*(English)*Zbl 0656.92004

The authors investigate the existence of positive solutions for reaction- diffusion systems of the following type:

Here ${\Omega}\subseteq {\mathbb{R}}^{n}$ is a bounded domain, ${\nu}_{i},{\mu}_{i}\in {\mathbb{R}}_{+}$ and ${g}_{i}(x,u)$ $(u\equiv ({u}_{1},{u}_{2})$; $i=1,2)$ are given functions of the form

It is assumed that the nonnegative exponents p, q, r, s satisfy the conditions

moreover, ${\sigma}_{i}$ and ${\rho}_{i}$ satisfy suitable assumptions concerning nonnegativity, smoothness and boundedness. The system (*) is complemented with no-flux boundary conditions and initial conditions. The motivation for this study comes from mathematical models describing pattern formation [see e.g. *H. Meinhardt*, Models of biological pattern formation. (1982)].

Under additional assumptions on p and r, the existence of a unique positive solution to the above problem is established. Results concerning equilibrium solutions of (*) (with no-flux boundary conditions) are also derived; in particular, nonexistence of patterns is shown under suitable assumptions. The relationship with earlier results on the same subjects [*W.-M. Ni* and *I. Takagi*: Trans. Am. Math. Soc. 297, 351-368 (1986; Zbl 0635.35031)] is discussed.

##### MSC:

92B05 | General biology and biomathematics |

35K57 | Reaction-diffusion equations |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35K60 | Nonlinear initial value problems for linear parabolic equations |