On the connection between two quasilinear elliptic problems with source terms of order 0 or 1

TL;DR

This paper establishes a detailed connection between two quasilinear elliptic problems with different source terms, leading to new insights into the existence, nonexistence, regularity, and multiplicity of solutions, especially for superlinear cases.

Contribution

It introduces a novel link between two elliptic problems with source terms of order 0 or 1, providing new results on solution properties and extremal solutions.

Findings

New existence and nonexistence results for solutions.

Regularity and multiplicity insights for the problems.

Enhanced understanding of extremal solutions when g is superlinear.

Abstract

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form \[ -\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x)+\alpha, \] involves a source gradient term with natural growth, where $\beta$ is nonnegative, $\lambda>0,f(x)\geqq0$, and $\alpha$ is a nonnegative measure. The second one, of the form \[ -\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}+\mu, \] presents a source term of order $0, $where $g$ is nondecreasing, and $\mu$ is a nonnegative measure. Here $\beta$ and $g$ can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when $g$ is superlinear.