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The relationships between equilibria and positive solutions of certain nonlinear elliptic systems. (English) Zbl 0884.35032
Author’s abstract: Let \(M(u,v)=0\), \(N(u,v)=0\) define two distinct phase curves \(\Gamma_1\), \(\Gamma_2\) in the \((u,v)\)-phase plane. This paper presents results on the relationships among the positive equilibria, the phase curves, and the existence of positive solutions to the PDE system \[ \Delta u+uM(u,v)=0, \quad \Delta v+vN(u,v)=0 \quad \text{in } \Omega \subset \mathbb{R}^n \] under Dirichlet boundary conditions, where \(\Omega\) is a bounded domain and \(M,N\) are monotone functions.

MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J60 Nonlinear elliptic equations
92D25 Population dynamics (general)
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