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

We study the Laplace equation in the half-space with a nonlinear supercritical Robin boundary condition on , where n a parts per thousand yen 3 and lambda a parts per thousand yen 0. Existence of solutions is obtained by means of a fixed point argument for a small data . The indexes p, q are chosen for the norm to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. . When f is radially symmetric, u is invariant under rotations around the axis {x (n) = 0}. Moreover, in a certain L (q) -norm, we show that solutions depend continuously on the parameter lambda a parts per thousand yen 0.