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Uniqueness and positivity issues in a quasilinear indefinite problem

We consider the problem $$ (P_λ)\quad -Δ_{p}u=λu^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }Ω$$ under Dirichlet or Neumann boundary conditions. Here $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ ($N\geq1$), $λ\in\mathbb{R}$, $1<q<p$, and $a\in C(\overlineΩ)$ changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for $λ<0$ the functional \[ I_λ(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{λ}{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] defined in $X=W_{0}^{1,p}(Ω)$ or $X=W^{1,p}(Ω)$, has \textit{exactly} one nonnegative global minimizer, and this one is the \textit{only} solution of $(P_λ)$ being positive in $Ω_{a}^{+}$ (the set where $a>0$). In particular, this problem has at most one positive solution for $λ<0$. Under some condition on $a$, the above uniqueness result fails for some values of $λ>0$ as we obtain, besides the ground state solution, a \textit{second} solution positive in $Ω_{a}^{+}$. We also provide conditions on $λ$, $a$ and $q$ such that these solutions become positive in $Ω$, and analyze the formation of dead cores for a generic solution.