Existence of weak solutions for the anisotropic $p(x)$-Laplacian via degree theory

TL;DR

This paper proves the existence of weak solutions for a boundary value problem involving the anisotropic p(x)-Laplacian, using topological degree theory, and addresses degenerate and singular cases in anisotropic Sobolev spaces.

Contribution

It introduces a novel application of degree theory to establish weak solutions for anisotropic p(x)-Laplacian problems, including degenerate and singular cases.

Findings

Existence of weak solutions under certain conditions.

Addresses degenerate and singular cases.

Highlights potential critical regions in the domain.

Abstract

In this paper, we consider Dirichlet boundary value problem involving the anisotropic $p(x)$-Laplacian, where $p(x)= (p_1(x), ..., p_n(x))$, with $p_i(x)> 1$ in $\overline{\Omega}$. Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions on the data, the existence of weak solutions for the given problem. An important contribution is that we are considering the degenerate and the singular cases in the discussion. Finally, according to the compact embedding for anisotropic Sobolev spaces, we point out that the considered boundaru value problem may be critical in some region of $\Omega$.