Multiplicity of dead core solutions in indefinite elliptic problems

We investigate nonnegative solutions of indefinite elliptic problems which enjoy the dead core phenomenon. Our model is the subhomogeneous problem $$-{\mathrm{\Delta }}_{p}u=(a^{+}(x)-\mu a^{-}(x))\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u,u\in W_0^{1,p}(\mathrm{\Omega }),$$ where $\mathrm{\Omega }$ 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.