JournalszaaVol. 20, No. 4pp. 915–928

Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in RNR^N

  • Michèle Grillot

    Université d'Orléans, France
  • Philippe Grillot

    Université d'Orléans, France
Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in $R^N$ cover
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Abstract

In this paper we study the behavior near infinity of non-negative solutions uC2(RN)u \in C^2(\mathbb R^N) of the semi-linear elliptic equation

Δu+uqup=0– \Delta u+u^q – u^p =0

where q(0,1),p>qq \in (0, 1), p > q and N2N ≥2. Especially, for a non-negative radial solution of this equation we prove the following alternative:
either uu has a compact support
or uu 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 RNR^N. Z. Anal. Anwend. 20 (2001), no. 4, pp. 915–928

DOI 10.4171/ZAA/1051