Critical boundary value problems in the upper half-space

Abstract

We consider the critical problem −Δu−12x⋅∇u=0 in R+N,∂u∂ν=λ|u|p−2u+|u|2∗−2u on ∂R+N,where R+N=(x′,xN):x′∈RN−1,xN>0 is the upper half-space, N≥4, ν is the outward normal vector at the boundary and 2≤p<2∗:=2(N−1)/(N−2). Using a variational approach, we obtain nonnegative nonzero solutions according to the value of the parameter λ>0.