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

The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations driven by the fractional Laplace operator in an open bounded set with continuous boundary and with a nonlinearity satisfying 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 the fractional Laplacian. At this purpose we employ a variational theorem of mixed type (one of the so-called $\nabla$-theorems).