Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms

TL;DR

This paper investigates the existence and multiplicity of solutions for a class of $p(x)$-Kirchhoff problems with nonstandard growth and nonlocal terms, using variational methods to establish multiple solutions.

Contribution

It introduces new results on the existence and multiplicity of solutions for $p(x)$-Kirchhoff problems with bi-nonlocal terms, extending previous work in the field.

Findings

Existence of two nontrivial solutions under certain conditions

Infinitely many solutions established using symmetric mountain pass and Clarke's theorems

Generalization of previous results on $p(x)$-Kirchhoff problems

Abstract

We study a class of $p(x)$-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results.