Bounded Solutions to nonlinear problems involving the fractional laplacian depending on parameters

TL;DR

This paper investigates the existence and multiplicity of bounded solutions for parameter-dependent nonlinear problems involving the fractional Laplacian in unbounded domains, using variational methods.

Contribution

It introduces a nonstandard variational approach to establish solutions for fractional Laplacian problems with parameters, including multiple solutions.

Findings

Existence of bounded solutions for nonlinear eigenvalue problems with fractional Laplacian.

Multiple solutions for fractional nonlinear problems with two parameters.

Application of variational methods to unbounded domain problems.

Abstract

The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.