On Neumann Boundary Value Problems for Some Quasilinear Elliptic Equations

We study the role played by the indefinite weight function a(x) on the existence of positive solutions to the problem  −div (|∇u|∇u) = λa(x)|u|u+ b(x)|u|u, x ∈ Ω, ∂u ∂n = 0, x ∈ ∂Ω , where Ω is a smooth bounded domain in Rn, b changes sign, 1 < p < N , 1 < γ < Np/(N − p) and γ 6= p. We prove that (i) if ∫ Ω a(x) dx 6= 0 and b satisfies another integral condition, then there exists some λ∗ such that λ∗ ∫ Ω a(x) dx < 0 and, for λ strictly between 0 and λ∗, the problem has a positive solution and (ii) if ∫ Ω a(x) dx = 0, then the problem has a positive solution for small λ provided that ∫ Ω b(x) dx < 0.