Elliptic problems in the half-space with nonlinear critical boundary conditions

We obtain multiple solutions for the nonlinear boundary value problem $$\begin{aligned} -\Delta u-\dfrac{1}{2}\left( x\cdot \nabla u\right) = f(u), \text{ in } {\mathbb {R}}_{+}^{N}, \qquad \dfrac{\partial u}{\partial \eta }= \beta |u|^{2/(N-2)}u, \text{ on } \partial {\mathbb {R}}_{+}^{N}, \end{aligned}$$ - Δ u - 1 2 x · ∇ u = f ( u ) , in R + N , ∂ u ∂ η = β | u | 2 / ( N - 2 ) u , on ∂ R + N , where $${\mathbb {R}}^N_+ = \{(x',x_N) \in {\mathbb {R}}^N_+ : x' \in {\mathbb {R}}^{N-1},\,x_N>0 \}$$ R + N = { ( x ′ , x N ) ∈ R + N : x ′ ∈ R N - 1 , x N > 0 } , $$\frac{\partial u}{\partial \eta }$$ ∂ u ∂ η is the partial outward normal derivative, $$\beta >0$$ β > 0 is a parameter and f is a superlinear function with subcritical growth.