Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms

TL;DR

This paper establishes the existence of finite energy solutions for a class of nonlinear elliptic equations with gradient, singular, and $L^1$ terms, using regularizing effects of certain functions.

Contribution

It demonstrates how regularizing terms $g$ and $h$ enable solutions with finite energy for equations with $L^1$ data, extending previous results to more singular and gradient-dependent cases.

Findings

Existence of finite energy solutions under $L^1$ data.

Regularizing effects of $g$ and $h$ facilitate solutions.

Applicable to equations with singular and gradient terms.

Abstract

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial \Omega$,} \ \end{cases} \end{equation*} in a domain $\Omega \subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(\Omega)$.