Nontrivial solutions of superlinear nonlocal problems

TL;DR

This paper investigates the existence of infinitely many weak solutions for superlinear nonlocal equations involving the fractional Laplacian, extending classical results to the fractional setting using variational methods.

Contribution

It introduces new existence results for nonlocal fractional problems with superlinear nonlinearities, broadening the scope of classical semilinear Laplacian equations.

Findings

Existence of infinitely many solutions under various superlinear growth conditions.

Extension of classical results to fractional Laplacian problems.

Application of Fountain Theorem to nonlocal equations.

Abstract

We study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti-Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.