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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.

35J65 Nonlinear boundary value problems for linear elliptic equations
35J45 Systems of elliptic equations, general (MSC2000)
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