Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation $\Delta u=|u|^{\sigma-1}u$ is considered in the exterior of a ball in $\mathbb{R}^n$, $n\ge3$. It is shown that if the exponent $\sigma$ is greater than a “critical” value ($=\frac{n}{n-2}$), then for $x\to\infty$ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that solutions with the indicated leading term in asymptotics of such a type exist.