Uniqueness and positivity issues in a quasilinear indefinite problem

Abstract

We consider the problem $$ (P_\lambda)\quad -\Delta_{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }\Omega $$ under Dirichlet or Neumann boundary conditions. Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$ ($N\geq1$), $\lambda\in\mathbb{R}$, $1 0$). In particular, this problem has at most one positive solution for $\lambda<0$. Under some condition on $a$, the above uniqueness result fails for some values of $\lambda>0$ as we obtain, besides the ground state solution, a \textit{second} solution positive in $\Omega_{a}^{+}$. We also provide conditions on $\lambda$, $a$ and $q$ such that these solutions become positive in $\Omega$, and analyze the formation of dead cores for a generic solution.