Simplicity of the principal eigenvalue for indefinite quasilinear problems

Abstract

Given any domain $\Omega \subseteq \mathbb R^N$, $w \in L^1_{loc} (\Omega)$ and a differentiable function $A: \mathbb R^N \to \mathbb [0,\infty)$ which is $p$-homogeneous and strictly convex, we consider the minimization problem $$ \inf \Big \{ \frac{\int_{\Omega} A( \nabla u ) } { \left( \int_{\Omega} w (x) |u|^q \right)^{\frac{p}{q} }} \, \colon \, u \in \mathcal{D}^{1,\mathcal{p}}_0 (\Omega) \quad 0< \int_{\Omega} w (x) |u|^q < \infty \Big \}. $$ If the infimum is achieved and $q =p >1$, without additional regularity assumptions on $\Omega$ or the weight function $w$, we show that the minimizer is unique up to a constant factor. The same conclusion holds when $A$ is allowed to depend on $x \in \Omega$ and satisfies natural growth assumptions. Some of our results also hold when $q < p$.