# Regularity of extremal solutions of nonlocal elliptic systems

## TL;DR

This paper investigates the regularity of extremal solutions for nonlinear nonlocal elliptic systems involving fractional Laplacians, establishing stability inequalities and regularity results in specific dimensions, and highlighting open problems for optimal dimensions.

## Contribution

It provides the first regularity results for extremal solutions of nonlocal elliptic systems with fractional operators, including stability inequalities and dimension bounds.

## Key findings

- Regularity of extremal solutions established for dimensions n<10s and n<2s+... for specific systems.
- Stability inequalities derived for minimal solutions with general nonlinearities and kernels.
- Open problem identified for achieving optimal dimension bounds when s in (0,1).

## Abstract

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray} \left\{ \begin{array}{lcl} \hfill \mathcal L u &=& \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill \mathcal L v &=& \gamma G(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill u,v &=&0 \qquad \qquad \text{on} \ \ \mathbb R^n\setminus\Omega , \end{array}\right. \end{eqnarray} with an integro-differential operator, including the fractional Laplacian, of the form \begin{equation*}\label{} \mathcal L(u (x))= \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*} when $J$ is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of $J(y)=\frac{a(y/|y|)}{|y|^{n+2s}}$ where $s\in (0,1)$ and $a$ is any nonnegative even measurable function in $L^1(\mathbb {S}^{n-1})$ that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions $n < 10s$ and $ n<2s+\frac{4s}{p\mp 1}[p+\sqrt{p(p\mp1)}]$ for the Gelfand and Lane-Emden systems when $p>1$ (with positive and negative exponents), respectively. When $s\to 1$, these dimensions are optimal. However, for the case of $s\in(0,1)$ getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions $n<4s$. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.