Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions

This paper, which is the follow-up to part I, concerns the equation $(-\Delta)^{s} v+G'(v)=0$ in $\mathbb{R}^{n}$, with $s \in (0,1)$, where $(-\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\'evy process. When $n=1$, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits $\pm 1$ at $\pm \infty$) if and only if the potential $G$ has only two absolute minima in $[-1,1]$, located at $\pm 1$ and satisfying $G'(-1)=G'(1)=0$. Under the additional hypothesis $G"(-1)>0$ and $G"(1)>0$, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution. For $n\geq 1$, we prove some results related to the one-dimensional symmetry of certain solutions ---in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.