Uniqueness and sign properties of minimizers in a quasilinear indefinite problem

Abstract

Let 1 < q < p and a ∈ C(Ω) be sign-changing, where Ω is abounded and smooth domain of RN . We show that the functionalIq (u) :=∫Ω( 1p |∇u|p − 1q a(x)|u|q),has exactly one nonnegative minimizer Uq (in W 1,p0 (Ω) or W 1,p(Ω)). In addi-tion, we prove that Uq is the only possible positive solution of the associatedEuler-Lagrange equation, which shows that this equation has at most one pos-itive solution. Furthermore, we show that if q is close enough to p then Uqis positive, which also guarantees that minimizers of Iq do not change sign.Several of these results are new even for p = 2.