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# Regularity of extremal solutions of nonlocal elliptic systems

• We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem

$\begin{eqnarray*} \left\{ \begin{array}{lcl} \hfill \mathcal L u & = & \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill \mathcal L v & = & \gamma G(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill u,v & = &0 \qquad \qquad \text{on} \ \ \mathbb R^{n} \backslash \Omega , \end{array}\right. \end{eqnarray*}$

with an integro-differential operator, including the fractional Laplacian, of the form

$\begin{equation*} \label{} \mathcal L(u (x)) = \lim\limits_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*}$

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.

Mathematics Subject Classification: 35R09, 35R11, 35B45, 35B65, 35J50.

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