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

Abstract

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $\Delta u+n(n-2)u^{\frac{n+2}{n-2}}=0$ 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 ${\xi }_1$ and ${\xi }_2$, respectively. Let $u$ be a positive $C^2$-function on $R^n$ such that $u=u_1$ in ${\Omega }_1$ and $u=u_2$ in ${\Omega }_2$, where ${\Omega }_1$ and ${\Omega }_2$ are disjoint non-empty open domains in ${\mathbb{R}}^n$. $u$ satisfies the equation $\Delta u+n(n-2)K{u}^{\frac{n+2}{n-2}}=0$ 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 $\underset{{\mathbb{R}}^n}{\sup }|K-1|.$ Relation with decay estimates at the isolated singularity via the Kelvin transform is also considered.

35 pages