Local existence conditions for an equations involving the $p(x)$-Laplacian with critical exponent in $\mathbb{R}^N$

TL;DR

This paper establishes local existence conditions for solutions to a critical $p(x)$-Laplacian equation in $ abla^N$, focusing on the behavior of variable exponents near critical points without symmetry assumptions.

Contribution

It introduces novel local conditions for existence based on localized Sobolev constants and refined concentration-compactness, without symmetry or decay assumptions.

Findings

Existence conditions depend only on local behavior of exponents near critical points.

No symmetry or periodicity assumptions on coefficients.

Results apply to equations with variable exponents and critical growth.

Abstract

The purpose of this paper is to formulate sufficient existence conditions for a critical equation involving the $p(x)$-Laplacian posed in $\mathbb{R}^N$. This equation is critical in the sense that the source term has the form $K(x)|u|^{q(x)-2}u$ with an exponent $q$ that can be equal to the critical exponent $p^*$ at some points of $\mathbb{R}^N$ including at infinity. The sufficient existence conditions we find are local in the sense that they depend only on the behaviour of the exponents $p$ and $q$ near these points. We stress that we do not assume any symmetry or periodicity of the coefficients of the equation and that $K$ is not required to vanish in some sense at infinity like in most existing results. The proof of these local existence conditions is based on a notion of localized best Sobolev constant at infinity and a refined concentration-compactness at infinity.