Existence of Solutions for Nonlocal Supercritical Elliptic Problems

Abstract

Utilizing a new variational principle, we prove the existence of a weak solution for the following nonlocal semilinear elliptic problem

$$\begin{aligned}\left\{ \begin{array}{ll} (-\Delta )^{s} u {=} \vert u \vert ^{p-2}u {+} f(x), &{} \text{ in } \Omega , \\ u {=} 0, &{} \text{ on } \mathbb {R}^{n}{\setminus } \Omega , \end{array} \right. \end{aligned}$$

where $(-\Delta )^s$ represents the fractional Laplace operator with $s \in (0,1], n > 2s, \Omega$ is an open bounded domain in $\mathbb {R}^n$ and $f \in L^d(\Omega )$ where $d \ge 2.$ We are particularly interested in problems where the nonlinear term is supercritical by means of fractional Sobolev spaces. As opposed to the usual standard variational methods, this new variational principle allows one to effectively work with problems beyond the standard weakly compact structure. Rather than working with the problem on the entire appropriate Sobolev space, this new principle enables one to deal with this problem on appropriate convex weakly compact subsets.