Quasilinear elliptic equations with Neumann boundary and singularity

Let Ω be a bounded domain with a smooth C ² boundary in ? ^N (N ≥ 3), 0 ∈ $ \bar \Omega $ \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(\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.