Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in ${\mathrm{R}}^N$

Abstract

In this paper we study the behavior near infinity of non-negative solutions $u\in C^2({\mathbb{R}}^N)$ of the semi-linear elliptic equation

where $q\in (0,1),p>q$ and $N\ge 2$. Especially, for a non-negative radial solution of this equation we prove the following alternative:
either $u$ has a compact support
or $u$ tends to one at infinity.
Moreover, we prove that if a general solution is sufficiently small in some sense, then it is compactly supported. To prove this result we use some inequalities between the solution and its spherical average at a shift point and consider a differential inequality. Finally, we prove the existence of non-trivial solutions which converge to one at infinity.