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References (71)
-
M. Guedda, L. Véron (1989)
Quasilinear elliptic equations involving critical Sobolev exponentsNonlinear Analysis-theory Methods & Applications, 13
-
G Franzina, G Palatucci (2014)
Fractional $$p$$ p -eigenvaluesRiv. Mat. Univ. Parma, 5
-
A Ambrosetti, M Struwe (1986)
A note on the problem $$-\Delta u=\lambda u+u\vert u\vert ^{2^\ast -2}$$ - Δ u = λ u + u | u | 2 * - 2Manuscr. Math., 54
-
D. Zhang (1989)
On multiple solutions of u+?u+u4/(n-2)u = 0Nonlinear Analysis-theory Methods & Applications
-
Tuomo Kuusi, G. Mingione, Y. Sire (2014)
Nonlocal Equations with Measure DataCommunications in Mathematical Physics, 337
-
L. Brasco, S. Mosconi, M. Squassina (2015)
Optimal decay of extremals for the fractional Sobolev inequalityCalculus of Variations and Partial Differential Equations, 55
-
A Iannizzotto, S Liu, K Perera, M Squassina (2016)
Existence results for fractional $$p$$ p -Laplacian problems via Morse theoryAdv. Calc. Var., 9
-
E. Lieb (1983)
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalitiesAnnals of Mathematics, 118
-
Giovanni Franzina, Giampiero Palatucci (2013)
Fractional p-eigenvaluesarXiv: Analysis of PDEs, 2
-
G Arioli, F Gazzola (1998)
Some results on $$p$$ p -Laplace equations with a critical growth termDiffer. Integr. Equations, 11
-
H. Ishii, G. Nakamura (2010)
A class of integral equations and approximation of p-Laplace equationsCalculus of Variations and Partial Differential Equations, 37
-
N. Ghoussoub, C. Yuan (2000)
Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponentsTransactions of the American Mathematical Society, 352
-
JP García Azorero, I Peral Alonso (1987)
Existence and nonuniqueness for the $$p$$ p -Laplacian: nonlinear eigenvaluesComm. Partial Differ. Equations, 12
-
A. Ambrosetti, Michael Struwe (1986)
A note on the problem −Δu=λu+u|u|2*−2manuscripta mathematica, 54
-
EAB Silva, SHM Soares (2001)
Quasilinear Dirichlet problems in $${\mathbb{R}}^n$$ R n with critical growthNonlinear Anal., 43
-
L. Brasco, E. Parini (2014)
The second eigenvalue of the fractional p-LaplacianAdvances in Calculus of Variations, 9
-
L Brasco, E Parini, M Squassina (2016)
Stability of variational eigenvalues for the fractional $$p$$ p -LaplacianDiscrete Contin. Dyn. Syst. Ser. A, 36
-
JV Gonçalves, CO Alves (1998)
Existence of positive solutions for $$m$$ m -Laplacian equations in $${\mathbb{R}}^N$$ R N involving critical Sobolev exponentsNonlinear Anal., 32
-
K. Perera, A. Szulkin (2005)
p-LAPLACIAN PROBLEMS WHERE THE NONLINEARITY CROSSES AN EIGENVALUEDiscrete and Continuous Dynamical Systems, 13
-
M Degiovanni, S Lancelotti (2007)
Linking over cones and nontrivial solutions for $$p$$ p -Laplace equations with $$p$$ p -superlinear nonlinearityAnn. Inst. H. Poincaré Anal. Non Linéaire, 24
-
Agnese Castro, Tuomo Kuusi, Giampiero Palatucci (2014)
Nonlocal Harnack inequalitiesJournal of Functional Analysis, 267
-
J. Azorero, I. Alonso (1991)
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric termTransactions of the American Mathematical Society, 323
-
M Degiovanni, S Lancelotti (2009)
Linking solutions for $$p$$ p -Laplace equations with nonlinearity at critical growthJ. Funct. Anal., 256
-
H. Brezis, L. Nirenberg (1983)
Positive solutions of nonlinear elliptic equations involving critical sobolev exponentsCommunications on Pure and Applied Mathematics, 36
-
L. Brasco, E. Lindgren (2015)
Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic caseAdvances in Mathematics, 304
-
Raffaella Servadei, E. Valdinoci (2013)
A Brezis-Nirenberg result for non-local critical equations in low dimensionCommunications on Pure and Applied Analysis, 12
-
Marco Degiovanni, Sergio Lancelotti (2009)
Linking solutions for p-Laplace equations with nonlinearity at critical growthJournal of Functional Analysis, 256
-
J. Azorero, I. Alonso (1987)
Existence and nonuniqueness for the p-Laplacian nonlinear EigenvaluesCommunications in Partial Differential Equations, 12
-
A. Iannizzotto, Shibo Liu, K. Perera, M. Squassina (2014)
Existence results for fractional p-Laplacian problems via Morse theoryAdvances in Calculus of Variations, 9
-
P. Rabinowitz (1978)
Some Critical Point Theorems and Applications to Semilinear Elliptic Partial Differential Equations.Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 5
-
L. Brasco, E. Parini, M. Squassina (2015)
Stability of variational eigenvalues for the fractional p−LaplacianDiscrete and Continuous Dynamical Systems, 36
-
F Andreu, JM Mazón, JD Rossi, J Toledo (2009)
A nonlocal $$p-$$ p - Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditionsSIAM J. Math. Anal., 40
-
K. Perera, R. Agarwal, D. O’Regan (2010)
Morse Theoretic Aspects of $p$-Laplacian Type Operators -
Elves Silva, M. Xavier (2003)
Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponentsAnnales De L Institut Henri Poincare-analyse Non Lineaire, 20
-
Henrik Egnell (1988)
Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponentsArchive for Rational Mechanics and Analysis, 104
-
Marco Degiovanni, Sergio Lancelotti (2007)
Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearityAnnales De L Institut Henri Poincare-analyse Non Lineaire, 24
-
A Capozzi, D Fortunato, G Palmieri (1985)
An existence result for nonlinear elliptic problems involving critical Sobolev exponentAnn. Inst. H. Poincaré Anal. Non Linéaire, 2
-
F. Andreu, J. Mazón, J. Rossi, J. Toledo (2009)
A Nonlocal p-Laplacian Evolution Equation with Nonhomogeneous Dirichlet Boundary ConditionsSIAM J. Math. Anal., 40
-
P Drábek, YX Huang (1997)
Multiplicity of positive solutions for some quasilinear elliptic equation in $${\mathbb{R}}^N$$ R N with critical Sobolev exponentJ. Differ. Equations, 140
-
K Perera, M Squassina, Y Yang (2015)
Bifurcation and multiplicity results for critical fractional $$p$$ p -Laplacian problemsMath. Nachr., 289
-
A. Iannizzotto, S. Mosconi, M. Squassina (2014)
Global H\"older regularity for the fractional $p$-LaplacianarXiv: Analysis of PDEs
-
K. Perera, M. Squassina, Yang Yang (2014)
Bifurcation and multiplicity results for critical fractional p‐Laplacian problemsMathematische Nachrichten, 289
-
D Zhang (1989)
On multiple solutions of $$\Delta u+\lambda u+\vert u\vert ^{4/(n-2)}u=0$$ Δ u + λ u + | u | 4 / ( n - 2 ) u = 0Nonlinear Anal., 13
-
H Egnell (1998)
Existence and nonexistence results for $$m$$ m -Laplace equations involving critical Sobolev exponentsArch. Ration. Mech. Anal., 104
-
P. Drábek, Y. Huang (1997)
Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in RNwith Critical Sobolev ExponentJournal of Differential Equations, 140
-
Raffaella Servadei (2016)
A critical fractional Laplace equation in the resonant caseTopological Methods in Nonlinear Analysis, 43
-
E. Lindgren, P. Lindqvist (2012)
Fractional eigenvaluesCalculus of Variations and Partial Differential Equations, 49
-
Raffaella Servadei, E. Valdinoci (2014)
The Brezis-Nirenberg result for the fractional LaplacianTransactions of the American Mathematical Society, 367
-
Raffaella Servadei (2013)
The Yamabe equation in a non-local setting, 2
-
A. Capozzi, D. Fortunato, G. Palmieri (1985)
An existence result for nonlinear elliptic problems involving critical Sobolev exponentAnnales De L Institut Henri Poincare-analyse Non Lineaire, 2
-
Zhihui Wei, Xinmin Wu (1992)
A multiplicity result for quasilinear elliptic equations involving critical Sobolev exponentsNonlinear Analysis-theory Methods & Applications, 18
-
E. Lindgren (2014)
Hölder estimates for viscosity solutions of equations of fractional p-Laplace typeNonlinear Differential Equations and Applications NoDEA, 23
-
A. Iannizzotto, M. Squassina (2013)
Weyl-type laws for fractional p-eigenvalue problemsAsymptot. Anal., 88
-
Elves Silva, S. Soares (2001)
Quasilinear Dirichlet problems in RN with critical growthNonlinear Analysis-theory Methods & Applications, 43
-
G. Arioli, F. Gazzola (1998)
Some results on $p$-Laplace equations with a critical growth termDifferential and Integral Equations
-
Agnese Castro, Tuomo Kuusi, Giampiero Palatucci (2015)
Local behavior of fractional $p$-minimizersarXiv: Analysis of PDEs
-
J. Goncalves, C. Alves (1998)
Existence of positive solutions for m -Laplacian equations in R N involving critical Sobolev exponentsNonlinear Analysis-theory Methods & Applications, 32
-
Yang Yang, K. Perera (2014)
(N,q)‐Laplacian problems with critical Trudinger‐Moser nonlinearitiesBulletin of the London Mathematical Society, 48
-
K. Perera (2003)
Nontrivial critical groups in $p$-Laplacian problems via the Yang indexTopological Methods in Nonlinear Analysis, 21
-
K Perera, A Szulkin (2005)
$$p$$ p -Laplacian problems where the nonlinearity crosses an eigenvalueDiscrete Contin. Dyn. Syst., 13
-
H Ishii, G Nakamura (2010)
A class of integral equations and approximation of $$p-$$ p - Laplace equationsCalc. Var. Partial Differ. Equations, 37
-
M. Comte (1991)
Solutions of elliptic equations with critical Sobolev exponent in dimension threeNonlinear Analysis-theory Methods & Applications, 17
-
E. Fadell, P. Rabinowitz (1977)
Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systemsInventiones mathematicae, 45
-
F. Gazzola, B. Ruf (1997)
Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equationsAdvances in Differential Equations
-
G Cerami, D Fortunato, M Struwe (1984)
Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponentsAnn. Inst. H. Poincaré Anal. Non Linéaire, 1
-
K Perera (2003)
Nontrivial critical groups in $$p$$ p -Laplacian problems via the Yang indexTopol. Methods Nonlinear Anal., 21
-
G. Cerami, D. Fortunato, M. Struwe (1984)
Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponentsAnnales De L Institut Henri Poincare-analyse Non Lineaire, 1
-
D. Costa, E. Silva (1995)
A note on problems involving critical Sobolev exponentsDifferential and Integral Equations
-
J. Azorero, I. Alonso (1987)
Existence and nonuniqueness for the p-laplacianCommunications in Partial Differential Equations, 12
-
L. Caffarelli (2012)
Non-local Diffusions, Drifts and Games -
EAB Silva, MS Xavier (2003)
Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponentsAnn. Inst. H. Poincaré Anal. Non Linéaire, 20
- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by Springer-Verlag Berlin Heidelberg
- Subject
- Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
- ISSN
- 0944-2669
- eISSN
- 1432-0835
- DOI
- 10.1007/s00526-016-1035-2
- Publisher site
- See Article on Publisher Site
Abstract
We obtain nontrivial solutions to the Brezis–Nirenberg problem for the fractional p-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when $$p \ne 2$$ p ≠ 2 . We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in (Brasco et al., Cal. Var. Partial Differ Equations 55:23, 2016). The second difficulty is the lack of a direct sum decomposition suitable for applying the classical linking theorem. We use an abstract linking theorem based on the cohomological index proved in (Yang and Perera, Ann. Sci. Norm. Super. Pisa Cl. Sci. doi: 10.2422/2036-2145.201406_004 , 2016) to overcome this difficulty.
Journal
Calculus of Variations and Partial Differential Equations – Springer Journals
Published: Jul 27, 2016