On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space

Abstract

We study the Laplace equation in the half-space \({\mathbb{R}_{+}^{n}}\) with a nonlinear supercritical Robin boundary condition \({\frac{\partial u}{\partial\eta }+\lambda u=u\left\vert u\right\vert^{\rho-1}+f(x)}\) on \({\partial \mathbb{R}_{+}^{n}=\mathbb{R}^{n-1}}\), where n ≥ 3 and λ ≥ 0. Existence of solutions \({u \in E_{pq}= \mathcal{D}^{1, p}(\mathbb{R}_{+}^{n}) \cap L^{q}(\mathbb{R}_{+}^{n})}\) is obtained by means of a fixed point argument for a small data \(f \in {L^{d}(\mathbb{R}^{n-1})}\). The indexes p, q are chosen for the norm \({\Vert\cdot\Vert_{E_{pq}}}\) to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. \({x\in\mathbb{R}^{n-1}}\). When f is radially symmetric, u is invariant under rotations around the axis {xn = 0}. Moreover, in a certain Lq-norm, we show that solutions depend continuously on the parameter λ ≥ 0.