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*−2u+f(x,u)   in Ω,

u≢0   in Ω,

∂u/∂ν=Q(x)|u|^2_*−2u   on ∂Ω,

where β≥0, Q is continuous on ∂Ω, p∈H¹(Ω) is continuous and positive in Ω^¯ and f is a lower-order perturbation of |u|^2*−1 with f(x,0)=0.