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

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)

Zbl 0833.92010