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

Abstract:

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 Lévy 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 hypotheses $G''(-1)>0$ 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.