# 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 $u \in C^2(\mathbb R^N)$ of the semi-linear elliptic equation

where $q \in (0, 1), p > q$ 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