Positive stationary solutions for p-Laplacian problems with nonpositive perturbation

TL;DR

This paper establishes the existence of positive stationary solutions for p-Laplacian problems with nonpositive perturbations using a novel topological degree approach, applicable to nonlinear elliptic PDEs with boundary conditions.

Contribution

It introduces a general topological degree framework for detecting solutions of nonlinear elliptic equations involving the p-Laplacian and applies it to PDEs with Dirichlet boundary conditions.

Findings

Proves existence of positive solutions under certain conditions.

Develops a new topological degree for maximal monotone operators.

Applies the theory to specific p-Laplacian PDEs.

Abstract

The paper is devoted to the existence of positive solutions of nonlinear elliptic equations with $p$-Laplacian. We provide a general topological degree that detects solutions of the problem $$ \{{array}{l} A(u)=F(u) u\in M {array}. $$ where $A:X\supset D(A)\to X^*$ is a maximal monotone operator in a Banach space $X$ and $F:M\to X^*$ is a continuous mapping defined on a closed convex cone $M\subset X$. Next, we apply this general framework to a class of partial differential equations with $p$-Laplacian under Dirichlet boundary conditions.