A nonlinear elliptic system with degenerate diffusions

In this paper we are interested in studying the existence of positive solutions to the following nonlinear system with degenerate diffusions: -4u,vD u=uMu,v inW -yu,vDv=vN u,va 16u6n+b u=0 * on6W a26v6n +gv=0 in a bounded region O in Rn with a smooth boundary, where the diffusion terms ϕ, y , : R ⊕ R → R are nonnegative functions and could be zero in certain values of u, v, and alphai ≥ 0 for i = 1, 2 and beta, gamma are convex, strictly increasing, beta(0) = gamma(0) = 0. Thus when alphai = 0, the boundary conditions are Dirichlet boundary conditions. The reaction terms M, N satisfy the conditions Mv < 0, Nu < 0 which are, mathematically and biologically, significantly different from [32] in which Nu > 0, Mv, < 0. They reflect different biological backgrounds and therefore contain totally different mathematical natures (which we will describe below). The positive coexistence of interacting system (*) means that there exist solutions u > 0, v > 0 to the system on O satisfying the boundary conditions. Let uo be the density of species u when v ≡ 0 in the system. Namely uo is the density of species u before the species v participates in the interaction. The function v o is defined similarly. It has been discovered that the positive coexistence of the system (*) can be characterized by uo, vo: the densities before interaction begins. More precisely, such positive coexistence is related to principal eigenvalues of two Shrodinger-type differential operators of which the functions uo, vo serve as roles of potential functions.; The case of Mv < 0, Nu > 0, as in [32], requires that these two principal eigenvalues are positive, which is a sufficient and necessary condition for positive solutions. By Theorem 13.2 of Amman [2], it turns out to be suitable to apply the compact solution operators which were originated by Blat-Brown and later developed by Li. However, this method does not work for our case of Mv < 0, Nu < 0, which actually allows the principal eigenvalues to be negative. So we turn to apply Leray-Schauder's topological degree theory and the index theorems in cones of Banach spaces given by Dancer and supplemented by Li. We also need to make use of bifurcation theorem by Crandall-Rabinowitz. We establish, in this investigation, some conditions under which the system (*) with Mv < 0, Nu < 0 will have positive coexistence.