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 $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial \Omega$,} \ \end{cases} \end{equation*} in a domain $\Omega \subset {\mathbb{R}}^N$ $(N\ge 2)$, 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 $L^1(\Omega )$.

References in corpus (6)

• Finite and Infinite energy solutions of singular elliptic problems: Existence and Uniqueness
• Anzellotti's pairing theory and the Gauss--Green theorem
• The Dirichlet problem for singular elliptic equations with general nonlinearities
• The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data
• Nonlinear diffusion in transparent media: the resolvent equation
• 1-Laplacian type problems with strongly singular nonlinearities and gradient terms