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ C (R; R) satisfies (f ), (f ) and (f ),then (P ) has a groundstate. 1 2 3 We also prove that any weak solution of (P ) has additional regularity properties. Theorem 2 (Local regularity). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ C (R; R) 2,q 1 N N satisfies (f ) and u ∈ H (R ) solves (P ), then for every q ≥ 1, u ∈ W (R ). loc In particular, Theorem 2 with the Morrey–Sobolev embeddings implies that solutions of (P ) are locally H¨ older continuous. If f has additional smoothness, then regularity of u could be further improved via Schauder estimates. Let us emphasize that Theorem 2 is established only under the growth assumption (f ) and does not require additional subcriticality assumption (f ). The regularity information of Theorem 2 allows us to establish a Pohoˇ zaev inte- gral identity for all finite energy solutions of (P ). Theorem 3 (Pohoˇ zaev identity). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ 1 N C (R; R) satisfies (f ) and u ∈ H (R ) solves (P ),then N − 2 N N + α 2 2 (1.5) |∇u| + |u| = I ∗ F (u) F (u). 2 N 2 N 2 N R R R In particular, (1.5) implies that if u = 0 is a solution of (P ), then α +2 α 2 2 I(u)= |∇u| + |u| > 0. 2(N + α) 2(N + α) N N R R Pohoˇ zaev identity (1.5) shows that our assumptions (f ), (f )and (f )are 1 2 3 “almost necessary” for the existence of nontrivial solutions to (P ). Indeed, if 1 p F (s)= |s| , then (1.5) implies that problem (P ) does not have nontrivial weak License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6560 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 1 N N +α N +α solutions in H (R )if p ∈ , (see also [31, Theorem 2] where super- N N −2 1 N critical ranges of p are included). If (f ) fails, then a solution u ∈ H (R )would satisfy −Δu + u = 0 and would then necessarily be trivial. Whereas the upper critical exponent (N + α)/(N − 2) appears as a natural extension of the critical Sobolev exponent 2N/(N − 2) for the local problem (1.2) with G = F , the lower critical exponent (N + α)/N in assumptions (f )and (f ) 1 2 is a new phenomenon. It is due to the effect of the nonlocal term in (P ) and has (N +α)/N no analogues in (1.2). The growth restriction |sf (s)|≤ c|s| for |s| < 1 occurs naturally in the application of the Hardy–Littlewood–Sobolev inequality to 1 1 N verify that I∈ C (H (R ); R). In fact, Pohoˇ zaev identity confirms that the power (N + α)/N is optimal for the existence of solutions, and in this respect it plays the role of the lower critical exponent for (P ). Finally, we obtain qualitative properties of groundstates of (P ), which are the counterpart of the properties obtained for solutions of the corresponding local equa- tion [10, 14, 16]. Theorem 4 (Qualitative properties of groundstates). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ C (R; R) satisfies (f ) and, in addition, f is odd and has constant sign on (0, ∞), then every groundstate of (P ) has constant sign and is radial ly symmetric with respect to some point in R . Before explaining the proofs of our results, we recall the strategy of H. Berestycki and P.-L. Lions’s proof of the existence of solutions to (1.2) [6, §3]. They consider the constrained minimization problem |u| 2 1 N (1.6) min |∇u| : u ∈ H (R )and G(u) − =1 ; N N 2 R R they first show that by the Poly ´ a–Szeg˝ o inequality for the Schwarz symmetrization, the minimum can be taken on radial and radially nonincreasing functions. Then 1 N they show the existence of a minimum v ∈ H (R ) by the direct method of the calculus of variations. This minimum v satisfies the equation −Δv = θ g(v) − v in R , 1 N with a Lagrange multiplier θ> 0. They conclude by noting that u ∈ H (R ) defined for x ∈ R by u(x)= v(x/ θ) solves (1.2). The approach of H. Berestycki and P.-L. Lions fails for nonlocal problem (P)for two different reasons. First, the nonlocal term will not be preserved or controlled under Schwarz symmetrization unless the nonlinearity f satisfies the more restric- tive assumption of Theorem 4. Second, the final scaling argument fails: the three terms in (P ) scale differently in space, so one cannot hope to get rid of a Lagrange multiplier by scaling in space. In general, a constrained minimization of type (1.6) cannot be used for the study of solutions of equations with multiple scaling rates. Similar issues of multiple scaling rates arise, for instance, in the study of nonlocal nonlinear Schr¨ odinger–Maxwell or Schr¨ odinger–Poisson equations. For instance, the existence of a radial groundstate solution to a class of Schr¨ odinger–Maxwell equations under general Berestycki–Lions type assumptions on the nonlinear term was established in [2] by applying the mountain–pass theorem to a family of trun- cated functionals and then by proving the convergence of the obtained sequence of radially symmetric critical points using the radial compactness lemma of Strauss. Despite some similarities, the structure of Schr¨ odinger–Maxwell equations is very License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6561 different with Choquard equations and thus new techniques are required for the study of (P ). Moreover, such results establish the existence of radial groundstates while we are interested in the construction of global groundstates. In the present work, in order to prove the existence of solutions of (P ), instead of the constrained minimization problem of type (1.6), we consider in section 2 the mountain pass level (1.7) b =inf sup I γ(t) , γ∈Γ t∈[0,1] where the set of paths is defined as 1 N (1.8) Γ = γ ∈ C [0, 1]; H (R ) : γ(0) = 0 and I(γ(1)) < 0 . Classically, in order to show that b is a critical level of the functional I, one con- structs a Palais–Smale sequence at the level b, that is, a sequence (u ) in n n∈N 1 N H (R ) such that I(u ) → b and I (u ) → 0as n →∞. Then one proves that n n the sequence (u ) converges up to translations and extraction of a subsequence n n∈N [37, 43]. The first step of this approach is to establish the boundedness of the se- 1 N quence (u ) in H (R ). Usually this involves an Ambrosetti–Rabinowitz type n n∈N superlinearity assumption, which in our setting would require the existence of μ> 1 + μ such that s ∈ R → F (s)/s is nondecreasing. In order to avoid an Ambrosetti–Rabinowitz type condition, in section 2 we employ a scaling technique introduced by L. Jeanjean. It consists in constructing a Palais–Smale sequence that satisfies asymptotical ly the Pohoˇ zaev identity [18] (see also [1, 2, 4, 17] ). This improvement is related to the monotonicity trick of M. Struwe [37, §II.9] and L. Jeanjean [19]. Next, we prove with a concentration compactness argument the existence of a nontrivial solution u to (P ) under the assumptions (f ), (f )and (f ) only. This combination of the scaling technique with a concentration- 2 3 compactness argument which does not rely on the radial compactness and a priori radial symmetry of the solution is a novelty in our proof. To conclude that such a constructed solution u is a groundstate, we first show that I(u)= b. This is a straightforward computation if u satisfies the Pohoˇ zaev identity (1.5) proved in section 3.3. This however brings a regularity issue, as the 1 N proof of the identity (1.5) requires a little more regularity than u ∈ H (R ). The growth assumption (f ) allows a critical growth of f and is too weak for a direct bootstrap argument. We study the delicate question of regularity of u in section 3.1 by introducing a new regularity result which can be thought of as a nonlocal coun- terpart of the critical Brezis–Kato regularity result [8]. Once additional regularity of the solution u is established, the Pohoˇ zaev identity (1.5) follows and can be em- ployed to estimate the critical level I(u). This is done using the construction of paths associated to critical points in section 4.1 following L. Jeanjean and K. Tanaka [20]. The qualitative properties of the groundstate of Theorem 4 are established in section 5. We show that the absolute value of a groundstate and its polarization are also groundstates. This leads to contradiction with the strong maximum principle if the solution is not invariant under these transformations. Finally in section 6 we explain how the proof of Theorem 1 can be simplified under the assumptions of Theorem 4 using symmetric mountain pass [40], adapting the original argument of Berestycki and Lions for (P ). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6562 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 2. Construction of a solution 2.1. Construction of a Pohoˇ zaev–Palais–Smale sequence. We first prove that there is a sequence of almost critical points at the level b defined in (1.7) that 1 N satisfies asymptotically (1.5). We define the Pohoˇ zaev functional P : H (R ) → R 1 N for u ∈ H (R )by N − 2 N N + α 2 2 P (u)= |∇u| + |u| − I ∗ F (u) F (u). 2 N 2 N 2 N R R R Proposition 2.1 (Construction of a Pohoˇ zaev–Palais–Smale sequence). If f ∈ 1 N C (R; R) satisfies (f ) and (f ), then there exists a sequence (u ) in H (R ) 1 3 n n∈N such that, as n →∞, I(u ) → b> 0, 1 N I (u ) → 0 strongly in H (R ) , P (u ) → 0. Proof. Our strategy consists in first proving in claims 1 and 2 that the functional I has the mountain pass geometry before concluding by a minimax principle. Claim 1. The critical level satisfies b< ∞. Proof of the claim. We need to show that the set of paths Γ is nonempty. In view 1 N of the definition of Γ, it is sufficient to construct u ∈ H (R ) such that I(u) < 0. If we choose s of assumption (f )so that F (s ) =0 and set w = s χ ,weobtain 0 3 0 0 B I ∗ F (w) F (w)= F (s ) I (x − y) > 0. α 0 α R B B 1 1 2N 2 N N 1 N N−2 By (f ) the left-hand side is continuous in L (R ) ∩ L (R ). Since H (R )is 2N 2 N N 1 N N−2 dense in L (R ) ∩ L (R ), there exists v ∈ H (R ) such that I ∗ F (v) F (v) > 0. 1 N We will take the function u in the family of functions u ∈ H (R ) defined for N x τ> 0and x ∈ R by u (x)= v . On this family, we compute for every τ> 0, N −2 N N +α τ τ τ 2 2 I(u )= |∇v| + |v| − I ∗ F (v) F (v), τ α 2 N 2 N 2 N R R R and observe that for τ> 0 large enough, I(u ) < 0. Claim 2. The critical level satisfies b> 0. Proof of the claim. Recall the Hardy–Littlewood–Sobolev inequality [23, theorem s N Ns/(N −αs) N 4.3]: if s ∈ (1, ), then for every v ∈ L (R ), I ∗ v ∈ L (R )and Ns N−αs N−αs (2.1) |I ∗ v| ≤ C |v| , N N R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6563 where C> 0 depends only on α, N and s. By the upper bound (f )on F , for every 1 N u ∈ H (R ), 1+ 2N N N +α I ∗ F (u) F (u) ≤ C |F (u)| N N R R 1+ 2N N N−2 ≤ C |u| + |u| α α+2 1+ 1+ N N−2 2 2 ≤ C |u| + |∇u| . N N R R 2 2 Hence there exists δ> 0 such that if |∇u| + |u| ≤ δ,then 2 2 I ∗ F (u) F (u) ≤ |∇u| + |u| , N 4 N R R and therefore 2 2 I(u) ≥ |∇u| + |u| . 4 N 2 2 2 In particular, if γ ∈ Γ, then |∇γ(0)| + |γ(0)| =0 <δ < |∇γ(1)| + N N R R |γ(1)| and by the intermediate value theorem there exists τ ¯ ∈ (0, 1) such that 2 2 |∇γ(¯ τ )| + |γ(¯ τ )| = δ.At the point τ¯, ≤I γ(¯ τ ) ≤ sup I(γ(τ )). τ ∈[0,1] Since γ ∈ Γ is arbitrary, this implies that b ≥ > 0. Conclusion. Following L. Jeanjean [18, §2] (see also [17, §4]), we define the map 1 N 1 N 1 N N Φ: R × H (R ) → H (R )for σ ∈ R, v ∈ H (R )and x ∈ R by −σ Φ(σ, v)(x)= v(e x). 1 N For every σ ∈ R and v ∈ H (R ), the functional I◦ Φ is computed as (N −2)σ Nσ (N +α)σ e e e 2 2 I Φ(σ, v) = |∇v| + |v| − I ∗ F (v) F (v). 2 N 2 N 2 N R R R 1 N In view of (f ), I◦ Φ is continuously Fr´ echet–differentiable on R × H (R ). We define the family of paths 1 N Γ= γ ˜ ∈ C [0, 1]; R × H (R ) :˜ γ(0) = (0, 0) and (I◦ Φ) γ ˜(1) < 0 . As Γ = {Φ ◦ γ ˜ :˜ γ ∈ Γ}, the mountain pass levels of I and I◦ Φcoincide: b =inf sup (I◦ Φ) γ ˜(τ ) . γ ˜∈Γ τ ∈[0,1] By the minimax principle [43, theorem 2.9], there exists a sequence (σ ,v ) n n n∈N 1 N in R × H (R ) such that as n →∞, (I◦ Φ)(σ ,v ) → b, n n 1 N (I◦ Φ) (σ ,v ) →0in R × H (R ) . n n 1 N Since for every (h, w) ∈ R × H (R ), (I◦ Φ) (σ ,v )[h, w]= I Φ(σ ,v ) [Φ(σ ,w)] + P Φ(σ ,v ) h, n n n n n n n we reach the conclusion by taking u =Φ(σ ,v ). n n n License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6564 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 2.2. Convergence of Pohoˇ zaev–Palais–Smale sequences. We will now show how a solution of problem (P ) can be constructed from the sequence given by Proposition 2.1. Proposition 2.2 (Convergence of Pohoˇ zaev–Palais–Smale sequences). Let f ∈ 1 N C (R; R) and (u ) be a sequence in H (R ).If f satisfies (f ) and (f ), n n∈N 1 2 I(u ) is bounded and, as n →∞, n∈N 1 N I (u ) → 0 strongly in (H (R )) , P (u ) → 0, then 1 N – either up to a subsequence u → 0 strongly in H (R ), 1 N –or there exists u ∈ H (R ) \{0} such that I (u)= 0 and a sequence (a ) of n n∈N N 1 N points in R such that up to a subsequence u (·− a ) u weakly in H (R ) n n as n →∞. Proof. Assume that the first part of the alternative does not hold, that is, 2 2 (2.2) lim inf |∇u | + |u | > 0. n n n→∞ We first establish in claim 1 the boundedness of the sequence and then the nonva- nishing of the sequence in claim 2. 1 N Claim 1. The sequence (u ) is bounded in H (R ). n n∈N Proof of claim 1. For every n ∈ N, α +2 α 1 2 2 |∇u | + |u | = I(u ) − P (u ). n n n n 2(N + α) 2(N + α) N + α N N R R As the right-hand side is bounded by our assumptions, the sequence (u ) is n n∈N 1 N bounded in H (R ). 2N Claim 2. For every p ∈ (2, ), N −2 lim inf sup |u | > 0. n→∞ a∈R B (a) Proof of claim 2. First, by (2.2) and the definition of the Pohoˇ zaev functional P we have (2.3) lim inf I ∗ F (u ) F (u ) α n n n→∞ N − 2 N 2 2 2 = lim inf |∇u| + |u| − P (u ) > 0. n→∞ N + α N N + α N N + α R R For every n ∈ N, the function u satisfies the inequality ([26, lemma I.1], [43, lemma 1.21], [31, lemma 2.3]) 1− p 2 2 p |u | ≤ C |∇u | + |u | sup |u | . n n n n N N N R R a∈R B (a) As F is continuous and satisfies (f ), for every > 0, there exists C such that for every s ∈ R, 2N 2N 2 p N +α N−2 |F (s)| ≤ |s| + |s| + C |s| . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6565 1 N Since (u ) is bounded in H (R ) and hence, by the Sobolev embedding, in n n∈N 2N N−2 L (R ), we have 1− 2N N +α lim inf |F (u )| ≤ C + C lim inf sup |u | . n n n→∞ n→∞ a∈R R B (a) Since > 0 is arbitrary, if lim inf sup N |u | =0, then n→∞ n a∈R B (a) 2N N +α lim inf |F (u )| =0, n→∞ and the Hardy–Littlewood–Sobolev inequality implies that lim inf I ∗ F (u ) F (u )= 0, α n n n→∞ in contradiction with (2.3). 2N Conclusion. Up to a translation, we can now assume that for some p ∈ (2, ), N −2 lim inf |u | > 0. n→∞ By Rellich’s theorem, this implies that up to a subsequence, (u ) converges n n∈N 1 N 1 N weakly in H (R )to u ∈ H (R ) \{0}. 1 N As the sequence (u ) is bounded in H (R ), by the Sobolev embedding, n n∈N 2N 2 N N N−2 it is also bounded in L (R ) ∩ L (R ). By (f ), the sequence (F ◦ u ) is 1 n n∈N 2N N +α therefore bounded in L (R ). Since the sequence (u ) converges weakly to n n∈N 1 N N u in H (R ), it converges up to a subsequence to u almost everywhere in R .By continuity of F,(F ◦ u ) converges almost everywhere to F ◦ u in R .This n n∈N 2N N +α implies that the sequence (F ◦ u ) converges weakly to F ◦ u in L (R ). As n n∈N 2N 2N N N N +α N−α the Riesz potential defines a linear continuous map from L (R )to L (R ), 2N N−α the sequence (I ∗ (F ◦ u )) converges weakly to I ∗ (F ◦ u)in L (R ). α n n∈N α On the other hand, in view of (f ) and by Rellich’s theorem, the sequence N 2N (f ◦ u ) converges strongly to f ◦ u in L (R ) for every p ∈ [1, ). We n n∈N loc α+2 conclude that p N I ∗ (F ◦ u ) (f ◦ u ) I ∗ (F ◦ u) (f ◦ u)weaklyin L (R ), α n n α 2N 1 N for every p ∈ [1, ). This implies in particular that for every ϕ ∈ C (R ), N +2 ∇u ·∇ϕ + uϕ − I ∗ (F ◦ u) (f ◦ u)ϕ N N R R = lim ∇u ·∇ϕ + uϕ − I ∗ (F ◦ u ) (f ◦ u )ϕ =0; α n n n→∞ N N R R that is, u is a weak solution of (P ). We point out that the assumption (f ) is only used in the proof of claim 2. Note also that without the additional assumptions of Theorem 4 we cannot rely on the Strauss radial compactness lemma [6, theorem A1] which is equivalent to the com- 1 N pactness of the embedding of the Sobolev subspace of radial functions H (R )into rad p N L (R )for 2 <p < 2N/(N − 2). Instead, our proof of convergence of Pohoˇ zaev– Palais–Smale sequences uses a direct concentration–compactness type argument of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6566 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Proposition 2.2. Such an approach could be useful for the study of other problems where radial symmetry of solutions either fails or is not readily available. Observe that in the limit α → 0, the assumptions (f ), (f )and (f ) do not allow 1 2 3 us to recover exactly (g ), (g )and (g ). The gap between (f )when α → 0and (g ) 1 2 3 1 1 2 2 is purely technical. When α → 0, (f ) gives the assumptions lim F (s) /|s| =0 2 s→0 2 2N/(N −2) and lim F (s) /|s| = 0, which is stronger than (g ). The first as- |s|→∞ sumption is not really surprising, as it can be observed that in (1.2) both g(u) and u have the same spatial homogeneity and therefore by scaling it could always be assumed that lim G(s)/s = 0. The second assumption is equivalent to s→0 2 2N/(N −2) 2 lim sup F (s) /|s| ≤ 0. Finally (f )gives G(s)= F (s) ≥ 0, which is |s|→∞ actually weaker than (g ). This weakening of the condition can also be explained by the difference between the various scalings of the problem (P ). 3. Regularity of solutions and Pohoˇ zaev identity The assumption (f ) is too weak for the standard bootstrap method as in [12, lemma A.1], [31, proposition 4.1]. Instead, in order to prove regularity of solutions of (P ) we shall rely on a nonlocal version of the Brezis–Kato estimate. 3.1. A nonlocal Brezis–Kato type regularity estimate. A special case of the 1 N regularity result of Brezis and Kato [8, theorem 2.3] states that if u ∈ H (R )is a solution of the linear elliptic equation (3.1) −Δu + u = Vu in R , ∞ N N p N and V ∈ L (R )+ L (R ), then u ∈ L (R ) for every p ≥ 1. We extend this result to a class of nonlocal linear equations. Proposition 3.1 (Improved integrability of solution of a nonlocal critical linear 2N 2N N N 1 N α+2 equation). If H, K ∈ L (R )+ L (R ) and u ∈ H (R ) solves (3.2) −Δu + u =(I ∗ Hu)K, N 2N p N then u ∈ L (R ) for every p ∈ [2, ). Moreover, there exists a constant C α N −2 independent of u such that 1 1 p 2 p 2 |u| ≤ C |u| . N N R R 2N/(α+2) N Note that the space L (R )is critical in this statement: starting from the 1 N 2N/(N −2) N information that u ∈ H (R ) ⊂ L (R ), a standard Hardy–Littlewood– 2N/(N −2) N Sobolev estimate would just show that u ∈ L (R ) and would thus give no additional regularity information. Instead, our proof of Proposition 3.1 follows the strategy of Brezis and Kato (see also Trudinger [39, Theorem 3]). The adaptation of the argument is complicated by the nonlocal effect of u on the right-hand side. Our main new tool for the proof of Proposition 3.1 is the following lemma, which ∞ N N is a nonlocal counterpart of the estimate [8, lemma 2.1]: if V ∈ L (R )+ L (R ), then for every > 0, there exists C such that 2 2 2 2 (3.3) V |u| ≤ |∇u| + C |u| . N N N R R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6567 2N α+2 Lemma 3.2. Let N ≥ 2, α ∈ (0, 2) and θ ∈ (0, 2).If H, K ∈ L (R )+ 2N α α L (R ) and <θ < 2 − , then for every > 0,there exists C ∈ R such ,θ N N 1 N that for every u ∈ H (R ), θ 2−θ 2 2 2 I ∗ (H |u| ) K |u| ≤ |∇u| + C |u| . α ,θ N N N R R R In the limit α = 0, this result is consistent with (3.3); the parameter θ only plays a role in the nonlocal case. In order to prove Lemma 3.2, we shall use several times the following inequality. Lemma 3.3. Let q, r, s, t ∈ [1, ∞) and λ ∈ [0, 2] such that α 1 1 λ 2 − λ 1+ − − = + . N s t q r If θ ∈ (0, 2) satisfies α 1 1 min(q, r) − <θ < max(q, r) 1 − , N s s α 1 1 min(q, r) − < 2 − θ< max(q, r) 1 − , N t t s N t N q N r N then for every H ∈ L (R ), K ∈ L (R ) and u ∈ L (R ) ∩ L (R ), 1 1 λ 2−λ s t q r θ 2−θ s t q r (I ∗ H |u| ) K |u| ≤ C |H | |K | |u| |u| . N N N N N R R R R R 1 1 α Proof. First observe that if s> ˜ 1, t> 1satisfy + =1 + ,the Hardy– ˜ s ˜ N Littlewood–Sobolev inequality is applicable and 1 1 s ˜ ˜ ˜ s ˜ t t θ 2−θ θ 2−θ I ∗ (H |u| ) K |u| ≤ C Hu Ku . N N N R R R Let μ ∈ R.Note that if 1 μ θ − μ 1 (3.4) 0 ≤ μ ≤ θ and := + + < 1, s ˜ q r s then by H¨ older’s inequality μ θ−μ s ˜ s ˜ s q r θ s q r Hu ≤ |H | |u| |u| . N N N N R R R R Similarly, if 1 λ − μ (2 − θ) − (λ − μ) 1 (3.5) λ − (2 − θ) ≤ μ ≤ λ and := + + < 1, t q r t then 2−θ−(λ−μ) 1 1 λ−μ ˜ ˜ t q r t t 2−θ t q r Ku ≤ |K | |u| |u| . N N N N R R R R It can be checked that (3.4) and (3.5) can be satisfied for some μ ∈ R if and only if 1 1 1 1 λ 2−λ α the assumptions of the lemma hold. In particular, + = + = + =1+ , ˜ s ˜ s t q r N so that we can conclude. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6568 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 2N ∗ ∗ ∗ ∗ N Proof of Lemma 3.2. Let H = H +H and K = K +K with H ,K ∈ L (R ) ∗ ∗ 2N N 2N 2N α+2 and H ,K ∈ L (R ). Applying Lemma 3.3, with q = r = , s = t = ∗ ∗ N −2 α+2 N −α and λ =0, we have since |θ − 1| < , N −2 θ 2−θ I ∗ (H |u| ) (K |u| ) α ∗ ∗ α+2 α+2 1− 2N 2N 2N 2N 2N N α+2 α+2 N−2 ≤ C |H | |K | |u| . ∗ ∗ N N N R R R 2N N −α Taking now s = t = , q = r =2 and λ =2, we have since |θ − 1| < , α N 2N 2N 2N 2N ∗ θ ∗ 2−θ ∗ ∗ 2 α α I ∗ (H |u| ) (K |u| ) ≤ C |H | |K | |u| . N N N N R R R R 2N 2N 2N Similarly, with s = , t = , q =2, r = and λ =1, α+2 α N −2 θ ∗ 2−θ I ∗ (H |u| ) (K |u| ) α ∗ α+2 1 1 2N 2N 2N 2N 2N 2 2 N ∗ 2 α+2 α N−2 ≤ C |H | |K | |u| |u| N N N N R R R R 2N 2N 2N and with s = , t = , q =2, r = and λ =1, α α+2 N −2 ∗ θ 2−θ I ∗ (H |u| ) (K |u| ) α ∗ α α+2 1 1 1 2N 2N 2N 2N 2N 2 2 N ∗ 2 α α+2 N−2 ≤ C |H | |K | |u| |u| . N N N N R R R R 1 N By the Sobolev inequality, we have thus proved that for every u ∈ H (R ), θ 2−θ I ∗ (H |u| ) (K |u| ) α+2 2N 2N 2N α+2 α+2 ≤ C |H | |K | |∇u| ∗ ∗ N N N R R R 2N 2N 2N ∗ ∗ 2 α α + |H | |K | |u| . N N N R R R ∗ ∗ The conclusion follows by choosing H and K such that α+2 2N 2N 2N α+2 α+2 C |H | |K | ≤ . ∗ ∗ N N R R Proof of Proposition 3.1. By Lemma 3.2 with θ =1, there exists λ> 0 such that 1 N for every ϕ ∈ H (R ), 1 λ 2 2 I ∗|Hϕ| |Kϕ|≤ |∇ϕ| + |ϕ| . 2 2 n n N R R R 2N Choose sequences (H ) and (K ) in L (R ) such that |H |≤ |H |, k k∈N k k∈N k |K |≤ |K |,and H → H and K → K almost everywhere in R .For each k ∈ N, k k k 1 N 1 N 1 N 1 N the form a : H (R ) × H (R ) → R defined for ϕ ∈ H (R )and ψ ∈ H (R ) by a (ϕ, ψ)= ∇ϕ ·∇ψ + λϕψ − (I ∗ H ϕ)K ψ k α k k N N R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6569 is bilinear and coercive; by the Lax–Milgram theorem [7, corollary 5.8], there exists 1 N a unique solution u ∈ H (R )of (3.6) −Δu + λu = I ∗ (H u ) K +(λ − 1)u, k k α k k k 1 N where u ∈ H (R ) is the given solution of (3.2). It can be proved that the sequence 1 N (u ) converges weakly to u in H (R )as k →∞. k k∈N N N For μ> 0, we define the truncation u : R → R for x ∈ R by k,μ ⎪ −μ if u (x) ≤−μ, u (x)= u (x)if −μ< u (x) <μ, k,μ k k μ if u (x) ≥ μ. p−2 1 N Since |u | u ∈ H (R ), we can take it as a test function in (3.6): k,μ k,μ p p 2 2 4(p−1) 2   2 ∇(u ) + |u | 2 k,μ k,μ p−2 p−2 ≤ (p − 1)|u | ∇u + |u | u u k,μ k,μ k,μ k,μ k p−2 p−2 = I ∗ (H u ) K |u | u +(λ − 1)u|u | u . α k k k k,μ k,μ k,μ k,μ 2N 2 If p< , by Lemma 3.2 with θ = ,there exists C> 0 such that α p p−2 I ∗|H u | |K ||u | u α k k,μ k k,μ k,μ p−1 ≤ I ∗ (|H ||u |) |K ||u | α k,μ k,μ p p 2 2 2(p−1) 2 2 ≤ ∇(u ) + C |u | . 2 k,μ k,μ N N R R We have thus 2(p−1) p p p−1 ∇(u ) ≤ C |u | + |u| + I ∗ (|K ||u | ) |H u |, 2 k,μ k α k k k k N N R R A k,μ where A = x ∈ R : |u (x)| >μ . k,μ k 2N Since p< , by the Hardy–Littlewood–Sobolev inequality, r r s p−1 p−1 s I ∗ (|K ||u | ) |H u |≤ C |K ||u | |H u | , α k k k k k k k k A R A k,μ k,μ 1 α 1 1 α 1 p N with = +1 − and = + .By Holder’s ¨ inequality, if u ∈ L (R ), then r 2N p s 2N p p−1 r N s N |K ||u | ∈ L (R )and |H u |∈ L (R ), whence by Lebesgue’s dominated k k k k convergence theorem p−1 lim I ∗ (|K ||u | ) |H u | =0. α k k k k μ→∞ k,μ In view of the Sobolev estimate, we have proved the inequality 1− pN N−2 lim sup |u | ≤ C lim sup |u | . k k N N k→∞ k→∞ R R N 2N By iterating over p a finite number of times we cover the range p ∈ [2, ). α N −2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6570 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Remark 3.1. A close inspection of the proofs of Lemma 3.2 and of Proposition 3.1 gives a more precise dependence of the constant C . Given a function M :(0, ∞) → N 2N (0, ∞)and p ∈ (2, ), there exists C such that if for every > 0, K and H p,M α N −2 ∗ ∗ can be decomposed as K = K + K and H = H + H with ∗ ∗ 2N 2N 2N 2N α+2 α α+2 α |K | ≤ and |K | ≤ M ( ), N N R R 2N 2N 2N α+2 2N α α+2 |H | ≤ and |H | ≤ M ( ), N N R R 1 N and if u ∈ H (R )satisfies −Δu + u =(I ∗ Hu)K, then one has p 2 p 2 |u| ≤ C |u| . p,M N N R R 3.2. Regularity of solutions. Now we are in a position to establish additional regularity of solutions of the nonlinear nonlocal problem (P ). N N N Proof of Theorem 2. Define H : R → R and K : R → R for x ∈ R by H (x)= F u(x) /u(x)and K (x)= f u(x) . Observe that for every x ∈ R , α α+2 N−2 |K (x)|≤ C |u(x)| + |u(x)| and α+2 N N −2 N N−2 |H (x)|≤ C |u(x)| + |u(x)| , N +α N +α 2N 2N N N p N α+2 so that K, H ∈ L (R )+ L (R ). By Proposition 3.1, u ∈ L (R ) for every N 2N 2N N 2N q N p ∈ [2, ). In view of (f ), F ◦ u ∈ L (R ) for every q ∈ [ , ). Since α N −2 N +α α N +α 2N N N 2N ∞ N < < ,we have I ∗ (F ◦ u) ∈ L (R ), and thus N +α α α N +α α+2 N N−2 |−Δu + u|≤ C |u| + |u| . By the classical bootstrap method for subcritical local problems in bounded do- 2,p mains, we deduce that u ∈ W (R ) for every p ≥ 1. loc 3.3. Pohozaev identity. The proof of Pohoˇ zaev identity (1.5) is a generalization of the argument for f (s)= s [31] (see also particular cases [29], [13, lemma 2.1]). The strategy is classical and consists in testing the equation against a suitable cut-off of x ·∇u(x) and integrating by parts ([21, proposition 6.2.1], [43, appendix B]). 2,2 N 1 N Proof of Theorem 3. By Theorem 2, u ∈ W (R ). Fix ϕ ∈ C (R ) such that loc 1,2 N ϕ = 1 in a neighbourhood of 0. The function v ∈ W (R ) defined for λ ∈ (0, ∞) and x ∈ R by v (x)= ϕ(λx) x ·∇u(x) can be used as a test function in the equation to obtain ∇u ·∇v + uv = I ∗ F (u) (f (u)v ). λ λ α λ N N N R R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6571 The left-hand side can be computed by integration by parts for every λ> 0as uv = u(x)ϕ(λx) x ·∇u(x)dx N N R R |u| = ϕ(λx) x ·∇ (x)dx |u(x)| = − Nϕ(λx)+ λx ·∇ϕ(λx) dx. N 2 Lebesgue’s dominated convergence theorem implies that lim uv = − |u| . λ→0 N 2 N R R 2,2 Similarly, as u ∈ W (R ), the gradient term can be written as loc |∇u| ∇u ·∇v = ϕ(λx) |∇u| + x ·∇ (x) dx N N R R |∇u(x)| = − (N − 2)ϕ(λx)+ λx ·∇ϕ(λx) dx. N 2 2 N Lebesgue’s dominated convergence again is applicable since ∇u ∈ L (R )and we obtain N − 2 lim ∇u ·∇v = − |∇u| . λ→0 N 2 N R R The last term can be rewritten by integration by parts for every λ> 0as I ∗ F (u) (f (u)v )= (F ◦ u)(y)I (x − y)ϕ(λx) x ·∇(F ◦ u)(x)dx dy α λ α N N N R R R = I (x − y) (F ◦ u)(y)ϕ(λx) x ·∇(F ◦ u)(x) 2 N N R R +(F ◦ u)(x)ϕ(λy) y ·∇(F ◦ u)(y) dx dy = − F u(y) I (x − y) Nϕ(λx)+ x ·∇ϕ(λx) F u(x) dx dy N N R R N − α + F u(y) I (x − y) 2 N N R R (x − y) · xϕ(λx) − yϕ(λy) F u(x) dx dy. |x − y| We can thus apply Lebesgue’s dominated convergence theorem to conclude that N + α lim I ∗ F (u) f (u) v = − (I ∗ F (u))F (u). α λ α λ→0 2 N N R R 4. From solutions to groundstates 4.1. Solutions and paths. One of the applications of the Pohoˇ zaev identity (1.5) is the possibility to associate to any variational solution of (P ) a path, following an argument of L. Jeanjean and K. Tanaka [20]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6572 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Proposition 4.1 (Lifting a solution to a path). If f ∈ C (R; R) satisfies (f ) and 1 N 1 N u ∈ H (R ) \{0} solves (P ), then there exists a path γ ∈ C [0, 1]; H (R ) such that γ(0) = 0, γ(1/2) = u, I γ(t) < I(u), for every t ∈ [0, 1] \{1/2}, I γ(1) < 0. Proof. The proof follows closely the arguments for the local problem developed by L. Jeanjean and K. Tanaka [20, lemma 2.1]. We define the path γ ˜ :[0, ∞) → 1 N H (R )by u(x/τ)if τ> 0, γ ˜(τ )(x)= 0if τ =0. The function γ ˜ is continuous on (0, ∞); for every τ> 0, 2 2 N −2 2 N 2 |∇γ ˜(τ )| + |γ ˜(τ )| = τ |∇u| + τ |u| , N N N R R R so that γ ˜ is continuous at 0. By the Pohoˇ zaev identity of Theorem 3, the functional can be computed for every τ> 0as N −2 N N +α τ τ τ 2 2 I γ ˜(τ ) = |∇u| + |u| − I ∗ F (u) F (u) 2 N 2 N 2 N R R R N −2 N +α N N +α τ (N − 2)τ τ Nτ 2 2 = − |∇u| + − |u| . 2 2(N + α) N 2 2(N + α) N R R It can be checked directly that I◦ γ ˜ achieves strict global maximum at 1: for every τ ∈ [0, ∞) \{1}, I γ ˜(τ ) < I(u). Since lim I γ ˜(τ ) = −∞, τ →∞ the path γ can then be defined by a suitable change of variable. 4.2. Minimality of the energy and existence of a groundstate. We now have all the tools available to show that the mountain-pass critical level b defined in (1.7) coincides with the groundstate energy level c defined in (1.4), which completes the proofofTheorem 1. Proof of Theorem 1. By Propositions 2.1 and 2.2, there exists a Pohoˇ zaev–Palais– 1 N Smale sequence (u ) in H (R )atthe mountain-pass level b> 0, that converges n n∈N 1 N weakly to some u ∈ H (R ) \{0} that solves (P ). Since lim P (u )=0, by n→∞ n the weak convergence of the sequence (u ) , the weak lower-semicontinuity of n n∈N the norm and the Pohoˇ zaev identity of Theorem 3, P (u) I(u)= I(u) − N + α α +2 α 2 2 = |∇u| + |u| 2(N + α) 2(N + α) N N R R (4.1) α +2 α 2 2 ≤ lim inf |∇u | + |u | n n n→∞ 2(N + α) N 2(N + α) N R R P (u ) = lim inf I(u ) − = lim inf I(u )= b. n n n→∞ n→∞ N + α License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6573 Since u is a nontrivial solution of (P ), we have I(u) ≥ c by definition of the groundstate energy level c, and hence c ≤ b. 1 N Let v ∈ H (R ) \{0} be another solution of (P ) such that I(v) ≤I(u). If we lift v to a path (Proposition 4.1) and recall the definition (1.7) of the mountain-pass level b, we conclude that I(v) ≥ b ≥I(u). We have thus proved that I(v)= I(u)= b = c, and this concludes the proof of Theorem 1. 4.3. Compactness of the set of groundstates. As a byproduct of the proof of Theorem 1, the weak convergence of the translated subsequence of Proposition 2.2 can be upgraded into strong convergence. Corollary 4.2 (Strong convergence of translated Pohoˇ zaev–Palais–Smale sequen- ces). Under the assumptions of Proposition 2.2,if 2 2 lim inf |∇u | + |u | > 0, n n n→∞ and if lim inf I(u ) ≤ c, n→∞ 1 N then there exists u ∈ H (R ) \{0} such that I (u)=0 andasequence (a ) of n n∈N N 1 N points in R such that up to a subsequence u (·− a ) → u strongly in H (R ) n n as n →∞. Proof. By Proposition 2.2, up to a subsequence and translations, we can assume that the sequence (u ) converges weakly to u. Since equality holds in (4.1), n n∈N α +2 α 2 2 |∇u| + |u| 2(N + α) N 2(N + α) N R R α +2 α 2 2 = lim inf |∇u | + |u | , n n n→∞ 2(N + α) N 2(N + α) N R R 1 N and hence (u ) converges strongly to u in H (R ). n n∈N As a direct consequence we have some information on the set of groundstates: Proposition 4.3 (Compactness of the set of groundstates). The set of ground- states 1 N S = u ∈ H (R ): I(u)= c and u is a weak solution of (P ) 1 N N is compact in H (R ) endowed with the strong topology up to translations in R . Proof. This is a direct consequence of Theorem 3 and Corollary 4.2. Remark 4.1 (Uniform regularity of groundstates). By the uniform regularity of solutions (Remark 3.1) and the compactness of the set of groundstates (Proposi- N 2N p N tion 4.3), for every p ∈ [2, ), S is bounded in L (R ). α N −2 5. Qualitative properties of groundstates 5.1. Paths achieving the mountain-pass level. Arguments in this section will use the following elementary property of the paths in the construction of the mountain-pass critical level b. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6574 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Lemma 5.1 (Optimal paths yield critical points). Let f ∈ C (R; R) satisfy (f ) and γ ∈ Γ,where Γ is defined in (1.8). If for every t ∈ [0, 1] \{t }, one has b = I γ(t ) > I γ(t) , then I γ(t ) =0. Proof. This can be deduced from the quantitative deformation lemma of M. Willem (see [43, lemma 2.3]). Assume that I (γ(t )) = 0. By continuity, it is possible to choose δ> 0and > 0 such that inf {I (v) : v − γ(t )≤ δ} > 8 /δ. 1 N With Willem’s notation, take X = H (R ), S = {γ(t )} and c = b. By the de- 1 N formation lemma, there exists η ∈ C ([0, 1]; H (R )) such that η(1,γ) ∈ Γand I η(1,γ(t )) ≤ b − <b and for every t ∈ [0, 1], we have I η(1,γ(t)) ≤ I γ(t) <b.Since [0, 1] is compact, we conclude with the contradiction that sup I η(1,γ(t)) <b. t∈[0,1] 5.2. Positivity of groundstates. We now prove that when f is odd, groundstates do not change sign. Proposition 5.2 (Groundstates do not change sign). Let f ∈ C (R; R) satisfy (f ). 1 N If f is odd and does not change sign on (0, ∞), then any groundstate u ∈ H (R ) of (P ) has constant sign. Proof. Without loss of generality, we can assume that f ≥ 0on(0, ∞). By Propo- sition 4.1, there exists an optimal path γ ∈ Γ on which the functional I achieves 1 N its maximum at 1/2. Since f is odd, F is even and thus for every v ∈ H (R ), I(|v|)= I(v). Hence, for every t ∈ [0, 1] \{1/2}, 1 1 I(|γ(t)|)= I(γ(t)) = I(γ( )) = I(|γ( )|). 2 2 By Lemma 5.1, |u| = |γ(1/2)| is also a groundstate. It satisfies the equation −Δ|u| + |u| = I ∗ F (|u|) f (|u|). Since u is continuous by Theorem 2, by the strong maximum principle we conclude that |u| > 0on R and thus u has constant sign. 5.3. Symmetry of groundstates. In this section, we now prove that ground- states are radial. Proposition 5.3 (Groundstates are symmetric). Let f ∈ C (R; R) satisfies (f ). 1 N (R ) If f is odd and does not change sign on (0, ∞), then any groundstate u ∈ H of (P ) is radial ly symmetric about a point. The argument relies on polarizations. It is intermediate between the argument based on equality cases in polarization inequalities [31] and the argument based on the Euler-Lagrange equation satisfied by polarizations [5, 42]. Before proving Proposition 5.3, we recall some elements of the theory of polar- ization ([3], [41], [9], [44, §8.3]). Assume that H ⊂ R is a closed half-space and that σ is the reflection with H N N N respect to ∂H.The polarization u : R → R of u : R → R is defined for x ∈ R by max u(x),u(σ (x)) if x ∈ H, u (x)= min u(x),u(σ (x)) if x ∈ H. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6575 We will use the following standard property of polarizations [9, lemma 5.3]. 1 N H Lemma 5.4 (Polarization and Dirichlet integrals). If u ∈ H (R ),then u ∈ 1 N H (R ) and H 2 2 |∇u | = |∇u| . N N R R We shall also use a polarization inequality with equality cases [31, lemma 5.3] (without the equality cases, see [3, corollary 4], [41, proposition 8]). 2N N +α Lemma 5.5 (Polarization and nonlocal integrals). Let α ∈ (0,N ), u ∈ L (R ) and H ⊂ R be a closed half-space. If u ≥ 0,then H H u(x) u(y) u (x) u (y) dx dy ≤ dx dy, N −α N −α N N |x − y| N N |x − y| R R R R H H with equality if and only if either u = u or u = u ◦ σ . The last tool that we need is a characterization of symmetric functions by po- larizations ([42, proposition 3.15], [31, lemma 5.4]). 2 N Lemma 5.6 (Symmetry and polarization). Assume that u ∈ L (R ) is nonneg- ative. There exist x ∈ R and a nonincreasing function v :(0, ∞) → R such that for almost every x ∈ R , u(x)= v(|x − x |) if and only if for every closed N H H half-space H ⊂ R , u = u or u = u ◦ σ . Proof of Proposition 5.3. The strategy consists in proving that u is also a ground- state (see corresponding results for the local problem [5, 42] and a weaker abstract H H result [35]) and to deduce therefrom that u = u or u = u ◦ σ . Without loss of generality, we can assume that f ≥ 0on(0, ∞). By Propo- sition 5.2, we can assume that u> 0. In view of Proposition 4.1, there exists an optimal path γ such that γ(1/2) = u and γ(t) ≥ 0 for every t ∈ [0, 1]. For H 1 N H H every half-space H define the path γ :[0, 1] → H (R )by γ (t)=(γ(t)) . H 1 N By Lemma 5.4, γ ∈ C ([0, 1]; H (R )). Observe that since F is nondecreasing, H H F (u )= F (u) , and therefore, for every t ∈ [0, 1], by Lemmas 5.4 and 5.5, I γ (t) ≤I γ(t) . Observe that γ ∈ Γsothat max I γ (t) ≥ b. t∈[0,1] Since for every t ∈ [0, 1] \{1/2}, I γ (t) ≤I γ(t) <b, we have 1 1 I γ ( ) = I γ( ) = b. 2 2 H H By Lemmas 5.4 and 5.5, we have either F (u) = F (u)or F (u )= F (u ◦ σ )in N H R . Assume that F (u) = F (u). Then, we have for every x ∈ H , u(x) f (s)ds = F u(x) − F u(σ (x)) ≥ 0; u(σ (x)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6576 VITALY MOROZ AND JEAN VAN SCHAFTINGEN this implies that either u(σ (x)) ≤ u(x)or f =0 on [u(x),u(σ (x))]. In particular, H H H N H  H f (u )= f (u)on R . By Lemma 5.1 applied to γ ,we have I (u ) = 0; and therefore, H H H H −Δu + u = I ∗ F (u ) f (u )= I ∗ F (u) f (u). α α Since u satisfies (P ), we conclude that u = u. H H If F (u )= F (u ◦ σ ), we conclude similarly that u = u ◦ σ . Since this H H holds for arbitrary H , we conclude by Lemma 5.6 that u is radial and radially decreasing. 6. Alternative proof of the existence In this section we sketch an alternative proof of the existence of a nontrivial 1 N solution u ∈ H (R ) \{0} such that c ≤I(u) ≤ b, under the additional symmetry assumption of Theorem 4 and in the spirit of the symmetrization arguments of H. Berestycki and P.-L. Lions [6, pp. 325-326]. The advantage of this approach is that it bypasses the concentration compactness argument and delays the Pohoˇ zaev identity which is still needed to prove that b ≤ c. Proof of Theorem 1 under the additional assumptions of Theorem 4. In addition to (f ), (f )and (f ), assume that f is an odd function which has constant sign on 1 2 3 (0, ∞). With this additional assumption, I◦ Φ(σ, |v| ) ≤I ◦ Φ(σ, v). Therefore, by the symmetric variational principle [40, theorem 3.2], we can prove as in the proof of Proposition 2.1 the existence of a sequence (u ) and (v ) n n∈N n n∈N such that as n →∞, I(u ) → b, 1 N I (u ) →0in H (R ) , P (u ) → 0, 2N ∗ 2 N N N−2 u − u →0in L (R ) ∩ L (R ), ∗ N where u : R → R is the Schwarz symmetrization of u ,thatis, forevery t> 0, N ∗ N {x ∈ R : u (x) >t} is a ball that has the same Lebesgue measure as {x ∈ R : |u (x)| >t} ([33, 34], [3, corollary 3], [9, §2], [23, §3.3], [44, definition 8.3.1]). 1 N As previously, the sequence (u ) is bounded in H (R ); by the Poly ´ a–Szeg˝ o n n∈N inequality ([33], [34], [9, theorem 8.2], [23, lemma 7.17], [44, theorem 8.3.14]), u ∈ 1,2 N W (R )and ∗ 2 2 |∇u | ≤ |∇u | , N N R R ∗ 1 N ∗ and thus the sequence (u ) is also bounded in H (R ). Since u is radial for n∈N n n ∗ p N 2N every n ∈ N, the sequence (u ) is compact in L (R ) for every p ∈ (2, ) n∈N N −2 ([36, lemmas 2 and 3], [25, proposition 1.1], [43, Corollary 1.26]). 2N ∗ 2 N N N−2 As u − u → 0as n →∞ in L (R ) ∩ L (R ), the sequence (u ) is also n n n∈N 2N p N compact L (R ) for every p ∈ (2, ). N −2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6577 2N N +α In view of (f ), this implies that F (u ) → F (u)as n →∞ in L (R )and 2 n thus (6.1) lim I ∗ F (u ) F (u )= I ∗ F (u) F (u) > 0. α n n α n→∞ N N R R 1 N Now one can prove that u (·− a ) converges to a nontrivial solution u ∈ H (R ) \ n n {0} as in the proof of Proposition 2.2. By (6.1), it also follows that c ≤I(u) ≤ b. Finally, employing the Pohoˇ zaev identity as in the proof of Theorem 1 allows us to conclude that c = b. References [1] A. Azzollini, P. d’Avenia, and A. 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MR3112778 Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, Wales, United Kingdom E-mail address: [email protected] ´ ´ Institut de Recherche en Mathematique et Physique, Universite Catholique de Lou- vain, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Transactions of the American Mathematical Society Unpaywall

Existence of groundstates for a class of nonlinear Choquard equations

Moroz, Vitaly; Van Schaftingen, Jean
Transactions of the American Mathematical Society – Dec 18, 2014
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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 9, September 2015, Pages 6557–6579 S 0002-9947(2014)06289-2 Article electronically published on December 18, 2014 EXISTENCE OF GROUNDSTATES FOR A CLASS OF NONLINEAR CHOQUARD EQUATIONS VITALY MOROZ AND JEAN VAN SCHAFTINGEN 1 N Abstract. We prove the existence of a nontrivial solution u ∈ H (R )to the nonlinear Choquard equation −Δu + u = I ∗ F (u) F (u)in R , where I is a Riesz potential, under almost necessary conditions on the non- linearity F in the spirit of Berestycki and Lions. This solution is a groundstate and has additional local regularity properties; if moreover F is even and mono- tone on (0, ∞), then u is of constant sign and radially symmetric. Contents 1. Introduction 6557 2. Construction of a solution 6562 3. Regularity of solutions and Pohoˇ zaev identity 6566 4. From solutions to groundstates 6571 5. Qualitative properties of groundstates 6573 6. Alternative proof of the existence 6576 References 6577 1. Introduction We consider the problem (P ) −Δu + u = I ∗ F (u) f (u)in R , where N ≥ 3, α ∈ (0,N ), I : R → R is the Riesz potential defined for every x ∈ R \{0} by N −α Γ( ) I (x)= , N/2 α N −α Γ( )π 2 |x| Received by the editors March 14, 2013 and, in revised form, September 22, 2013. 2010 Mathematics Subject Classification. Primary 35J61; Secondary 35B33, 35B38, 35B65, 35Q55, 45K05. Key words and phrases. Stationary Choquard equation, stationary nonlinear Schr¨ odinger– Newton equation, stationary Hartree equation, Riesz potential, nonlocal semilinear elliptic prob- lem, Pohoˇ zaev identity, existence, variational method, groundstate, mountain pass, symmetry, polarization. The second author was supported by the Grant n. 2.4550.10 “Etude qualitative des solutions d’´ equations aux d´ eriv´ ees partielles elliptiques” of the Fonds de la Recherche Fondatementale Col- lective (F´ ed´ eration Wallonie–Bruxelles). c 2014 American Mathematical Society Reverts to public domain 28 years from publication License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6558 VITALY MOROZ AND JEAN VAN SCHAFTINGEN F ∈ C (R; R)and f := F . Solutions of (P ) are formally critical points of the functional defined by 1 1 2 2 I(u)= |∇u| + |u| − I ∗ F (u) F (u). 2 2 N N R R We are interested in the existence and some qualitative properties of solutions to (P ). Problem (P ) is a semilinear elliptic equation with a nonlocal nonlinearity. For N =3, α =2 and F (s)= it covers in particular the Choquard–Pekar equation 2 3 (1.1) −Δu + u =(I ∗|u| )u in R , introduced at least in 1954, in a work by S. I. Pekar describing the quantum me- chanics of a polaron at rest [32]. In 1976 P. Choquard used (1.1) to describe an electron trapped in its own hole, in a certain approximation to Hartree–Fock the- ory of one component plasma [22]. In 1996 R. Penrose proposed (1.1) as a model of self-gravitating matter [30]. In this context equation (1.1) is usually called the non- linear Schr¨ odinger–Newton equation. Note that if u solves (1.1), then the function it ψ defined by ψ(t, x)= e u(x) is a solitary wave of the focusing time-dependent Hartree equation 2 N iψ +Δψ = −(I ∗|ψ| )ψ in R × R . t 2 + In this context (1.1) is also known as the stationary nonlinear Hartree equation. The existence of solutions for stationary equation (1.1) was proved by variational methods by E. H. Lieb, P.-L. Lions and G. Menzala [22, 24, 28] and also by ordinary differential equations techniques [11, 30, 38]. In the more general case of equation 1 p (P)with F (s)= |s| ,problem (P ) is known to have a solution if and only if N +α N +α <p < ([27, p. 457], [31, Theorem 1]; see also [15, Lemma 2.7]). N N −2 The existence results for (P ) up until now were only available when the nonlin- earity F is homogeneous. This situation contrasts with the striking existence result for the corresponding local problem (1.2) −Δu + u = g(u)in R , which can be considered as a limiting problem of (P)when α → 0, with g = Ff . H. Berestycki and P.-L. Lions [6, Theorem 1] have proved that (1.2) has a nontrivial solution if nonlinearity g ∈ C (R; R) satisfies the assumptions 2N N−2 (g ) there exists C> 0 such that for every s ∈ R, sg(s) ≤ C |s| + |s| , G(s) 1 G(s) (g ) lim < and lim sup ≤ 0, 2N s→0 |s| 2 N−2 |s|→∞ |s| (g ) there exists s ∈ R \{0} such that G(s ) > , 3 0 0 where G(s)= g(σ)dσ (and if g = Ff,then G = ). They also proved that 0 2 ∞ N if u ∈ L (R ) is a finite energy solution of (1.2), then u satisfies the Pohoˇ zaev loc identity [6, Proposition 1] N − 2 N 2 2 (1.3) |∇u| + |u| = N G(u). 2 2 N N N R R R This, in particular, implies that assumptions (g ), (g )and (g ) are “almost nec- 1 2 3 essary” for the existence of nontrivial finite energy solutions of (1.2). Indeed, the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6559 necessity of (g ) follows directly from (1.3). For (g )and (g ), if f (s)= s with 3 1 2 N +2 s ∈ (1, ), then (1.3) immediately implies that (1.2) does not have any bounded N −2 finite-energy nontrivial solution. In this spirit, we prove the existence of solutions to Choquard equation (P ), assuming that nonlinearity f ∈ C (R; R) satisfies the growth assumption: N +α N +α N N−2 (f )there exists C> 0 such that for every s ∈ R, |sf (s)|≤ C |s| + |s| , its antiderivative F : s ∈ R → f (σ)dσ is subcritical : F (s) F (s) (f ) lim = 0 and lim =0, N +α N +α s→0 |s|→∞ N N−2 |s| |s| and nontrivial : (f)thereexists s ∈ R \{0} such that F (s ) =0. 3 0 0 It is standard to check using Hardy–Littlewood–Sobolev inequality that if f ∈ C (R; R) satisfies growth assumption (f ), then I defines on the Sobolev space 1 N H (R ) a continuously differentiable functional and critical points of I are weak solutions of equation (P ). In what follows, solutions of (P ) are always understood in the weak sense. 1 N We say u ∈ H (R ) \{0} is a groundstate of (P)if u is a solution of (P)and 1 N (1.4) I(u)= c := inf I(v): v ∈ H (R ) \{0} is a solution of (P ) . Our main result in this paper is the following. Theorem 1 (Existence of a groundstate). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ C (R; R) satisfies (f ), (f ) and (f ),then (P ) has a groundstate. 1 2 3 We also prove that any weak solution of (P ) has additional regularity properties. Theorem 2 (Local regularity). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ C (R; R) 2,q 1 N N satisfies (f ) and u ∈ H (R ) solves (P ), then for every q ≥ 1, u ∈ W (R ). loc In particular, Theorem 2 with the Morrey–Sobolev embeddings implies that solutions of (P ) are locally H¨ older continuous. If f has additional smoothness, then regularity of u could be further improved via Schauder estimates. Let us emphasize that Theorem 2 is established only under the growth assumption (f ) and does not require additional subcriticality assumption (f ). The regularity information of Theorem 2 allows us to establish a Pohoˇ zaev inte- gral identity for all finite energy solutions of (P ). Theorem 3 (Pohoˇ zaev identity). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ 1 N C (R; R) satisfies (f ) and u ∈ H (R ) solves (P ),then N − 2 N N + α 2 2 (1.5) |∇u| + |u| = I ∗ F (u) F (u). 2 N 2 N 2 N R R R In particular, (1.5) implies that if u = 0 is a solution of (P ), then α +2 α 2 2 I(u)= |∇u| + |u| > 0. 2(N + α) 2(N + α) N N R R Pohoˇ zaev identity (1.5) shows that our assumptions (f ), (f )and (f )are 1 2 3 “almost necessary” for the existence of nontrivial solutions to (P ). Indeed, if 1 p F (s)= |s| , then (1.5) implies that problem (P ) does not have nontrivial weak License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6560 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 1 N N +α N +α solutions in H (R )if p ∈ , (see also [31, Theorem 2] where super- N N −2 1 N critical ranges of p are included). If (f ) fails, then a solution u ∈ H (R )would satisfy −Δu + u = 0 and would then necessarily be trivial. Whereas the upper critical exponent (N + α)/(N − 2) appears as a natural extension of the critical Sobolev exponent 2N/(N − 2) for the local problem (1.2) with G = F , the lower critical exponent (N + α)/N in assumptions (f )and (f ) 1 2 is a new phenomenon. It is due to the effect of the nonlocal term in (P ) and has (N +α)/N no analogues in (1.2). The growth restriction |sf (s)|≤ c|s| for |s| < 1 occurs naturally in the application of the Hardy–Littlewood–Sobolev inequality to 1 1 N verify that I∈ C (H (R ); R). In fact, Pohoˇ zaev identity confirms that the power (N + α)/N is optimal for the existence of solutions, and in this respect it plays the role of the lower critical exponent for (P ). Finally, we obtain qualitative properties of groundstates of (P ), which are the counterpart of the properties obtained for solutions of the corresponding local equa- tion [10, 14, 16]. Theorem 4 (Qualitative properties of groundstates). Assume that N ≥ 3 and α ∈ (0,N ).If f ∈ C (R; R) satisfies (f ) and, in addition, f is odd and has constant sign on (0, ∞), then every groundstate of (P ) has constant sign and is radial ly symmetric with respect to some point in R . Before explaining the proofs of our results, we recall the strategy of H. Berestycki and P.-L. Lions’s proof of the existence of solutions to (1.2) [6, §3]. They consider the constrained minimization problem |u| 2 1 N (1.6) min |∇u| : u ∈ H (R )and G(u) − =1 ; N N 2 R R they first show that by the Poly ´ a–Szeg˝ o inequality for the Schwarz symmetrization, the minimum can be taken on radial and radially nonincreasing functions. Then 1 N they show the existence of a minimum v ∈ H (R ) by the direct method of the calculus of variations. This minimum v satisfies the equation −Δv = θ g(v) − v in R , 1 N with a Lagrange multiplier θ> 0. They conclude by noting that u ∈ H (R ) defined for x ∈ R by u(x)= v(x/ θ) solves (1.2). The approach of H. Berestycki and P.-L. Lions fails for nonlocal problem (P)for two different reasons. First, the nonlocal term will not be preserved or controlled under Schwarz symmetrization unless the nonlinearity f satisfies the more restric- tive assumption of Theorem 4. Second, the final scaling argument fails: the three terms in (P ) scale differently in space, so one cannot hope to get rid of a Lagrange multiplier by scaling in space. In general, a constrained minimization of type (1.6) cannot be used for the study of solutions of equations with multiple scaling rates. Similar issues of multiple scaling rates arise, for instance, in the study of nonlocal nonlinear Schr¨ odinger–Maxwell or Schr¨ odinger–Poisson equations. For instance, the existence of a radial groundstate solution to a class of Schr¨ odinger–Maxwell equations under general Berestycki–Lions type assumptions on the nonlinear term was established in [2] by applying the mountain–pass theorem to a family of trun- cated functionals and then by proving the convergence of the obtained sequence of radially symmetric critical points using the radial compactness lemma of Strauss. Despite some similarities, the structure of Schr¨ odinger–Maxwell equations is very License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6561 different with Choquard equations and thus new techniques are required for the study of (P ). Moreover, such results establish the existence of radial groundstates while we are interested in the construction of global groundstates. In the present work, in order to prove the existence of solutions of (P ), instead of the constrained minimization problem of type (1.6), we consider in section 2 the mountain pass level (1.7) b =inf sup I γ(t) , γ∈Γ t∈[0,1] where the set of paths is defined as 1 N (1.8) Γ = γ ∈ C [0, 1]; H (R ) : γ(0) = 0 and I(γ(1)) < 0 . Classically, in order to show that b is a critical level of the functional I, one con- structs a Palais–Smale sequence at the level b, that is, a sequence (u ) in n n∈N 1 N H (R ) such that I(u ) → b and I (u ) → 0as n →∞. Then one proves that n n the sequence (u ) converges up to translations and extraction of a subsequence n n∈N [37, 43]. The first step of this approach is to establish the boundedness of the se- 1 N quence (u ) in H (R ). Usually this involves an Ambrosetti–Rabinowitz type n n∈N superlinearity assumption, which in our setting would require the existence of μ> 1 + μ such that s ∈ R → F (s)/s is nondecreasing. In order to avoid an Ambrosetti–Rabinowitz type condition, in section 2 we employ a scaling technique introduced by L. Jeanjean. It consists in constructing a Palais–Smale sequence that satisfies asymptotical ly the Pohoˇ zaev identity [18] (see also [1, 2, 4, 17] ). This improvement is related to the monotonicity trick of M. Struwe [37, §II.9] and L. Jeanjean [19]. Next, we prove with a concentration compactness argument the existence of a nontrivial solution u to (P ) under the assumptions (f ), (f )and (f ) only. This combination of the scaling technique with a concentration- 2 3 compactness argument which does not rely on the radial compactness and a priori radial symmetry of the solution is a novelty in our proof. To conclude that such a constructed solution u is a groundstate, we first show that I(u)= b. This is a straightforward computation if u satisfies the Pohoˇ zaev identity (1.5) proved in section 3.3. This however brings a regularity issue, as the 1 N proof of the identity (1.5) requires a little more regularity than u ∈ H (R ). The growth assumption (f ) allows a critical growth of f and is too weak for a direct bootstrap argument. We study the delicate question of regularity of u in section 3.1 by introducing a new regularity result which can be thought of as a nonlocal coun- terpart of the critical Brezis–Kato regularity result [8]. Once additional regularity of the solution u is established, the Pohoˇ zaev identity (1.5) follows and can be em- ployed to estimate the critical level I(u). This is done using the construction of paths associated to critical points in section 4.1 following L. Jeanjean and K. Tanaka [20]. The qualitative properties of the groundstate of Theorem 4 are established in section 5. We show that the absolute value of a groundstate and its polarization are also groundstates. This leads to contradiction with the strong maximum principle if the solution is not invariant under these transformations. Finally in section 6 we explain how the proof of Theorem 1 can be simplified under the assumptions of Theorem 4 using symmetric mountain pass [40], adapting the original argument of Berestycki and Lions for (P ). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6562 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 2. Construction of a solution 2.1. Construction of a Pohoˇ zaev–Palais–Smale sequence. We first prove that there is a sequence of almost critical points at the level b defined in (1.7) that 1 N satisfies asymptotically (1.5). We define the Pohoˇ zaev functional P : H (R ) → R 1 N for u ∈ H (R )by N − 2 N N + α 2 2 P (u)= |∇u| + |u| − I ∗ F (u) F (u). 2 N 2 N 2 N R R R Proposition 2.1 (Construction of a Pohoˇ zaev–Palais–Smale sequence). If f ∈ 1 N C (R; R) satisfies (f ) and (f ), then there exists a sequence (u ) in H (R ) 1 3 n n∈N such that, as n →∞, I(u ) → b> 0, 1 N I (u ) → 0 strongly in H (R ) , P (u ) → 0. Proof. Our strategy consists in first proving in claims 1 and 2 that the functional I has the mountain pass geometry before concluding by a minimax principle. Claim 1. The critical level satisfies b< ∞. Proof of the claim. We need to show that the set of paths Γ is nonempty. In view 1 N of the definition of Γ, it is sufficient to construct u ∈ H (R ) such that I(u) < 0. If we choose s of assumption (f )so that F (s ) =0 and set w = s χ ,weobtain 0 3 0 0 B I ∗ F (w) F (w)= F (s ) I (x − y) > 0. α 0 α R B B 1 1 2N 2 N N 1 N N−2 By (f ) the left-hand side is continuous in L (R ) ∩ L (R ). Since H (R )is 2N 2 N N 1 N N−2 dense in L (R ) ∩ L (R ), there exists v ∈ H (R ) such that I ∗ F (v) F (v) > 0. 1 N We will take the function u in the family of functions u ∈ H (R ) defined for N x τ> 0and x ∈ R by u (x)= v . On this family, we compute for every τ> 0, N −2 N N +α τ τ τ 2 2 I(u )= |∇v| + |v| − I ∗ F (v) F (v), τ α 2 N 2 N 2 N R R R and observe that for τ> 0 large enough, I(u ) < 0. Claim 2. The critical level satisfies b> 0. Proof of the claim. Recall the Hardy–Littlewood–Sobolev inequality [23, theorem s N Ns/(N −αs) N 4.3]: if s ∈ (1, ), then for every v ∈ L (R ), I ∗ v ∈ L (R )and Ns N−αs N−αs (2.1) |I ∗ v| ≤ C |v| , N N R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6563 where C> 0 depends only on α, N and s. By the upper bound (f )on F , for every 1 N u ∈ H (R ), 1+ 2N N N +α I ∗ F (u) F (u) ≤ C |F (u)| N N R R 1+ 2N N N−2 ≤ C |u| + |u| α α+2 1+ 1+ N N−2 2 2 ≤ C |u| + |∇u| . N N R R 2 2 Hence there exists δ> 0 such that if |∇u| + |u| ≤ δ,then 2 2 I ∗ F (u) F (u) ≤ |∇u| + |u| , N 4 N R R and therefore 2 2 I(u) ≥ |∇u| + |u| . 4 N 2 2 2 In particular, if γ ∈ Γ, then |∇γ(0)| + |γ(0)| =0 <δ < |∇γ(1)| + N N R R |γ(1)| and by the intermediate value theorem there exists τ ¯ ∈ (0, 1) such that 2 2 |∇γ(¯ τ )| + |γ(¯ τ )| = δ.At the point τ¯, ≤I γ(¯ τ ) ≤ sup I(γ(τ )). τ ∈[0,1] Since γ ∈ Γ is arbitrary, this implies that b ≥ > 0. Conclusion. Following L. Jeanjean [18, §2] (see also [17, §4]), we define the map 1 N 1 N 1 N N Φ: R × H (R ) → H (R )for σ ∈ R, v ∈ H (R )and x ∈ R by −σ Φ(σ, v)(x)= v(e x). 1 N For every σ ∈ R and v ∈ H (R ), the functional I◦ Φ is computed as (N −2)σ Nσ (N +α)σ e e e 2 2 I Φ(σ, v) = |∇v| + |v| − I ∗ F (v) F (v). 2 N 2 N 2 N R R R 1 N In view of (f ), I◦ Φ is continuously Fr´ echet–differentiable on R × H (R ). We define the family of paths 1 N Γ= γ ˜ ∈ C [0, 1]; R × H (R ) :˜ γ(0) = (0, 0) and (I◦ Φ) γ ˜(1) < 0 . As Γ = {Φ ◦ γ ˜ :˜ γ ∈ Γ}, the mountain pass levels of I and I◦ Φcoincide: b =inf sup (I◦ Φ) γ ˜(τ ) . γ ˜∈Γ τ ∈[0,1] By the minimax principle [43, theorem 2.9], there exists a sequence (σ ,v ) n n n∈N 1 N in R × H (R ) such that as n →∞, (I◦ Φ)(σ ,v ) → b, n n 1 N (I◦ Φ) (σ ,v ) →0in R × H (R ) . n n 1 N Since for every (h, w) ∈ R × H (R ), (I◦ Φ) (σ ,v )[h, w]= I Φ(σ ,v ) [Φ(σ ,w)] + P Φ(σ ,v ) h, n n n n n n n we reach the conclusion by taking u =Φ(σ ,v ). n n n License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6564 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 2.2. Convergence of Pohoˇ zaev–Palais–Smale sequences. We will now show how a solution of problem (P ) can be constructed from the sequence given by Proposition 2.1. Proposition 2.2 (Convergence of Pohoˇ zaev–Palais–Smale sequences). Let f ∈ 1 N C (R; R) and (u ) be a sequence in H (R ).If f satisfies (f ) and (f ), n n∈N 1 2 I(u ) is bounded and, as n →∞, n∈N 1 N I (u ) → 0 strongly in (H (R )) , P (u ) → 0, then 1 N – either up to a subsequence u → 0 strongly in H (R ), 1 N –or there exists u ∈ H (R ) \{0} such that I (u)= 0 and a sequence (a ) of n n∈N N 1 N points in R such that up to a subsequence u (·− a ) u weakly in H (R ) n n as n →∞. Proof. Assume that the first part of the alternative does not hold, that is, 2 2 (2.2) lim inf |∇u | + |u | > 0. n n n→∞ We first establish in claim 1 the boundedness of the sequence and then the nonva- nishing of the sequence in claim 2. 1 N Claim 1. The sequence (u ) is bounded in H (R ). n n∈N Proof of claim 1. For every n ∈ N, α +2 α 1 2 2 |∇u | + |u | = I(u ) − P (u ). n n n n 2(N + α) 2(N + α) N + α N N R R As the right-hand side is bounded by our assumptions, the sequence (u ) is n n∈N 1 N bounded in H (R ). 2N Claim 2. For every p ∈ (2, ), N −2 lim inf sup |u | > 0. n→∞ a∈R B (a) Proof of claim 2. First, by (2.2) and the definition of the Pohoˇ zaev functional P we have (2.3) lim inf I ∗ F (u ) F (u ) α n n n→∞ N − 2 N 2 2 2 = lim inf |∇u| + |u| − P (u ) > 0. n→∞ N + α N N + α N N + α R R For every n ∈ N, the function u satisfies the inequality ([26, lemma I.1], [43, lemma 1.21], [31, lemma 2.3]) 1− p 2 2 p |u | ≤ C |∇u | + |u | sup |u | . n n n n N N N R R a∈R B (a) As F is continuous and satisfies (f ), for every > 0, there exists C such that for every s ∈ R, 2N 2N 2 p N +α N−2 |F (s)| ≤ |s| + |s| + C |s| . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6565 1 N Since (u ) is bounded in H (R ) and hence, by the Sobolev embedding, in n n∈N 2N N−2 L (R ), we have 1− 2N N +α lim inf |F (u )| ≤ C + C lim inf sup |u | . n n n→∞ n→∞ a∈R R B (a) Since > 0 is arbitrary, if lim inf sup N |u | =0, then n→∞ n a∈R B (a) 2N N +α lim inf |F (u )| =0, n→∞ and the Hardy–Littlewood–Sobolev inequality implies that lim inf I ∗ F (u ) F (u )= 0, α n n n→∞ in contradiction with (2.3). 2N Conclusion. Up to a translation, we can now assume that for some p ∈ (2, ), N −2 lim inf |u | > 0. n→∞ By Rellich’s theorem, this implies that up to a subsequence, (u ) converges n n∈N 1 N 1 N weakly in H (R )to u ∈ H (R ) \{0}. 1 N As the sequence (u ) is bounded in H (R ), by the Sobolev embedding, n n∈N 2N 2 N N N−2 it is also bounded in L (R ) ∩ L (R ). By (f ), the sequence (F ◦ u ) is 1 n n∈N 2N N +α therefore bounded in L (R ). Since the sequence (u ) converges weakly to n n∈N 1 N N u in H (R ), it converges up to a subsequence to u almost everywhere in R .By continuity of F,(F ◦ u ) converges almost everywhere to F ◦ u in R .This n n∈N 2N N +α implies that the sequence (F ◦ u ) converges weakly to F ◦ u in L (R ). As n n∈N 2N 2N N N N +α N−α the Riesz potential defines a linear continuous map from L (R )to L (R ), 2N N−α the sequence (I ∗ (F ◦ u )) converges weakly to I ∗ (F ◦ u)in L (R ). α n n∈N α On the other hand, in view of (f ) and by Rellich’s theorem, the sequence N 2N (f ◦ u ) converges strongly to f ◦ u in L (R ) for every p ∈ [1, ). We n n∈N loc α+2 conclude that p N I ∗ (F ◦ u ) (f ◦ u ) I ∗ (F ◦ u) (f ◦ u)weaklyin L (R ), α n n α 2N 1 N for every p ∈ [1, ). This implies in particular that for every ϕ ∈ C (R ), N +2 ∇u ·∇ϕ + uϕ − I ∗ (F ◦ u) (f ◦ u)ϕ N N R R = lim ∇u ·∇ϕ + uϕ − I ∗ (F ◦ u ) (f ◦ u )ϕ =0; α n n n→∞ N N R R that is, u is a weak solution of (P ). We point out that the assumption (f ) is only used in the proof of claim 2. Note also that without the additional assumptions of Theorem 4 we cannot rely on the Strauss radial compactness lemma [6, theorem A1] which is equivalent to the com- 1 N pactness of the embedding of the Sobolev subspace of radial functions H (R )into rad p N L (R )for 2 <p < 2N/(N − 2). Instead, our proof of convergence of Pohoˇ zaev– Palais–Smale sequences uses a direct concentration–compactness type argument of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6566 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Proposition 2.2. Such an approach could be useful for the study of other problems where radial symmetry of solutions either fails or is not readily available. Observe that in the limit α → 0, the assumptions (f ), (f )and (f ) do not allow 1 2 3 us to recover exactly (g ), (g )and (g ). The gap between (f )when α → 0and (g ) 1 2 3 1 1 2 2 is purely technical. When α → 0, (f ) gives the assumptions lim F (s) /|s| =0 2 s→0 2 2N/(N −2) and lim F (s) /|s| = 0, which is stronger than (g ). The first as- |s|→∞ sumption is not really surprising, as it can be observed that in (1.2) both g(u) and u have the same spatial homogeneity and therefore by scaling it could always be assumed that lim G(s)/s = 0. The second assumption is equivalent to s→0 2 2N/(N −2) 2 lim sup F (s) /|s| ≤ 0. Finally (f )gives G(s)= F (s) ≥ 0, which is |s|→∞ actually weaker than (g ). This weakening of the condition can also be explained by the difference between the various scalings of the problem (P ). 3. Regularity of solutions and Pohoˇ zaev identity The assumption (f ) is too weak for the standard bootstrap method as in [12, lemma A.1], [31, proposition 4.1]. Instead, in order to prove regularity of solutions of (P ) we shall rely on a nonlocal version of the Brezis–Kato estimate. 3.1. A nonlocal Brezis–Kato type regularity estimate. A special case of the 1 N regularity result of Brezis and Kato [8, theorem 2.3] states that if u ∈ H (R )is a solution of the linear elliptic equation (3.1) −Δu + u = Vu in R , ∞ N N p N and V ∈ L (R )+ L (R ), then u ∈ L (R ) for every p ≥ 1. We extend this result to a class of nonlocal linear equations. Proposition 3.1 (Improved integrability of solution of a nonlocal critical linear 2N 2N N N 1 N α+2 equation). If H, K ∈ L (R )+ L (R ) and u ∈ H (R ) solves (3.2) −Δu + u =(I ∗ Hu)K, N 2N p N then u ∈ L (R ) for every p ∈ [2, ). Moreover, there exists a constant C α N −2 independent of u such that 1 1 p 2 p 2 |u| ≤ C |u| . N N R R 2N/(α+2) N Note that the space L (R )is critical in this statement: starting from the 1 N 2N/(N −2) N information that u ∈ H (R ) ⊂ L (R ), a standard Hardy–Littlewood– 2N/(N −2) N Sobolev estimate would just show that u ∈ L (R ) and would thus give no additional regularity information. Instead, our proof of Proposition 3.1 follows the strategy of Brezis and Kato (see also Trudinger [39, Theorem 3]). The adaptation of the argument is complicated by the nonlocal effect of u on the right-hand side. Our main new tool for the proof of Proposition 3.1 is the following lemma, which ∞ N N is a nonlocal counterpart of the estimate [8, lemma 2.1]: if V ∈ L (R )+ L (R ), then for every > 0, there exists C such that 2 2 2 2 (3.3) V |u| ≤ |∇u| + C |u| . N N N R R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6567 2N α+2 Lemma 3.2. Let N ≥ 2, α ∈ (0, 2) and θ ∈ (0, 2).If H, K ∈ L (R )+ 2N α α L (R ) and <θ < 2 − , then for every > 0,there exists C ∈ R such ,θ N N 1 N that for every u ∈ H (R ), θ 2−θ 2 2 2 I ∗ (H |u| ) K |u| ≤ |∇u| + C |u| . α ,θ N N N R R R In the limit α = 0, this result is consistent with (3.3); the parameter θ only plays a role in the nonlocal case. In order to prove Lemma 3.2, we shall use several times the following inequality. Lemma 3.3. Let q, r, s, t ∈ [1, ∞) and λ ∈ [0, 2] such that α 1 1 λ 2 − λ 1+ − − = + . N s t q r If θ ∈ (0, 2) satisfies α 1 1 min(q, r) − <θ < max(q, r) 1 − , N s s α 1 1 min(q, r) − < 2 − θ< max(q, r) 1 − , N t t s N t N q N r N then for every H ∈ L (R ), K ∈ L (R ) and u ∈ L (R ) ∩ L (R ), 1 1 λ 2−λ s t q r θ 2−θ s t q r (I ∗ H |u| ) K |u| ≤ C |H | |K | |u| |u| . N N N N N R R R R R 1 1 α Proof. First observe that if s> ˜ 1, t> 1satisfy + =1 + ,the Hardy– ˜ s ˜ N Littlewood–Sobolev inequality is applicable and 1 1 s ˜ ˜ ˜ s ˜ t t θ 2−θ θ 2−θ I ∗ (H |u| ) K |u| ≤ C Hu Ku . N N N R R R Let μ ∈ R.Note that if 1 μ θ − μ 1 (3.4) 0 ≤ μ ≤ θ and := + + < 1, s ˜ q r s then by H¨ older’s inequality μ θ−μ s ˜ s ˜ s q r θ s q r Hu ≤ |H | |u| |u| . N N N N R R R R Similarly, if 1 λ − μ (2 − θ) − (λ − μ) 1 (3.5) λ − (2 − θ) ≤ μ ≤ λ and := + + < 1, t q r t then 2−θ−(λ−μ) 1 1 λ−μ ˜ ˜ t q r t t 2−θ t q r Ku ≤ |K | |u| |u| . N N N N R R R R It can be checked that (3.4) and (3.5) can be satisfied for some μ ∈ R if and only if 1 1 1 1 λ 2−λ α the assumptions of the lemma hold. In particular, + = + = + =1+ , ˜ s ˜ s t q r N so that we can conclude. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6568 VITALY MOROZ AND JEAN VAN SCHAFTINGEN 2N ∗ ∗ ∗ ∗ N Proof of Lemma 3.2. Let H = H +H and K = K +K with H ,K ∈ L (R ) ∗ ∗ 2N N 2N 2N α+2 and H ,K ∈ L (R ). Applying Lemma 3.3, with q = r = , s = t = ∗ ∗ N −2 α+2 N −α and λ =0, we have since |θ − 1| < , N −2 θ 2−θ I ∗ (H |u| ) (K |u| ) α ∗ ∗ α+2 α+2 1− 2N 2N 2N 2N 2N N α+2 α+2 N−2 ≤ C |H | |K | |u| . ∗ ∗ N N N R R R 2N N −α Taking now s = t = , q = r =2 and λ =2, we have since |θ − 1| < , α N 2N 2N 2N 2N ∗ θ ∗ 2−θ ∗ ∗ 2 α α I ∗ (H |u| ) (K |u| ) ≤ C |H | |K | |u| . N N N N R R R R 2N 2N 2N Similarly, with s = , t = , q =2, r = and λ =1, α+2 α N −2 θ ∗ 2−θ I ∗ (H |u| ) (K |u| ) α ∗ α+2 1 1 2N 2N 2N 2N 2N 2 2 N ∗ 2 α+2 α N−2 ≤ C |H | |K | |u| |u| N N N N R R R R 2N 2N 2N and with s = , t = , q =2, r = and λ =1, α α+2 N −2 ∗ θ 2−θ I ∗ (H |u| ) (K |u| ) α ∗ α α+2 1 1 1 2N 2N 2N 2N 2N 2 2 N ∗ 2 α α+2 N−2 ≤ C |H | |K | |u| |u| . N N N N R R R R 1 N By the Sobolev inequality, we have thus proved that for every u ∈ H (R ), θ 2−θ I ∗ (H |u| ) (K |u| ) α+2 2N 2N 2N α+2 α+2 ≤ C |H | |K | |∇u| ∗ ∗ N N N R R R 2N 2N 2N ∗ ∗ 2 α α + |H | |K | |u| . N N N R R R ∗ ∗ The conclusion follows by choosing H and K such that α+2 2N 2N 2N α+2 α+2 C |H | |K | ≤ . ∗ ∗ N N R R Proof of Proposition 3.1. By Lemma 3.2 with θ =1, there exists λ> 0 such that 1 N for every ϕ ∈ H (R ), 1 λ 2 2 I ∗|Hϕ| |Kϕ|≤ |∇ϕ| + |ϕ| . 2 2 n n N R R R 2N Choose sequences (H ) and (K ) in L (R ) such that |H |≤ |H |, k k∈N k k∈N k |K |≤ |K |,and H → H and K → K almost everywhere in R .For each k ∈ N, k k k 1 N 1 N 1 N 1 N the form a : H (R ) × H (R ) → R defined for ϕ ∈ H (R )and ψ ∈ H (R ) by a (ϕ, ψ)= ∇ϕ ·∇ψ + λϕψ − (I ∗ H ϕ)K ψ k α k k N N R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6569 is bilinear and coercive; by the Lax–Milgram theorem [7, corollary 5.8], there exists 1 N a unique solution u ∈ H (R )of (3.6) −Δu + λu = I ∗ (H u ) K +(λ − 1)u, k k α k k k 1 N where u ∈ H (R ) is the given solution of (3.2). It can be proved that the sequence 1 N (u ) converges weakly to u in H (R )as k →∞. k k∈N N N For μ> 0, we define the truncation u : R → R for x ∈ R by k,μ ⎪ −μ if u (x) ≤−μ, u (x)= u (x)if −μ< u (x) <μ, k,μ k k μ if u (x) ≥ μ. p−2 1 N Since |u | u ∈ H (R ), we can take it as a test function in (3.6): k,μ k,μ p p 2 2 4(p−1) 2   2 ∇(u ) + |u | 2 k,μ k,μ p−2 p−2 ≤ (p − 1)|u | ∇u + |u | u u k,μ k,μ k,μ k,μ k p−2 p−2 = I ∗ (H u ) K |u | u +(λ − 1)u|u | u . α k k k k,μ k,μ k,μ k,μ 2N 2 If p< , by Lemma 3.2 with θ = ,there exists C> 0 such that α p p−2 I ∗|H u | |K ||u | u α k k,μ k k,μ k,μ p−1 ≤ I ∗ (|H ||u |) |K ||u | α k,μ k,μ p p 2 2 2(p−1) 2 2 ≤ ∇(u ) + C |u | . 2 k,μ k,μ N N R R We have thus 2(p−1) p p p−1 ∇(u ) ≤ C |u | + |u| + I ∗ (|K ||u | ) |H u |, 2 k,μ k α k k k k N N R R A k,μ where A = x ∈ R : |u (x)| >μ . k,μ k 2N Since p< , by the Hardy–Littlewood–Sobolev inequality, r r s p−1 p−1 s I ∗ (|K ||u | ) |H u |≤ C |K ||u | |H u | , α k k k k k k k k A R A k,μ k,μ 1 α 1 1 α 1 p N with = +1 − and = + .By Holder’s ¨ inequality, if u ∈ L (R ), then r 2N p s 2N p p−1 r N s N |K ||u | ∈ L (R )and |H u |∈ L (R ), whence by Lebesgue’s dominated k k k k convergence theorem p−1 lim I ∗ (|K ||u | ) |H u | =0. α k k k k μ→∞ k,μ In view of the Sobolev estimate, we have proved the inequality 1− pN N−2 lim sup |u | ≤ C lim sup |u | . k k N N k→∞ k→∞ R R N 2N By iterating over p a finite number of times we cover the range p ∈ [2, ). α N −2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6570 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Remark 3.1. A close inspection of the proofs of Lemma 3.2 and of Proposition 3.1 gives a more precise dependence of the constant C . Given a function M :(0, ∞) → N 2N (0, ∞)and p ∈ (2, ), there exists C such that if for every > 0, K and H p,M α N −2 ∗ ∗ can be decomposed as K = K + K and H = H + H with ∗ ∗ 2N 2N 2N 2N α+2 α α+2 α |K | ≤ and |K | ≤ M ( ), N N R R 2N 2N 2N α+2 2N α α+2 |H | ≤ and |H | ≤ M ( ), N N R R 1 N and if u ∈ H (R )satisfies −Δu + u =(I ∗ Hu)K, then one has p 2 p 2 |u| ≤ C |u| . p,M N N R R 3.2. Regularity of solutions. Now we are in a position to establish additional regularity of solutions of the nonlinear nonlocal problem (P ). N N N Proof of Theorem 2. Define H : R → R and K : R → R for x ∈ R by H (x)= F u(x) /u(x)and K (x)= f u(x) . Observe that for every x ∈ R , α α+2 N−2 |K (x)|≤ C |u(x)| + |u(x)| and α+2 N N −2 N N−2 |H (x)|≤ C |u(x)| + |u(x)| , N +α N +α 2N 2N N N p N α+2 so that K, H ∈ L (R )+ L (R ). By Proposition 3.1, u ∈ L (R ) for every N 2N 2N N 2N q N p ∈ [2, ). In view of (f ), F ◦ u ∈ L (R ) for every q ∈ [ , ). Since α N −2 N +α α N +α 2N N N 2N ∞ N < < ,we have I ∗ (F ◦ u) ∈ L (R ), and thus N +α α α N +α α+2 N N−2 |−Δu + u|≤ C |u| + |u| . By the classical bootstrap method for subcritical local problems in bounded do- 2,p mains, we deduce that u ∈ W (R ) for every p ≥ 1. loc 3.3. Pohozaev identity. The proof of Pohoˇ zaev identity (1.5) is a generalization of the argument for f (s)= s [31] (see also particular cases [29], [13, lemma 2.1]). The strategy is classical and consists in testing the equation against a suitable cut-off of x ·∇u(x) and integrating by parts ([21, proposition 6.2.1], [43, appendix B]). 2,2 N 1 N Proof of Theorem 3. By Theorem 2, u ∈ W (R ). Fix ϕ ∈ C (R ) such that loc 1,2 N ϕ = 1 in a neighbourhood of 0. The function v ∈ W (R ) defined for λ ∈ (0, ∞) and x ∈ R by v (x)= ϕ(λx) x ·∇u(x) can be used as a test function in the equation to obtain ∇u ·∇v + uv = I ∗ F (u) (f (u)v ). λ λ α λ N N N R R R License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6571 The left-hand side can be computed by integration by parts for every λ> 0as uv = u(x)ϕ(λx) x ·∇u(x)dx N N R R |u| = ϕ(λx) x ·∇ (x)dx |u(x)| = − Nϕ(λx)+ λx ·∇ϕ(λx) dx. N 2 Lebesgue’s dominated convergence theorem implies that lim uv = − |u| . λ→0 N 2 N R R 2,2 Similarly, as u ∈ W (R ), the gradient term can be written as loc |∇u| ∇u ·∇v = ϕ(λx) |∇u| + x ·∇ (x) dx N N R R |∇u(x)| = − (N − 2)ϕ(λx)+ λx ·∇ϕ(λx) dx. N 2 2 N Lebesgue’s dominated convergence again is applicable since ∇u ∈ L (R )and we obtain N − 2 lim ∇u ·∇v = − |∇u| . λ→0 N 2 N R R The last term can be rewritten by integration by parts for every λ> 0as I ∗ F (u) (f (u)v )= (F ◦ u)(y)I (x − y)ϕ(λx) x ·∇(F ◦ u)(x)dx dy α λ α N N N R R R = I (x − y) (F ◦ u)(y)ϕ(λx) x ·∇(F ◦ u)(x) 2 N N R R +(F ◦ u)(x)ϕ(λy) y ·∇(F ◦ u)(y) dx dy = − F u(y) I (x − y) Nϕ(λx)+ x ·∇ϕ(λx) F u(x) dx dy N N R R N − α + F u(y) I (x − y) 2 N N R R (x − y) · xϕ(λx) − yϕ(λy) F u(x) dx dy. |x − y| We can thus apply Lebesgue’s dominated convergence theorem to conclude that N + α lim I ∗ F (u) f (u) v = − (I ∗ F (u))F (u). α λ α λ→0 2 N N R R 4. From solutions to groundstates 4.1. Solutions and paths. One of the applications of the Pohoˇ zaev identity (1.5) is the possibility to associate to any variational solution of (P ) a path, following an argument of L. Jeanjean and K. Tanaka [20]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6572 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Proposition 4.1 (Lifting a solution to a path). If f ∈ C (R; R) satisfies (f ) and 1 N 1 N u ∈ H (R ) \{0} solves (P ), then there exists a path γ ∈ C [0, 1]; H (R ) such that γ(0) = 0, γ(1/2) = u, I γ(t) < I(u), for every t ∈ [0, 1] \{1/2}, I γ(1) < 0. Proof. The proof follows closely the arguments for the local problem developed by L. Jeanjean and K. Tanaka [20, lemma 2.1]. We define the path γ ˜ :[0, ∞) → 1 N H (R )by u(x/τ)if τ> 0, γ ˜(τ )(x)= 0if τ =0. The function γ ˜ is continuous on (0, ∞); for every τ> 0, 2 2 N −2 2 N 2 |∇γ ˜(τ )| + |γ ˜(τ )| = τ |∇u| + τ |u| , N N N R R R so that γ ˜ is continuous at 0. By the Pohoˇ zaev identity of Theorem 3, the functional can be computed for every τ> 0as N −2 N N +α τ τ τ 2 2 I γ ˜(τ ) = |∇u| + |u| − I ∗ F (u) F (u) 2 N 2 N 2 N R R R N −2 N +α N N +α τ (N − 2)τ τ Nτ 2 2 = − |∇u| + − |u| . 2 2(N + α) N 2 2(N + α) N R R It can be checked directly that I◦ γ ˜ achieves strict global maximum at 1: for every τ ∈ [0, ∞) \{1}, I γ ˜(τ ) < I(u). Since lim I γ ˜(τ ) = −∞, τ →∞ the path γ can then be defined by a suitable change of variable. 4.2. Minimality of the energy and existence of a groundstate. We now have all the tools available to show that the mountain-pass critical level b defined in (1.7) coincides with the groundstate energy level c defined in (1.4), which completes the proofofTheorem 1. Proof of Theorem 1. By Propositions 2.1 and 2.2, there exists a Pohoˇ zaev–Palais– 1 N Smale sequence (u ) in H (R )atthe mountain-pass level b> 0, that converges n n∈N 1 N weakly to some u ∈ H (R ) \{0} that solves (P ). Since lim P (u )=0, by n→∞ n the weak convergence of the sequence (u ) , the weak lower-semicontinuity of n n∈N the norm and the Pohoˇ zaev identity of Theorem 3, P (u) I(u)= I(u) − N + α α +2 α 2 2 = |∇u| + |u| 2(N + α) 2(N + α) N N R R (4.1) α +2 α 2 2 ≤ lim inf |∇u | + |u | n n n→∞ 2(N + α) N 2(N + α) N R R P (u ) = lim inf I(u ) − = lim inf I(u )= b. n n n→∞ n→∞ N + α License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6573 Since u is a nontrivial solution of (P ), we have I(u) ≥ c by definition of the groundstate energy level c, and hence c ≤ b. 1 N Let v ∈ H (R ) \{0} be another solution of (P ) such that I(v) ≤I(u). If we lift v to a path (Proposition 4.1) and recall the definition (1.7) of the mountain-pass level b, we conclude that I(v) ≥ b ≥I(u). We have thus proved that I(v)= I(u)= b = c, and this concludes the proof of Theorem 1. 4.3. Compactness of the set of groundstates. As a byproduct of the proof of Theorem 1, the weak convergence of the translated subsequence of Proposition 2.2 can be upgraded into strong convergence. Corollary 4.2 (Strong convergence of translated Pohoˇ zaev–Palais–Smale sequen- ces). Under the assumptions of Proposition 2.2,if 2 2 lim inf |∇u | + |u | > 0, n n n→∞ and if lim inf I(u ) ≤ c, n→∞ 1 N then there exists u ∈ H (R ) \{0} such that I (u)=0 andasequence (a ) of n n∈N N 1 N points in R such that up to a subsequence u (·− a ) → u strongly in H (R ) n n as n →∞. Proof. By Proposition 2.2, up to a subsequence and translations, we can assume that the sequence (u ) converges weakly to u. Since equality holds in (4.1), n n∈N α +2 α 2 2 |∇u| + |u| 2(N + α) N 2(N + α) N R R α +2 α 2 2 = lim inf |∇u | + |u | , n n n→∞ 2(N + α) N 2(N + α) N R R 1 N and hence (u ) converges strongly to u in H (R ). n n∈N As a direct consequence we have some information on the set of groundstates: Proposition 4.3 (Compactness of the set of groundstates). The set of ground- states 1 N S = u ∈ H (R ): I(u)= c and u is a weak solution of (P ) 1 N N is compact in H (R ) endowed with the strong topology up to translations in R . Proof. This is a direct consequence of Theorem 3 and Corollary 4.2. Remark 4.1 (Uniform regularity of groundstates). By the uniform regularity of solutions (Remark 3.1) and the compactness of the set of groundstates (Proposi- N 2N p N tion 4.3), for every p ∈ [2, ), S is bounded in L (R ). α N −2 5. Qualitative properties of groundstates 5.1. Paths achieving the mountain-pass level. Arguments in this section will use the following elementary property of the paths in the construction of the mountain-pass critical level b. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6574 VITALY MOROZ AND JEAN VAN SCHAFTINGEN Lemma 5.1 (Optimal paths yield critical points). Let f ∈ C (R; R) satisfy (f ) and γ ∈ Γ,where Γ is defined in (1.8). If for every t ∈ [0, 1] \{t }, one has b = I γ(t ) > I γ(t) , then I γ(t ) =0. Proof. This can be deduced from the quantitative deformation lemma of M. Willem (see [43, lemma 2.3]). Assume that I (γ(t )) = 0. By continuity, it is possible to choose δ> 0and > 0 such that inf {I (v) : v − γ(t )≤ δ} > 8 /δ. 1 N With Willem’s notation, take X = H (R ), S = {γ(t )} and c = b. By the de- 1 N formation lemma, there exists η ∈ C ([0, 1]; H (R )) such that η(1,γ) ∈ Γand I η(1,γ(t )) ≤ b − <b and for every t ∈ [0, 1], we have I η(1,γ(t)) ≤ I γ(t) <b.Since [0, 1] is compact, we conclude with the contradiction that sup I η(1,γ(t)) <b. t∈[0,1] 5.2. Positivity of groundstates. We now prove that when f is odd, groundstates do not change sign. Proposition 5.2 (Groundstates do not change sign). Let f ∈ C (R; R) satisfy (f ). 1 N If f is odd and does not change sign on (0, ∞), then any groundstate u ∈ H (R ) of (P ) has constant sign. Proof. Without loss of generality, we can assume that f ≥ 0on(0, ∞). By Propo- sition 4.1, there exists an optimal path γ ∈ Γ on which the functional I achieves 1 N its maximum at 1/2. Since f is odd, F is even and thus for every v ∈ H (R ), I(|v|)= I(v). Hence, for every t ∈ [0, 1] \{1/2}, 1 1 I(|γ(t)|)= I(γ(t)) = I(γ( )) = I(|γ( )|). 2 2 By Lemma 5.1, |u| = |γ(1/2)| is also a groundstate. It satisfies the equation −Δ|u| + |u| = I ∗ F (|u|) f (|u|). Since u is continuous by Theorem 2, by the strong maximum principle we conclude that |u| > 0on R and thus u has constant sign. 5.3. Symmetry of groundstates. In this section, we now prove that ground- states are radial. Proposition 5.3 (Groundstates are symmetric). Let f ∈ C (R; R) satisfies (f ). 1 N (R ) If f is odd and does not change sign on (0, ∞), then any groundstate u ∈ H of (P ) is radial ly symmetric about a point. The argument relies on polarizations. It is intermediate between the argument based on equality cases in polarization inequalities [31] and the argument based on the Euler-Lagrange equation satisfied by polarizations [5, 42]. Before proving Proposition 5.3, we recall some elements of the theory of polar- ization ([3], [41], [9], [44, §8.3]). Assume that H ⊂ R is a closed half-space and that σ is the reflection with H N N N respect to ∂H.The polarization u : R → R of u : R → R is defined for x ∈ R by max u(x),u(σ (x)) if x ∈ H, u (x)= min u(x),u(σ (x)) if x ∈ H. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6575 We will use the following standard property of polarizations [9, lemma 5.3]. 1 N H Lemma 5.4 (Polarization and Dirichlet integrals). If u ∈ H (R ),then u ∈ 1 N H (R ) and H 2 2 |∇u | = |∇u| . N N R R We shall also use a polarization inequality with equality cases [31, lemma 5.3] (without the equality cases, see [3, corollary 4], [41, proposition 8]). 2N N +α Lemma 5.5 (Polarization and nonlocal integrals). Let α ∈ (0,N ), u ∈ L (R ) and H ⊂ R be a closed half-space. If u ≥ 0,then H H u(x) u(y) u (x) u (y) dx dy ≤ dx dy, N −α N −α N N |x − y| N N |x − y| R R R R H H with equality if and only if either u = u or u = u ◦ σ . The last tool that we need is a characterization of symmetric functions by po- larizations ([42, proposition 3.15], [31, lemma 5.4]). 2 N Lemma 5.6 (Symmetry and polarization). Assume that u ∈ L (R ) is nonneg- ative. There exist x ∈ R and a nonincreasing function v :(0, ∞) → R such that for almost every x ∈ R , u(x)= v(|x − x |) if and only if for every closed N H H half-space H ⊂ R , u = u or u = u ◦ σ . Proof of Proposition 5.3. The strategy consists in proving that u is also a ground- state (see corresponding results for the local problem [5, 42] and a weaker abstract H H result [35]) and to deduce therefrom that u = u or u = u ◦ σ . Without loss of generality, we can assume that f ≥ 0on(0, ∞). By Propo- sition 5.2, we can assume that u> 0. In view of Proposition 4.1, there exists an optimal path γ such that γ(1/2) = u and γ(t) ≥ 0 for every t ∈ [0, 1]. For H 1 N H H every half-space H define the path γ :[0, 1] → H (R )by γ (t)=(γ(t)) . H 1 N By Lemma 5.4, γ ∈ C ([0, 1]; H (R )). Observe that since F is nondecreasing, H H F (u )= F (u) , and therefore, for every t ∈ [0, 1], by Lemmas 5.4 and 5.5, I γ (t) ≤I γ(t) . Observe that γ ∈ Γsothat max I γ (t) ≥ b. t∈[0,1] Since for every t ∈ [0, 1] \{1/2}, I γ (t) ≤I γ(t) <b, we have 1 1 I γ ( ) = I γ( ) = b. 2 2 H H By Lemmas 5.4 and 5.5, we have either F (u) = F (u)or F (u )= F (u ◦ σ )in N H R . Assume that F (u) = F (u). Then, we have for every x ∈ H , u(x) f (s)ds = F u(x) − F u(σ (x)) ≥ 0; u(σ (x)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6576 VITALY MOROZ AND JEAN VAN SCHAFTINGEN this implies that either u(σ (x)) ≤ u(x)or f =0 on [u(x),u(σ (x))]. In particular, H H H N H  H f (u )= f (u)on R . By Lemma 5.1 applied to γ ,we have I (u ) = 0; and therefore, H H H H −Δu + u = I ∗ F (u ) f (u )= I ∗ F (u) f (u). α α Since u satisfies (P ), we conclude that u = u. H H If F (u )= F (u ◦ σ ), we conclude similarly that u = u ◦ σ . Since this H H holds for arbitrary H , we conclude by Lemma 5.6 that u is radial and radially decreasing. 6. Alternative proof of the existence In this section we sketch an alternative proof of the existence of a nontrivial 1 N solution u ∈ H (R ) \{0} such that c ≤I(u) ≤ b, under the additional symmetry assumption of Theorem 4 and in the spirit of the symmetrization arguments of H. Berestycki and P.-L. Lions [6, pp. 325-326]. The advantage of this approach is that it bypasses the concentration compactness argument and delays the Pohoˇ zaev identity which is still needed to prove that b ≤ c. Proof of Theorem 1 under the additional assumptions of Theorem 4. In addition to (f ), (f )and (f ), assume that f is an odd function which has constant sign on 1 2 3 (0, ∞). With this additional assumption, I◦ Φ(σ, |v| ) ≤I ◦ Φ(σ, v). Therefore, by the symmetric variational principle [40, theorem 3.2], we can prove as in the proof of Proposition 2.1 the existence of a sequence (u ) and (v ) n n∈N n n∈N such that as n →∞, I(u ) → b, 1 N I (u ) →0in H (R ) , P (u ) → 0, 2N ∗ 2 N N N−2 u − u →0in L (R ) ∩ L (R ), ∗ N where u : R → R is the Schwarz symmetrization of u ,thatis, forevery t> 0, N ∗ N {x ∈ R : u (x) >t} is a ball that has the same Lebesgue measure as {x ∈ R : |u (x)| >t} ([33, 34], [3, corollary 3], [9, §2], [23, §3.3], [44, definition 8.3.1]). 1 N As previously, the sequence (u ) is bounded in H (R ); by the Poly ´ a–Szeg˝ o n n∈N inequality ([33], [34], [9, theorem 8.2], [23, lemma 7.17], [44, theorem 8.3.14]), u ∈ 1,2 N W (R )and ∗ 2 2 |∇u | ≤ |∇u | , N N R R ∗ 1 N ∗ and thus the sequence (u ) is also bounded in H (R ). Since u is radial for n∈N n n ∗ p N 2N every n ∈ N, the sequence (u ) is compact in L (R ) for every p ∈ (2, ) n∈N N −2 ([36, lemmas 2 and 3], [25, proposition 1.1], [43, Corollary 1.26]). 2N ∗ 2 N N N−2 As u − u → 0as n →∞ in L (R ) ∩ L (R ), the sequence (u ) is also n n n∈N 2N p N compact L (R ) for every p ∈ (2, ). N −2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXISTENCE OF GROUNDSTATES 6577 2N N +α In view of (f ), this implies that F (u ) → F (u)as n →∞ in L (R )and 2 n thus (6.1) lim I ∗ F (u ) F (u )= I ∗ F (u) F (u) > 0. α n n α n→∞ N N R R 1 N Now one can prove that u (·− a ) converges to a nontrivial solution u ∈ H (R ) \ n n {0} as in the proof of Proposition 2.2. By (6.1), it also follows that c ≤I(u) ≤ b. Finally, employing the Pohoˇ zaev identity as in the proof of Theorem 1 allows us to conclude that c = b. References [1] A. Azzollini, P. d’Avenia, and A. 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MR3112778 Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, Wales, United Kingdom E-mail address: [email protected] ´ ´ Institut de Recherche en Mathematique et Physique, Universite Catholique de Lou- vain, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

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