Nonlinear elliptic equations with measure valued absorption potential

Abstract

We study the semilinear elliptic equation --$\Delta$u + g(u)$\sigma$ = $\mu$ with Dirichlet boundary condition in a smooth bounded domain where $\sigma$ is a nonnegative Radon measure, $\mu$ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that $\sigma$ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| ^{q--1} r, with q > 1 and $\mu$ satisfies an absolute continuity condition expressed in terms of some capacities involving $\sigma$. 2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05 .