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Existence and uniqueness for some competition models with diffusion. (English. Abridged French version) Zbl 0741.92018
Let \(D\) be a bounded domain in \(R^ n\) with smooth boundary, and let \(a\), \(b\), \(c\), \(d\) be real constants with \(b\), \(c\) positive. Consider the stationary Lotka-Volterra system for two competing species: \[ -\Delta u=au-u^ 2-buv, \quad -\Delta v=dv-cuv-v^ 2\hbox{ in } D,\tag{*} \] under homogeneous Dirichlet boundary conditions. The authors prove an existence and uniqueness result for positive solutions (coexistence states) to (*) provided the constants \(a,\dots,d\) satisfy certain inequalities involving the positive solutions of the equation \(-\Delta w=kw-w^ 2\), \(k\in \)R. In particular, these inequalities are satisfied if \(b\) and \(c\) are sufficiently small.

92D40 Ecology
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)