Symmetry results in the half-space for a semi-linear fractional Laplace equation

Abstract

In this paper, we analyze the semi-linear fractional Laplace equation (−)s u = f (u) in RN +, u = 0 in RN \RN +, where RN + = {x = (x , xN ) ∈ RN : xN > 0} stands for the half-space and f is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if u is a bounded solution with ρ := supRN u verifying f (ρ) = 0, then u is necessarily one dimensional.