A counterexample on competing species equations. (English) Zbl 0842.35033

Dancer, E. N.

Differ. Integral Equ. 9, No. 2, 239-246 (1996).

The author considers the competing species system \[ - \Delta u = u(a - u - bv), \quad - \Delta v = v(d - v - cu) \quad \text{in } \Omega \] with boundary condition \(u = v = 0\) on \(\partial \Omega\). Using calculations from [J. C. Eilbeck, J. E. Furter and J. Lopez-Gomez, J. Differ. Equ. 107, No. 1, 96-139 (1994; Zbl 0833.92010)], he shows that there is no positive solution of this system for certain parameter values \(a,b,c,d\), where \(a\) and \(d\) are close to the first eigenvalue of \(-\Delta\) with homogeneous boundary condition.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J55	Systems of elliptic equations, boundary value problems (MSC2000)

Keywords:

nonexistence of positive solutions; bifurcation equation

Citations:

Zbl 0833.92010