Uniqueness and positivity issues in a quasilinear indefinite problem

Abstract

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 $\Omega$ 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 \[ I_λ(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{λ}{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] defined in $X=W_0^{1,p}(\Omega )$ 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.

References in corpus (1)

• Uniqueness and sign properties of minimizers in a quasilinear indefinite problem