Funções de Green e aplicações a problemas elípticos

Abstract

In this work, we study the Laplace equation in the half-space RN + with a nonlinear supercritical Robin boundary condition ¶u¶h + lu = ujujr??1 + f (x) on ¶RN+ = RN??1, where N _ 3and l _ 0. Existence of solution u 2 Ep,q = D1,p(RN +) \ Lq(RN+) is obtained by means of a fixed point argument for a small data f 2 Ld(RN??1). The indexes p, q are chosen for the norm k _ kEp,q to be invariant by scaling of the boundary problem. The solution u is positive whether f (x) > 0 a.e. x 2 RN??1. When f is radially symmetric, u is invariant under rotations around the axis fxN = 0g. Moreover, in a certain Lq-norm, we show that solutions depend continuosly on the parameter l _ 0.