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Non-constant steady-state solutions for Brusselator type systems. (English) Zbl 1156.35022

Summary: We are concerned with the following stationary system:

$\begin{array}{cc}\hfill -\theta {\Delta }u=\lambda \left(1-\left(b+1\right)u+b{u}^{m}v\right)\phantom{\rule{1.em}{0ex}}& \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\hfill \\ \hfill -{\Delta }v=\lambda {a}^{2}\left(u-{u}^{m}v\right)\phantom{\rule{1.em}{0ex}}& \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\hfill \end{array}$

subject to homogeneous Neumann boundary conditions. Here ${\Omega }\subset {ℝ}^{N}$ $\left(N\ge 1\right)$ is a smooth and bounded domain and $a,b,m,\lambda ,\theta$ are positive parameters. The particular case $m=2$ corresponds to the steady-state Brusselator system. We establish existence and non-existence results for non-constant positive classical solutions. In particular, we provide upper and lower bounds for solutions which allows us to extend the previous works in the literature without any restriction on the dimension $N\ge 1$. Our analysis also emphasizes the role played by the nonlinearity ${u}^{m}$. The proofs rely essentially on various types of a priori estimates.

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35J60 Nonlinear elliptic equations 35B45 A priori estimates for solutions of PDE