Regularity of extremal solutions of nonlocal elliptic systems

Abstract

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{document}$ \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*} $\end{document} with an integro-differential operator, including the fractional Laplacian, of the form \begin{document}$ \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*} $\end{document} when \begin{document}$ J $\end{document} is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of \begin{document}$ J(y) = \frac{a(y/|y|)}{|y|^{n+2s}} $\end{document} where \begin{document}$ s\in (0,1) $\end{document} and \begin{document}$ a $\end{document} is any nonnegative even measurable function in \begin{document}$ L^1(\mathbb {S}^{n-1}) $\end{document} that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions \begin{document}$ n and \begin{document}$ n for the Gelfand and Lane-Emden systems when \begin{document}$ p>1 $\end{document} (with positive and negative exponents), respectively. When \begin{document}$ s\to 1 $\end{document} , these dimensions are optimal. However, for the case of \begin{document}$ s\in(0,1) $\end{document} getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions \begin{document}$ n . As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.