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 $\Delta u + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ 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 $\xi_1$ 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 ${\R}^n$. $u$ satisfies the equation $\Delta u + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ 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.