A nonlinear general Neumann problem involving two critical exponents

Abstract

We discuss the existence of solutions to the following nonlinear problem involving two critical Sobolev exponents ⎧ ⎪ ⎪ − div(p(x)∇u) = β|u| 2 ∗ −2 u + f (x, u )i nΩ, u � 0i nΩ, ∂u ∂ν = Q(x)|u| 2∗−2 u on ∂Ω, where β 0, Q is continuous on ∂Ω, p ∈ H 1 (Ω) is continuous and positive in ¯ Ω and f is a lower-order perturbation of |u| 2 ∗ −1 with f (x ,0 )= 0.