Existence of solution to a critical trace equation with variable exponent

TL;DR

This paper establishes local conditions ensuring the existence of non-trivial solutions to a boundary-critical p(x)-Laplacian equation using advanced variational methods and a generalized concentration-compactness principle.

Contribution

It introduces a generalized concentration-compactness principle for variable exponent Sobolev spaces and applies it to prove solution existence for a boundary-critical p(x)-Laplacian problem.

Findings

Existence of solutions under specific local conditions.

Extension of concentration-compactness principle to variable exponent trace embeddings.

Application of mountain pass theorem to boundary-critical problems.

Abstract

In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the $p(x)-$Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration--compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain pass theorem.