Infinitely many solutions for non-local problems with broken symmetry

Abstract

AbstractThe aim of this paper is to investigate the existence of solutions of the non-local elliptic problem\left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u\ =\lvert u\rvert^{p-2}u+h(% x)&&\displaystyle\text{in }\Omega,\\ &\displaystyle{u=0}&&\displaystyle\text{on }\mathbb{R}^{n}\setminus\Omega,\end% {aligned}\right.where {s\in(0,1)}, {n>2s}, Ω is an open bounded domain of {\mathbb{R}^{n}} with Lipschitz boundary {\partial\Omega}, {(-\Delta)^{s}} is the non-local Laplacian operator, {2<p<2_{s}^{\ast}} and {h\in L^{2}(\Omega)}. This problem requires the study of the eigenvalue problem related to the fractional Laplace operator, with or without potential.