Asymptotic symmetries for fractional operators

TL;DR

This paper investigates the symmetry properties of solutions to non-local fractional equations involving integrodifferential operators, extending known results from the local case and analyzing the uniqueness of ground state and nodal solutions near the linear regime.

Contribution

It establishes symmetry and uniqueness results for solutions to fractional non-local equations, generalizing classical local operator results to fractional and integrodifferential operators.

Findings

Solutions inherit symmetries of eigenfunctions for p close to 2.

Ground state and nodal solutions are unique up to multiplication.

Results extend symmetry properties known from local operators to fractional cases.

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 \[ \begin{aligned} &- \mathcal{L}_K u + V(x)u = |u|^{p-2}u, &&\text{in } \Omega, \newline &u=0, &&\text{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \] where $2 < p < 2^{*}_s = \frac{2N}{N-2s}$, $\Omega$ 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 \[ \begin{aligned} &(- \Delta)^s u + V(x)u = |u|^{p-2}u, &&\text{in } \Omega, \newline &u=0, &&\text{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \] where $(-\Delta)^s$ denotes the fractional Laplacian (with $0<s<1$). We give assumptions on $V$, $\Omega$ 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.