Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

TL;DR

This paper establishes existence and uniqueness of solutions for nonlinear elliptic equations with measure data and possibly singular lower order terms, extending classical results to more general and singular cases.

Contribution

It provides new existence and uniqueness results for nonlinear elliptic equations with measure data and singular nonlinearities, broadening the scope of solvable problems.

Findings

Existence of solutions under broad conditions.

Uniqueness of solutions in the singular setting.

Extension of classical results to measure data and singular nonlinearities.

Abstract

We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{cases} -\Delta_p u = H(u)\mu & \text{in}\ \Omega,\\ u>0 &\text{in}\ \Omega,\\ u=0 &\text{on}\ \partial\Omega. \end{cases}$$ Here $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ ($1<p<N$) is the $p$-laplacian operator, $\mu$ is a nonnegative bounded Radon measure on $\Omega$ and $H(s)$ is a continuous, positive and finite function outside the origin which grows at most as $s^{-\gamma}$, with $\gamma\ge0$, near zero.