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

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $\mathrm{\Delta }u+n(n-2)u^{\frac{n+2}{n-2}}=0$ in $R^n (n \ge 3).$ByaresultofGidas,NiandNirenberg,$u_1$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 ${\mathrm{\Omega }}_1$ 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 $\Delta u + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$in$R^n.$BythesameresultofGidas,NiandNirenberg,$K \not\equiv 1$in$R^n$\mathrm{.}Inthispaperwediscusslowerboundson$\displaystyle{\sup_{\R^n} |K - 1|} .$ Relation with decay estimates at the isolated singularity via the Kelvin transform is also considered.