Simplicity of the principal eigenvalue for indefinite quasilinear problems

Given any domain Ω ⊆ RN, w ∈ L¹loc (Ω) and a differentiable function A: RN → [0,∞) which is p-homogeneous and strictly convex, we consider the minimization problem $$ inf \{ {∫Ω A( ∇ u ) } { ( ∫Ω w (x) |u|^q )p/q} \, \, u ∈ D1,p_0 (Ω) 0< ∫Ω w (x) |u|^q < ∞ \}. $$ If the infimum is achieved and $q =p >1$, without additional regularity assumptions on $Ω$ 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 $x ∈ Ω$ and satisfies natural growth assumptions. Some of our results also hold when $q < p$.