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

We prove the existence of sign-changing solution to the problem

$$\begin{aligned} -\Delta u-\dfrac{1}{2}\left( x\cdot \nabla u\right) =\lambda u, \hbox { in }\mathbb {R}_{+}^{N}, \qquad \dfrac{\partial u}{\partial \nu }=|u|^{2_*-2}u, \hbox { on } \partial \mathbb {R}_{+}^{N}, \end{aligned}$$

where \(\mathbb {R}^N_+ = \{(x',x_N): x' \in \mathbb {R}^{N-1},\,x_N>0 \}\) is the upper half-space, \(2_*:=2(N-1)/(N-2)\), \(N \ge 7\), \(\frac{\partial u}{\partial \nu }\) is the partial outward normal derivative and the parameter \(\lambda >0\) interacts with the spectrum of the linearized problem. In the proof, we apply variational methods.