Nonlinear Elliptic Equations With Variable Exponents Satisfying Cerami Condition

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 Ambrossetti-Rabinowitz condition.

Contribution

It extends solution existence results to (p(x), q(x))-equations on manifolds under Cerami condition, without relying on the classical AR condition.

Findings

Established existence and multiplicity results using mountain pass and Fountain theorems.

Provided an example verifying the theoretical results.

Extended analysis 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(x), q(x))-equation and the nonlinearity is superlinear but does not satisfy the usual Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in Complete manifolds. The main results are established by means of the mountain pass theorem and Fountain theorem with Cerami condition. Moreover, we are giving an example of a (p(x), q(x)) equation that verifies all our demonstrated results.