Existence and multiplicity of solutions for $m(x)-$polyharmonic elliptic Kirchhoff type equations without Ambrosetti-Rabinowitz conditions

TL;DR

This paper establishes the existence of infinitely many solutions for a class of $m(x)$-polyharmonic Kirchhoff equations without relying on the Ambrosetti-Rabinowitz condition, using a Schauder basis and variational methods.

Contribution

It introduces a novel approach employing Schauder bases to verify mountain pass geometry and extends results to variable exponent and sublinear cases.

Findings

Proves existence of infinitely many solutions for $m(x)$-polyharmonic Kirchhoff equations.

Introduces a new eigenvalue-like quantity $ ext{lambda}_M$ for constant $m(x)$ cases.

Extends previous results to more general nonlinearities and growth conditions.

Abstract

In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, for the case $m(x)\equiv Const$, we introduce a positive quantity $\lambda_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, $2011$) and Bae-Kim (Mathematical Methods in the Applied Sciences, $2020$).