On multiple solutions for nonlocal fractional problems via $\nabla$-theorems

TL;DR

This paper proves the existence of multiple solutions for nonlocal fractional equations involving the fractional Laplacian, using variational methods and $ abla$-theorems, near eigenvalues of the operator.

Contribution

It introduces a novel application of $ abla$-theorems to establish multiple solutions for fractional Laplace problems near eigenvalues.

Findings

At least three non-trivial solutions near each eigenvalue.

Solutions exist under superlinear and subcritical growth conditions.

Employs variational methods with $ abla$-theorems for nonlocal problems.

Abstract

The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus \Omega\,, \end{array} \right. $$ where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda$ is a real parameter, $\Omega\subset \mathbb{R}^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$. At this purpose we employ a variational theorem of mixed type (one of the so-called $\nabla$-theorems).