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

Abstract

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\hat{I}\text{c◯}$,} \newline u\geq 0 & \text{in $\hat{I}\text{c◯}$,} \newline u=0 & \text{on $\partial \hat{I}\text{c◯}$,} \ \end{cases} \end{equation*} in a domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ $(N\ge 2)$, where $1\le p<N$, $g$ 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 $L^1(\hat{I}\text{c◯})$.