We study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$ where $\lambda >0$ is a positive parameter, Ω is a bounded domain in $\mathbb{R}^{N}$ $(N > 1)$ with smooth boundary ∂Ω, or $\Omega =(0,1)$ , $\frac{\partial z}{\partial \eta }$ is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that $f(0) = 0 = g(0)$ and $\lim_{s \rightarrow \infty } \frac{f(Mg(s))}{s} = 0$ for all $M>0$ . In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system.

中文翻译：

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分析反应扩散系统，其中参数同时影响反应项和边界

我们研究形式为$$ \ textstyle \ begin {cases}-\ Delta u = \ lambda f（v）; \ quad \ Omega，\\-\ Delta v = \ lambda的稳态反应扩散模型的正解g（u）; \ quad \ Omega，\\ \ frac {\ partial u} {\ partial \ eta} + \ sqrt {\ lambda} u = 0; \ quad \ partial \ Omega，\\ \ frac {\ partial v} {\ partial \ eta} + \ sqrt {\ lambda} v = 0; \ quad \ partial \ Omega，\ end {cases} $$，其中$ \ lambda> 0 $是一个正参数，Ω是$ \ mathbb {R} ^ {N} $ $（N> 1）$中的有界域具有平滑边界∂Ω或$ \ Omega =（0,1）$的情况，$ \ frac {\ partial z} {\ partial \ eta} $是z的向外法线导数。我们假设f和g是连续增加的函数，使得$ f（0）= 0 = g（0）$和$ \ lim_ {s \ rightarrow \ infty} \ frac {f（Mg（s））} {s}对于所有$ M> 0 $ = 0 $。特别是，我们将（Fonseka等人，J。Math。Anal。应用 476（2）：480-494，2019）到上述系统。