The authors investigate the existence of positive solutions for reaction- diffusion systems of the following type:
Here is a bounded domain, and ; are given functions of the form
It is assumed that the nonnegative exponents p, q, r, s satisfy the conditions
moreover, and 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.