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Diffusion, self-diffusion and cross-diffusion. (English) Zbl 0867.35032
The following system, which determines steady-state solutions for a corresponding parabolic system, is considered: $$\align \Delta[(d_1+\alpha_{11}u_1+\alpha_{12}u_2)u_1] &+u_1(a_1-b_1u_1-c_1u_2)=0,\\ \Delta[(d_2+\alpha_{21}u_1+\alpha_{22}u_2)u_2] &+u_2(a_2-b_2u_1-c_2u_2)=0\quad\text{in }\Omega,\endalign$$ $${\partial u_1\over\partial\nu}={\partial u_2\over\partial\nu}=0\quad\text{on }\partial\Omega,\quad u_2>0,\quad u_2>0\quad\text{in }\Omega,$$ where $u_1$, $u_2$ represent the densities of two competing species, $\Omega$ is a bounded smooth domain of $\bbfR^N$ with $N\ge 1$, $\partial\Omega$ is the boundary of $\Omega$, $\nu$ is the outward unit normal vector on $\partial\Omega$, $d_i$, $a_i$, $b_i$, $c_i$ $(i=1,2)$ are positive constants, $\alpha_{ij}$ $(i,j=1,2)$ are nonnegative constants. The results of this paper are concerning the existence and the nonexistence of non-constant solutions of this system.

35J65Nonlinear boundary value problems for linear elliptic equations
35J45Systems of elliptic equations, general (MSC2000)
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