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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:

$\begin{array}{cc}\hfill {\Delta }\left[\left({d}_{1}+{\alpha }_{11}{u}_{1}+{\alpha }_{12}{u}_{2}\right){u}_{1}\right]& +{u}_{1}\left({a}_{1}-{b}_{1}{u}_{1}-{c}_{1}{u}_{2}\right)=0,\hfill \\ \hfill {\Delta }\left[\left({d}_{2}+{\alpha }_{21}{u}_{1}+{\alpha }_{22}{u}_{2}\right){u}_{2}\right]& +{u}_{2}\left({a}_{2}-{b}_{2}{u}_{1}-{c}_{2}{u}_{2}\right)=0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\hfill \end{array}$
$\frac{\partial {u}_{1}}{\partial \nu }=\frac{\partial {u}_{2}}{\partial \nu }=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },\phantom{\rule{1.em}{0ex}}{u}_{2}>0,\phantom{\rule{1.em}{0ex}}{u}_{2}>0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },$

where ${u}_{1}$, ${u}_{2}$ represent the densities of two competing species, ${\Omega }$ is a bounded smooth domain of ${ℝ}^{N}$ with $N\ge 1$, $\partial {\Omega }$ is the boundary of ${\Omega }$, $\nu$ is the outward unit normal vector on $\partial {\Omega }$, ${d}_{i}$, ${a}_{i}$, ${b}_{i}$, ${c}_{i}$ $\left(i=1,2\right)$ are positive constants, ${\alpha }_{ij}$ $\left(i,j=1,2\right)$ 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