$p$-Laplace equations with singular weights

TL;DR

This paper investigates p-Laplace boundary value problems with singular weights, employing Morse theory and cohomological methods to establish the existence of nontrivial solutions despite the lack of direct sum decompositions.

Contribution

It introduces a cohomological local splitting technique to estimate critical groups for p-Laplacian problems with boundary singular weights, advancing solution existence theory.

Findings

Existence of nontrivial solutions for p-Laplace equations with singular boundary weights.

Development of a cohomological local splitting method for critical group estimation.

Application of Morse theory in a non-decomposable functional setting.

Abstract

We study a class of $p$-Laplacian Dirichlet problems with weights that are possibly singular on the boundary of the domain, and obtain nontrivial solutions using Morse theory. In the absence of a direct sum decomposition, we use a cohomological local splitting to get an estimate of the critical groups.