Weak solvability of nonlinear elliptic equations involving variable exponents

TL;DR

This paper investigates the existence and multiplicity of solutions for nonlinear elliptic equations with variable exponents on manifolds, using variational methods without the Ambrosetti-Rabinowitz condition.

Contribution

It establishes new existence and multiplicity results for variable exponent elliptic equations on manifolds without relying on the Ambrosetti-Rabinowitz condition.

Findings

Proves solutions exist using mountain pass and Fountain theorems

Provides an example demonstrating applicability of results

Extends theory to variable exponent Sobolev spaces on manifolds

Abstract

We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the $( p( m ), \, q( m ) )-$ equation and the nonlinearity is superlinear but does not fulfil the Ambrosetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a $( p( m ), \, q( m ) )$ equation that highlights the applicability of our theoretical results is also provided.