Nonlinear perturbations of a periodic fractional Laplacian with supercritical growth
DOI:
https://doi.org/10.12775/TMNA.2020.073Słowa kluczowe
Variational methods, supercritical exponent, fractional equationAbstrakt
Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.Bibliografia
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis. 14 (1973), 349–381.
C.O. Alves, P.C. Carrião and O.H. Miyagaki, Nonlinear perturbations of a periodic elliptic problem with critical growth, J. Math. Anal. Appl. 260 (2001), 133–146.
C.O. Alves and G.M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in RN , Nonlinear Anal. 75 (2012), 2750–2759.
C.O. Alves, S.H.M. Soares and M.A.S. Souto, Schrödinger–Poisson equations with supercritical growth, Eletron. J. Differ. Equ. 1 (2011), 1–11.
V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Annali di Matematica 196 (2017), 2043–2062.
G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in RN , J. Differential Equations. 225 (2013), 2340–2362.
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A. 143 (2013), 39–71.
L.A. Caffarelli and L. Silvestre, An extension problems related to the fractional Laplacian, Comm. Partial Differential Equations 36 (2007), 1245–1260.
M. Del Pino and P.L. Felmer, Local mountain pass for semilinear elliptic in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull Sci Math. 136 (2012), 521–573.
J.M. do Ó, O.H. Miyagaki and M. Squassina, Critical and subcritical fractional problems with vanishing potentials, Commun. Contemp. Math. 18 (2016), 1550063, 20 pp.
G.M. Figueiredo, O.H. Miyagaki and S.I. Moreira, Nonlinear perturbations of a periodic Schrödinger equation with supercritical growth, Z. Angew. Math. Phys. 66 (2016), 2379–2394.
O. Kavian, Introduction à la Théorie des Points Critiques et Applications aux Problèms Elliptiques, Springer, New York, Berlin, 1994.
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A. 268 (2000), 298–305.
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002), 056108, 7 pp.
G. Li and C. Wang, The existence of a nontrivial solution to p-Laplacian equations in RN with supercritical growth, Math. Methods Appl. Sci. 36 (2013), 69–79.
Q. Li, K. Teng, X. Wu and W. Wang, Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth, Math. Methods Appl. Sci. 42 (2019), 1480–1487.
P.L. Lions, The concentration-compacteness principle in the calculus of variations. The locally compact case, Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.
G. Molica Bisci, V.D. Rădulescu A and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University, 2016.
J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468.
M. Souza and Y.L. Araújo, A class of asymptotically periodic fractional Schrödinger equations with critical growth, Commun. Contemp. Math. 3 (2018), 1750011, 31 pp.
M. Xiang, B. Zhang and V.R. Rădulescu, Existence of solutions for perturbed fractional p-Laplacian equations, J. Differential Equations 260 (2016), 1392–1413.
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Prawa autorskie (c) 2021 Topological Methods in Nonlinear Analysis
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