Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

TL;DR

This paper compares two quasilinear elliptic problems with Dirichlet boundary conditions, establishing a correlation that yields new insights into their existence, nonexistence, and multiplicity of solutions.

Contribution

It introduces a novel comparison between two elliptic problems involving gradient and source terms, leading to new existence and multiplicity results.

Findings

New existence results for the first problem.

Nonexistence conditions for the second problem.

Multiplicity of solutions under certain parameters.

Abstract

Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form $-\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x),$ where $\beta$ is nonnegative, involves a gradient term with natural growth. The second one, of the form $-\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}$ where $g$ is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems.