Steady-states of the Gierer-Meinhardt system in exterior domains
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| Format: | Preprint |
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2024
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| author | Ghergu, Marius McNicholl, Jack |
| author_facet | Ghergu, Marius McNicholl, Jack |
| contents | We discuss the existence and nonexistence of solutions to the steady-state Gierer-Meinhardt system $$ \begin{cases} \displaystyle -Δu=\frac{u^p}{v^q}+λρ(x) \,, u>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle -Δv=\frac{u^m}{v^s} \,, v>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle \;\;\; \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0 &\quad\mbox{ on }\partial K,\\[0.1in] \displaystyle \;\;\; u(x), v(x)\to 0 &\quad\mbox{ as }|x|\to \infty, \end{cases} $$ where $K\subset \mathbb{R}^N$ $(N\geq 2)$ is a compact set, $ρ\in C^{0,γ}_{loc}(\overline{\mathbb{R}^N\setminus K})$, $γ\in (0,1)$, is a nonnegative function and $p,q,m,s, λ>0$. Combining fixed point arguments with suitable barrier functions, we construct solutions with a prescribed asymptotic growth at infinity. Our approach can be extended to many other classes of semilinear elliptic systems with various sign of exponents. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_13603 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Steady-states of the Gierer-Meinhardt system in exterior domains Ghergu, Marius McNicholl, Jack Analysis of PDEs 35J47, 35B45, 35J75, 35B40 We discuss the existence and nonexistence of solutions to the steady-state Gierer-Meinhardt system $$ \begin{cases} \displaystyle -Δu=\frac{u^p}{v^q}+λρ(x) \,, u>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle -Δv=\frac{u^m}{v^s} \,, v>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle \;\;\; \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0 &\quad\mbox{ on }\partial K,\\[0.1in] \displaystyle \;\;\; u(x), v(x)\to 0 &\quad\mbox{ as }|x|\to \infty, \end{cases} $$ where $K\subset \mathbb{R}^N$ $(N\geq 2)$ is a compact set, $ρ\in C^{0,γ}_{loc}(\overline{\mathbb{R}^N\setminus K})$, $γ\in (0,1)$, is a nonnegative function and $p,q,m,s, λ>0$. Combining fixed point arguments with suitable barrier functions, we construct solutions with a prescribed asymptotic growth at infinity. Our approach can be extended to many other classes of semilinear elliptic systems with various sign of exponents. |
| title | Steady-states of the Gierer-Meinhardt system in exterior domains |
| topic | Analysis of PDEs 35J47, 35B45, 35J75, 35B40 |
| url | https://arxiv.org/abs/2403.13603 |