Multiplicity of dead core solutions in indefinite elliptic problems

Abstract

We investigate nonnegative solutions of indefinite elliptic problems which enjoy the dead core phenomenon. Our model is the subhomogeneous problem $$ -\Delta_p u = (a^+(x) - \mu a^-(x))|u|^{q-2}u, \quad u \in W_0^{1,p}(\Omega), $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $1<q<p$, and $\mu > 0$. Thanks to a dead core formation property, we obtain multiple nonnegative dead core solutions of this problem for $\mu$ large enough. This assertion, in combination with a uniqueness feature, yields an exact multiplicity result, which gives a precise description of the nonnegative solutions set of this problem for large values of $\mu$. We also extend the multiplicity result to other classes of problems.