Existence of solutions for a class of $p(x)$-laplacian equations involving a concave-convex nonlinearity with critical growth in $\mathbb{R}^{N}$

TL;DR

This paper establishes the existence of solutions for a class of variable exponent p(x)-Laplacian equations with critical growth nonlinearities in Euclidean space using variational methods.

Contribution

It introduces new existence results for p(x)-Laplacian equations with critical growth nonlinearities employing variational techniques.

Findings

Existence of solutions proven for the class of equations.

Application of Ekeland's Variational Principle and Mountain Pass Theorem.

Results extend understanding of variable exponent problems with critical growth.

Abstract

We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle and the Mountain Pass Theorem.