Multiplicity of dead core solutions in indefinite elliptic problems

TL;DR

This paper studies the existence and multiplicity of dead core solutions in indefinite elliptic problems, showing that for large parameters, multiple solutions with dead cores exist and providing a detailed characterization.

Contribution

It introduces new multiplicity results for nonnegative dead core solutions in indefinite elliptic problems, extending previous findings to broader classes of problems.

Findings

Multiple dead core solutions exist for large parameter values.

A precise description of the solution set is provided.

The results extend to other classes of 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.