Nonlinear elliptic equations with measures revisited

Abstract

We study the existence of solutions of the nonlinear problem $$\{\begin{array}{rlrl}-\Delta u+g(u)& =\mu & & \text{in }\Omega ,\\ u& =0& & \text{on }\partial \Omega ,\end{array}$$ where $\mu$ is a Radon measure and $g:\mathbb{R}\to \mathbb{R}$ is a nondecreasing continuous function with $g(0)=0$. This equation need not have a solution for every measure $\mu$, and we say that $\mu$ is a good measure if the Dirichlet problem above admits a solution. We show that for every $\mu$ there exists a largest good measure ${\mu }^{\ast }\le \mu$. This reduced measure has a number of remarkable properties.

Cited by in corpus (1)

• Existence of multiple solutions to an elliptic problem with measure data