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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: \begin{aligned} \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,\end{aligned} ${\partial u_1\over\partial\nu}={\partial u_2\over\partial\nu}=0\quad\text{on }\partial\Omega,\quad u_2>0,\quad u_2>0\quad\text{in }\Omega,$ where $$u_1$$, $$u_2$$ represent the densities of two competing species, $$\Omega$$ is a bounded smooth domain of $$\mathbb{R}^N$$ with $$N\geq 1$$, $$\partial\Omega$$ 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.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J45 Systems of elliptic equations, general (MSC2000)
##### Keywords:
existence; nonexistence; non-constant solutions
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