Regularity of solutions for a system of integral equations
Regularity of solutions for a system of integral equations
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DOI：
10.3934/cpaa.2005.4.1
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发表时间：
2004-12
影响因子：
1
通讯作者：
Wenxiong Chen;Congming Li
中科院分区：
文献类型：
--
作者：
Wenxiong Chen;Congming Li
本文研究如下积分方程组的正解：$R^n$：$u(X)={int_{R^{n}}|x-y|^{α-n}v(Y)^q dy$，$v(X)=\int_{R^{n}}|x-y|^{α-n}u(Y)^p dy$$\frac{1}{q+1}+\frac{1}{p+1}=\frac{n-\alpha}{n}$.在前文中，我们证明了在L^{p+1}(R^n)中的自然可积条件u和L^{q+1}(R^n)中的自然可积条件下，所有解都是径向对称的，并且都是单调递减的。在本文中，我们进一步研究了解的正则性。我们证明了解是有界的，因此是连续光滑的。我们还证明了：如果$p=q$，则$u=v$，且它们都必须假设标准形式$c(\frac{t}{t^2+|x-x_o|^2})^{(n-\α)/2}$具有某个常数$c=c(n，\α)$，并且对于R^n$中的某些$t>；0$和$x_o\.
In this paper, we study positive solutions of the following system of integral equations in $R^n$: $u(x) = \int_{R^{n}} |x-y|^{\alpha -n} v(y)^q dy$, $ v(x) = \int_{R^{n}} |x-y|^{\alpha -n} u(y)^p dy$ with $\frac{1}{q+1}+\frac{1}{p+1}=\frac{n-\alpha}{n}$. In our previous paper, under the natural integrability conditions $u \in L^{p+1} (R^n)$ and $v \in L^{q+1} (R^n)$, we prove that all the solutions are radially symmetric and monotone decreasing about some point. In this paper, we go further to study the regularity of the solutions. We show that the solutions are bounded, and hence continuous and smooth. We also prove that if $p = q$, then $u = v$, and they both must assume the standard form $ c(\frac{t}{t^2 + |x - x_o|^2})^{(n-\alpha)/2} $ with some constant $c = c(n, \alpha)$, and for some $t > 0$ and $x_o \in R^n$.