Asymptotic symmetries for fractional operators

Abstract

In this paper, we study equations driven by a non-local integrodifferential operator $\mathcal{L}_K$ with homogeneous Dirichlet boundary conditions. More precisely, we study the problem \[ \begin{aligned} &- \mathcal{L}_K u + V(x)u = |u|^{p-2}u, &&\text{in } Ω, \newline &u=0, &&\text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \] where $2 < p < 2^{*}_s = \frac{2N}{N-2s}$, $Ω$ is an open bounded domain in $\mathbb{R}^{N}$ for $N\ge 2$ and $V$ is a $L^\infty$ potential such that $-\mathcal{L}_K + V$ is positive definite. As a particular case, we study the problem \[ \begin{aligned} &(- Δ)^s u + V(x)u = |u|^{p-2}u, &&\text{in } Ω, \newline &u=0, &&\text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \] where $(-Δ)^s$ denotes the fractional Laplacian (with $0<s<1$). We give assumptions on $V$, $Ω$ and $K$ such that ground state solutions (resp. least energy nodal solutions) respect the symmetries of some first (resp. second) eigenfunctions of $-\mathcal{L}_K + V$, at least for $p$ close to $2$. We study the uniqueness, up to a multiplicative factor, of those types of solutions. The results extend those obtained for the local case.