arXiv 1902.046402019-02math.AP

非局部椭圆型方程组极值解的正则性

Regularity of extremal solutions of nonlocal elliptic systems

Mostafa Fazly

备注：: 20 pages. Comments are welcome. To appear in Discrete Continuous Dynamical Systems

摘要：

研究了具有积分-微分算子的非线性非局部特征值问题\begin{eqnarray} \left\{ \begin{array}{lcl} \hfill \mathcal L u &=& λF(u,v) \qquad \text{in} \ \ Ω, \\ \hfill \mathcal L v &=& γG(u,v) \qquad \text{in} \ \ Ω, \\ \hfill u,v &=&0 \qquad \qquad \text{on} \ \ \mathbb R^n\setminusΩ, \end{array}\right. \end{eqnarray}的极值解的正则性，当$J$是非负可测偶跳核时，其形式为\begin{equation*}\label{} \mathcal L(u (x))= \lim_{ε\to 0} \int_{\mathbb R^n\setminus B_ε(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*}.特别是，我们认为跳跃内核的形式$J(y)=\frac{a(y/|y|)}{|y|^{n+2s}}$的$s\in (0,1)$和$a$是任何非负的甚至可测量的功能，在$L^1(\mathbb {S}^{n-1})$，满足椭圆假设。首先，我们建立了上述系统的一般非线性和一般核的最小解的稳定性不等式。然后，我们证明了当$p>1$（具有正和负指数）时，Gelfand和Lane-Emden系统的$n < 10s$和$ n<2s+\frac{4s}{p\mp 1}[p+\sqrt{p(p\mp1)}]$维极值解的正则性。当$s\to 1$时，这些尺寸是最佳的。然而，对于$s\in(0,1)$的情况下，获得最佳尺寸仍然是一个悬而未决的问题。此外，对于一般的非线性，我们考虑梯度系统，我们建立的极值解的尺寸$n<4s$的规律性。据我们所知，这是关于非局部方程组极值解的第一个正则性结果。

英文摘要：

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray} \left\{ \begin{array}{lcl} \hfill \mathcal L u &=& λF(u,v) \qquad \text{in} \ \ Ω, \\ \hfill \mathcal L v &=& γG(u,v) \qquad \text{in} \ \ Ω, \\ \hfill u,v &=&0 \qquad \qquad \text{on} \ \ \mathbb R^n\setminusΩ, \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_{ε\to 0} \int_{\mathbb R^n\setminus B_ε(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.

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