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

In this paper, we analyze the semi-linear fractional Laplace equation (-Δ)su=f(u)inR+N,u=0inRN\R+N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^s u = f(u) \quad \text { in } {\mathbb {R}}^N_+,\quad u=0 \quad \text { in } {\mathbb {R}}^N{\setminus } {\mathbb {R}}^N_+, \end{aligned}$$\end{document}where R+N={x=(x′,xN)∈RN:xN>0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N_+=\{x=(x',x_N)\in {\mathbb {R}}^N:\ x_N>0\}$$\end{document} 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:=supRNu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho :=\sup _{{\mathbb {R}}^N}u$$\end{document} verifying f(ρ)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\rho )=0$$\end{document}, then u is necessarily one dimensional.