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Zeitschrift für Analysis und ihre Anwendungen

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Volume 20, Issue 4, 2001, pp. 915–928
DOI: 10.4171/ZAA/1051

Published online: 2001-12-31

Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in $R^N$

Michèle Grillot[1] and Philippe Grillot[2]

(1) Université d'Orléans, France
(2) Université d'Orléans, France

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 $$– \Delta u+u^q – u^p =0$$ 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.

Keywords: Laplacian, non-linearity, asymptotical behavior

Grillot Michèle, Grillot Philippe: Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in $R^N$. Z. Anal. Anwend. 20 (2001), 915-928. doi: 10.4171/ZAA/1051