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 $$\begin{aligned} (-\Delta )^s u = f(u) \quad \text { in } {\mathbb {R}}^N_+,\quad u=0 \quad \text { in } {\mathbb {R}}^N{\setminus } {\mathbb {R}}^N_+, \end{aligned}$$ where $${\mathbb {R}}^N_+=\{x=(x',x_N)\in {\mathbb {R}}^N:\ x_N>0\}$$ 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 $$\rho :=\sup _{{\mathbb {R}}^N}u$$ verifying $$f(\rho )=0$$ , then u is necessarily one dimensional.