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

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.