Neumann problems for nonlinear elliptic equations with $L^1$ data

TL;DR

This paper establishes existence and stability results for solutions to nonlinear elliptic Neumann problems with $L^1$ data, extending the theory to include zero order terms and renormalized solutions.

Contribution

It introduces new existence and stability results for nonlinear elliptic Neumann problems with minimal data regularity, including zero order terms.

Findings

Existence of solutions for $L^1$ data in nonlinear elliptic Neumann problems.

Stability results for renormalized solutions.

Extension to operators with zero order terms.

Abstract

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is \begin{equation*} \begin{cases} -\Delta_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ \Omega, \\ \left( |\nabla u|^{p-2}\nabla u+ c(x)|u|^{p-2}u \right)\cdot\underline n=0 & \text{on}\ \partial \Omega \,, \end{cases} \end{equation*} when $f$ is just a summable function. Our approach allows also to deduce a stability result for renormalized solutions and an existence result for operator with a zero order term.