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 $$-\hat{I}{”}_{p}u=(a^{+}(x)-\hat{I}¼a^{-}(x))|u{|}^{q-2}u,u\in W_0^{1,p}(\hat{I}\text{c◯}),$$ where $\hat{I}\text{c◯}$ is a bounded domain in ${\mathbb{R}}^N$, $1<q<p$, and $\hat{I}¼>0$. Thanks to a dead core formation property, we obtain multiple nonnegative dead core solutions of this problem for $\hat{I}¼$ 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 $\hat{I}¼$. We also extend the multiplicity result to other classes of problems.