### Local behaviour of the solutions of a class of nonlinear elliptic systems

Marie-Francoise Bidaut-Veron

#### Abstract

Here we study the behaviour near a punctual singularity of the positive solutions of semilinear elliptic systems in ${\mathbb R}^{N}(N\geq 3)$ given by $\left\{ \begin{array}{c} \Delta u+\left| x\right| ^{a}u^{s}v^{p}=0, \\ \Delta v+\left| x\right| ^{b}u^{q}v^{t}=0, \end{array} \right.$ (where $a,b,p,q,s,t\in$ ${\mathbb R}$ , $p,q>0,s,t\geq 0$). We describe the first undercritical case, and the sublinear and linear cases. The proofs do not use any variational methods, but lie essentially upon comparison properties between the two solutions $u$ and $v$, and the properties of the subsolutions and supersolutions of the scalar equation $\Delta f+\left| x\right| ^{\sigma }f^{\eta }=0$ ($\sigma ,\eta \in$ ${\mathbb R}$ , $\eta >0$). This extends the classical study of the scalar equation when $0 <\eta <\max$ $(N,(N+\sigma ))/(N-2)$.

#### Article information

Source
Adv. Differential Equations, Volume 5, Number 1-3 (2000), 147-192.

Dates
First available in Project Euclid: 27 December 2012