Quasilinear elliptic equations with Neumann boundary and singularity

Abstract

Let Ω be a bounded domain with a smooth C 2 boundary in ℝ N (N ≥ 3), 0 ∈ $$ \bar \Omega $$ , and n denote the unit outward normal to ∂Ω. We are concerned with the Neumann boundary problems: −div(|x| α |∇u|p−2∇u) =|x| β u p(α,β)−1 − λ|xβ γ u p−1, u(x) > 0, x ∈ Ω, ∂u/∂n = 0 on ∂Ω, where 1 < p < N and α < 0, β < 0 such that $$ p(\alpha ,\beta ) \triangleq \frac{{p(N + \beta )}} {{N - p + \alpha }} $$ > p, γ > α−p. For various parameters α, β or γ, we establish certain existence results of the solutions in the case 0 ∈ Ω or 0 ∈ ∂Ω.