# Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in $R_{N}$

### Michèle Grillot

Université d'Orléans, France### Philippe Grillot

Université d'Orléans, France

## Abstract

In this paper we study the behavior near infinity of non-negative solutions $u∈C_{2}(R_{N})$ of the semi-linear elliptic equation

where $q∈(0,1),p>q$ and $N≥2$. 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.

## Cite this article

Michèle Grillot, Philippe Grillot, Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in $R_{N}$. Z. Anal. Anwend. 20 (2001), no. 4, pp. 915–928

DOI 10.4171/ZAA/1051