Uniqueness and positivity issues in a quasilinear indefinite problem

Abstract

We consider the problem(Pλ ) − p u = λu p−1 + a(x)u q−1, u ≥ 0 in under Dirichlet or Neumann boundary conditions. Here is a smooth bounded domain ofRN (N ≥ 1), λ ∈ R, 1 < q < p, and a ∈ C() changes sign. These conditions enablethe existence of dead core solutions for this problem, which may admit multiple nontrivialsolutions. We show that for λ < 0 the functionalIλ (u) :=∫( 1p |∇u|p − λp |u|p − 1q a(x)|u|q),defined in X = W 1, p0 () or X = W 1, p(), has exactly one nonnegative global minimizer,and this one is the only solution of (Pλ ) being positive in +a (the set where a > 0). Inparticular, this problem has at most one positive solution for λ < 0. Under some conditionon a, the above uniqueness result fails for some values of λ > 0 as we obtain, besides theground state solution, a second solution positive in +a . We also provide conditions on λ,a and q such that these solutions become positive in , and analyze the formation of deadcores for a generic solution.