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

We prove the existence of sign-changing solution to the problem \(\begin{aligned} -\Delta u-\dfrac{1}{2}\left( x\cdot \nabla u\right) =\lambda u, \hbox { in }\mathbb {R}_{+}^{N}, \qquad \dfrac{\partial u}{\partial \nu }=|u|^{2_*-2}u, \hbox { on } \partial \mathbb {R}_{+}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u-{\displaystyle \frac{1}{2}}\left(x,·,\mathrm{\nabla },u\right)=\lambda u,\text{in}{\mathbb{R}}_{+}^N,{\displaystyle \frac{\partial u}{\partial \nu }}=|u|_{}^{2_{\ast }^{}-2}u,\text{on}\partial {\mathbb{R}}_{+}^N,\end{array}$$ where \(\mathbb {R}^N_+ = \{(x',x_N): x' \in \mathbb {R}^{N-1},\,x_N>0 \}\) ${\mathbb{R}}_{+}^N=\{(x^{\prime },x_N):x^{\prime }\in {\mathbb{R}}^{N-1},x_N>0\}$ is the upper half-space, \(2_*:=2(N-1)/(N-2)\) $2_{\ast }^{}:=2(N-1)/(N-2)$ , \(N \ge 7\) $N\ge 7$ , \(\frac{\partial u}{\partial \nu }\) $\frac{\partial u}{\partial \nu }$ is the partial outward normal derivative and the parameter \(\lambda >0\) $\lambda >0$ interacts with the spectrum of the linearized problem. In the proof, we apply variational methods.