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 $$\{\begin{array}{ll}-{\Delta }_{p}u=H(u)\mu & \text{in}\Omega ,\\ u>0& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega .\end{array}$$ Here $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$ ($N\ge 2$), ${\Delta }_{p}u:=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$ ($1<p<N$) 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 $\gamma \ge 0$, near zero.