Given $$\rho >0$$ ρ > 0 , we study the elliptic problem $$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$ find ( U , λ ) ∈ H 0 1 ( Ω ) × R such that - Δ U + λ U = | U | p - 1 U ∫ Ω U 2 d x = ρ , where $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a bounded domain and $$p>1$$ p > 1 is Sobolev-subcritical, searching for conditions (about $$\rho $$ ρ , N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is $$L^2$$ L 2 -subcritical, i.e. $$1<p<1+4/N$$ 1 < p < 1 + 4 / N , the problem admits solutions for every $$\rho >0$$ ρ > 0 . In the $$L^2$$ L 2 -critical and supercritical case, i.e. when $$1+4/N \le p < 2^*-1$$ 1 + 4 / N ≤ p < 2 ∗ - 1 , we show that, for any $$k\in {\mathbb {N}}$$ k ∈ N , the problem admits solutions having Morse index bounded above by k only if $$\rho $$ ρ is sufficiently small. Next we provide existence results for certain ranges of $$\rho $$ ρ , which can be estimated in terms of the Dirichlet eigenvalues of $$-\Delta $$ - Δ in $$H^1_0(\Omega )$$ H 0 1 ( Ω ) , extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.
Calculus of Variations and Partial Differential Equations – Springer Journals
Published: Aug 23, 2017
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