Advances in Differential Equations

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

Marie-Francoise Bidaut-Veron

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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

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

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35J60: Nonlinear elliptic equations


Bidaut-Veron, Marie-Francoise. Local behaviour of the solutions of a class of nonlinear elliptic systems. Adv. Differential Equations 5 (2000), no. 1-3, 147--192.

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