Sign-changing solution for an elliptic equation with critical growth at the boundary

We prove the existence of sign-changing solution to the problem - Δ u - 1 2 x · ∇ u = λ u , in R + N , ∂ u ∂ ν = | u | 2 ∗ - 2 u , on ∂ R + N , where R + N = { ( x ′ , x N ) : x ′ ∈ R N - 1 , x N > 0 } is the upper half-space, 2 ∗ : = 2 ( N - 1 ) / ( N - 2 ) , N ≥ 7 , ∂ u ∂ ν is the partial outward normal derivative and the parameter λ > 0 interacts with the spectrum of the linearized problem. In the proof, we apply variational methods.