Uniqueness and positivity issues in a quasilinear indefinite problem

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 $\mathbb{R}^{N}$ ($N\geq1$), $\lambda\in\mathbb{R}$, $1<q<p$, and $a\in C(\overline{\Omega})$ changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for $\lambda<0$ the functional $$I_{\lambda}(u):=\int_{\Omega}\left( \frac{1}{p}|\nabla u|^{p}-\frac{\lambda }{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) ,$$ defined in $X=W_{0}^{1,p}(\Omega)$ or $X=W^{1,p}(\Omega)$, has \textit{exactly} one nonnegative global minimizer, and this one is the \textit{only} solution of $(P_{\lambda})$ being positive in $\Omega_{a}^{+}$ (the set where $a>0$). 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.