Infinitely many positive solutions for a nonlocal problem

Abstract In this paper, we obtain infinitely many small positive solutions of the following nonlocal problem − L K u = f ( x , u ) in Ω , u = 0 in R N ∖ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω , and L K is an integrodifferential operator of fractional Laplacian type. The character of this work is that we do not require any growth condition on f for u large.

[1]  F. Zhou,et al.  Infinitely many small solutions for a modified nonlinear Schrödinger equations , 2014 .

[2]  Giovanni Molica Bisci,et al.  Variational Methods for Nonlocal Fractional Problems , 2016 .

[3]  E. Valdinoci,et al.  Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators , 2013 .

[4]  Giovanni Molica Bisci,et al.  Superlinear nonlocal fractional problems with infinitely many solutions , 2015 .

[5]  E. Valdinoci,et al.  Dislocation Dynamics in Crystals: A Macroscopic Theory in a Fractional Laplace Setting , 2013 .

[6]  N. Laskin.  Fractional quantum mechanics and Lévy path integrals , 1999, hep-ph/9910419.

[7]  Enrico Valdinoci,et al.  Mountain Pass solutions for non-local elliptic operators , 2012 .

[8]  M. Willem.  Minimax Theorems , 1997 .

[9]  Zhi-Qiang Wang,et al.  On Clark's theorem and its applications to partially sublinear problems , 2015 .

[10]  Wei Zhang,et al.  Infinitely many sign-changing solutions for a nonlocal problem , 2018, Annali di Matematica Pura ed Applicata (1923 -).

[11]  Enrico Valdinoci,et al.  The Brezis-Nirenberg result for the fractional Laplacian , 2014 .

[12]  E. Valdinoci,et al.  Nonlocal Diffusion and Applications , 2015, 1504.08292.

[13]  Zhi-Qiang Wang,et al.  Nonlinear boundary value problems with concave nonlinearities near the origin , 2001 .

[14]  Enrico Valdinoci,et al.  Variational methods for non-local operators of elliptic type , 2012 .

[15]  Asymptotically linear problems driven by fractional Laplacian operators , 2014 .

[16]  Leiga Zhao,et al.  The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system , 2017 .

[17]  Yuanyang Yu,et al.  The least energy sign-changing solution for a nonlocal problem , 2017 .

[18]  Ryuji Kajikiya,et al.  A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations , 2005 .

[19]  David Clark,et al.  A Variant of the Lusternik-Schnirelman Theory , 1972 .

[20]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[21]  Stefan Thurner,et al.  Anomalous diffusion in view of Einstein's 1905 theory of Brownian motion , 2005 .

[22]  P. Rabinowitz,et al.  Dual variational methods in critical point theory and applications , 1973 .

[23]  Two nontrivial solutions for an elliptic problem involving some nonlocal integro-differential operators , 2015 .

[24]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .