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

Abstract

The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-Δ)^s u-λu=f(x,u) & {\mbox{ in }} Ω\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus Ω\,, \end{array} \right. $$ 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).