On multiple solutions to a nonlocal fractional p(cdot )-Laplacian problem with concave\u2013convex nonlinearities

Jongrak Lee,Yun-Ho Kim,Jae-Myoung Kim,Andrea Scapellato

doi:10.1186/s13662-022-03689-6

Abstract

The aim of this paper is to examine the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional p(cdot )-Laplacian with concave–convex nonlinearities when, in general, the nonlinear term does not satisfy the Ambrosetti–Rabinowitz condition. The main tools for obtaining this result are the mountain pass theorem and a modified version of Ekeland’s variational principle for an energy functional with the compactness condition of the Palais–Smale type, namely the Cerami condition. Also we discuss several existence results of a sequence of infinitely many solutions to our problem. To achieve these results, we employ the fountain theorem and the dual fountain theorem as main tools.

Highlights

1 Introduction In the last years the study of problems involving differential equations and variational problems associated with the p(·)-Laplacian operator has been paid to an increasing deal of attention because they can be viewed as a model for many physical phenomena which arise in several investigations related to elastic mechanics, electro-rheological fluid (“smart fluids”), image processing, etc
A natural question is to understand if some results can be recovered when we change the local p(·)-Laplacian, defined as – div(|∇u|p(x)–2∇u), into the nonlocal fractional p(·)-Laplacian
As far as we are aware, Kaumann et al [31] defined a new class of fractional Sobolev spaces with variable exponents that takes a fractional variable exponent operator into consideration

Summary

Introduction

In the last years the study of problems involving differential equations and variational problems associated with the p(·)-Laplacian operator has been paid to an increasing deal of attention because they can be viewed as a model for many physical phenomena which arise in several investigations related to elastic mechanics, electro-rheological fluid (“smart fluids”), image processing, etc. This paper is devoted to the study of the existence of nontrivial solutions for the following Schrödinger-type problem involving the nonlocal fractional p(·)-Laplacian:

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Conclusion