Some eigenvalue problems involving the (p(x),q(x))-Laplacian

TL;DR

This paper investigates eigenvalue problems involving the (p(x),q(x))-Laplacian with Robin and Neumann boundary conditions, proving the existence of solutions using variational methods under certain conditions.

Contribution

It introduces new existence results for eigenvalue problems with variable exponent Laplacians using advanced variational techniques.

Findings

Existence of solutions established for the (p(x),q(x))-Laplacian problems.

Application of Mountain Pass theorem and Ekeland's variational principle.

Solutions obtained under specific conditions on the data.

Abstract

In this work, we are concerned with a Robin and Neumann problem with (p(x),q(x))-Laplacian. Under some appropriate conditions on the data involved in the elliptic problem, we prove the existence of solutions applying two versions of Mountain Pass theorem, Ekeland's variational principle and Lagrange multiplier rule.