Uniqueness and positivity issues in a quasilinear indefinite problem

Abstract

We consider the problem

$$\begin{aligned} (P_\lambda )\quad -\Delta _{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\ge 0\quad \text{ in } \Omega \end{aligned}$$

under Dirichlet or Neumann boundary conditions. Here $\Omega$ is a smooth bounded domain of ${\mathbb {R}}^{N}$ ($N\ge 1$), $\lambda \in {\mathbb {R}}$, $1<q<p$, 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

$$\begin{aligned} I_{\lambda }(u):=\int _{\Omega }\left( \frac{1}{p}|\nabla u|^{p}-\frac{\lambda }{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \end{aligned}$$

defined in $X=W_{0}^{1,p}(\Omega )$ or $X=W^{1,p}(\Omega )$, has exactly one nonnegative global minimizer, and this one is the 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 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.