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 $(-\Delta {)}^{s}v+G^{\prime }(v)=0$ 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 $G^{\prime }(-1)=G^{\prime }(1)=0$. 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 $n\ge 1$, 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.

References in corpus (2)

• Local and global minimizers for a variational energy involving a fractional norm
• Sharp energy estimates for nonlinear fractional diffusion equations

Cited by in corpus (2)

• Local and global minimizers for a variational energy involving a fractional norm
• Qualitative properties of positive solutions for mixed integro-differential equations