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arXiv math/0202243math.APmath.DG
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R^n中具有临界指数的半线性偏微分方程的组合解

Combining solutions of semilinear partial differential equations in R^n with critical exponent

Man Chun Leung

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35 pages
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摘要：

设$u_1$和$u_2$是方程$Δu + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$在$R^n (n \ge 3).$中的两个不同的正光滑解。根据Gidas，Ni和Nirenberg的结果，$u_1$和$u_2$分别在点$ξ_1$和$ξ_2$上径向对称。令$u$是$R^n$上的正$C^2$ -函数，使得$Ω_1$中的$u = u_1$和$Ω_2$中的$u = u_2$，其中$Ω_1$和$Ω_2$是${\R}^n$中不相交的非空开放结构域。$u$满足$R^n.$中的方程$Δu + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$。根据Gidas，Ni和Nirenberg的相同结果，$R^n$中的$K \not\equiv 1$。本文讨论了$\displaystyle{\sup_{\R^n} |K - 1|} .$的下界，并考虑了孤立奇点处的Kelvin变换与衰变估计的关系。

英文摘要：

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $Δu + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ in $R^n (n \ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the points $ξ_1$ and $ξ_2$, respectively. Let $u$ be a positive $C^2$-function on $R^n$ such that $u = u_1$ in $Ω_1$ and $u = u_2$ in $Ω_2$, where $Ω_1$ and $Ω_2$ are disjoint non-empty open domains in ${\R}^n$. $u$ satisfies the equation $Δu + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ in $R^n.$ By the same result of Gidas, Ni and Nirenberg, $K \not\equiv 1$ in $R^n$. In this paper we discuss lower bounds on $\displaystyle{\sup_{\R^n} |K - 1|} .$ Relation with decay estimates at the isolated singularity via the Kelvin transform is also considered.

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