Nonlinear perturbations of a $p(x)$-Laplacian equation with critical growth in $\mathbb{R}^N$

TL;DR

This paper establishes the existence of solutions for a class of $p(x)$-Laplacian equations with critical growth, considering both periodic and nonperiodic variable exponents, advancing understanding of nonlinear PDEs with variable exponents.

Contribution

It introduces new existence results for $p(x)$-Laplacian equations with critical growth, covering both periodic and nonperiodic exponent cases.

Findings

Existence of solutions in periodic exponent case

Existence of solutions in nonperiodic perturbation case

Extension of critical growth analysis to variable exponent equations

Abstract

We prove the existence of solution for a class of $p(x)$-Laplacian equations where the nonlinearity has a critical growth. Here, we consider two cases: the first case involves the situation where the variable exponents are periodic functions. The second one involves the case where the variable exponents are nonperiodic perturbations.