Comparison principle for elliptic equations with mixed singular nonlinearities

TL;DR

This paper establishes a comparison principle for elliptic equations with mixed singular and nonlinear terms, proving existence and uniqueness of positive solutions under certain conditions.

Contribution

It introduces a comparison principle for elliptic equations with singular nonlinearities and proves existence and uniqueness of solutions in this context.

Findings

Proved a comparison principle for elliptic equations with mixed nonlinearities.

Established existence of positive solutions under specified conditions.

Demonstrated uniqueness of solutions for the considered boundary value problem.

Abstract

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by \begin{equation*} \begin{cases} \displaystyle -\Delta_p u= \frac{f}{u^\gamma} + g u^q & \mbox{in $\Omega$,} \\ u = 0 & \mbox{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega$ is an open bounded subset of $\mathbb{R}^N$, $\Delta_p u:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-Laplacian operator, $\gamma\geq 0$ and $0\leq q\leq p-1$; $f$ and $g$ are nonnegative functions belonging to suitable Lebesgue spaces.