On some quasilinear equations involving the p-Laplacian with Robin boundary conditions

摘要

Let Ω{is contained in}R~N be a smooth-bounded domain, let f and g be continuous functions on Ω{top}-, g≥0. Let a>0 be some real number and let p ∈ (1, N) and q ∈ (p, p~*] be two real numbers, where p~*=np/(n-p). We are interested in proving the results of existence and non-existence for the non-negative solution of the equation -div(|{nabla}u|~(p-2){nabla}u)+(g-λ)|u|~(p-2)u in Ω; |{nabla}u|~(p-2){nabla}u·n{top}→=-a|u|~(p-2)u on {partial deriv}Ω. We consider the case λ<λ_1 (coercive case) and the case λ≥λ_1, where λ_1 is the principal eigenvalue for the operator Δ_p+g with Robin boundary conditions from which we recall the definition and some properties.