Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914645417132032 |
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| author | Umezu, Kenichiro |
| author_facet | Umezu, Kenichiro |
| contents | Let $0<q<1<p$. In this study, we investigate positive solutions of the logistic elliptic equation $-Δu = u(1-u^{p-1})$ in a smooth bounded domain $Ω$ of $\mathbb{R}^N$, $N\geq1$, with the nonlinear boundary condition $\frac{\partial u}{\partial ν}=-λu^q$ on $\partialΩ$. This nonlinear boundary condition arises from coastal fishery harvesting. When $p>1$ is subcritical, we prove that in the case of $λ_Ω>1$, there exist at least two positive solutions for $λ>0$ sufficiently small but no positive solutions for $λ>0$ large enough. In the case of $λ_Ω<1$, there exists at least one positive solution for every $λ>0$. Here, $λ_Ω>0$ is the smallest eigenvalue of $-Δ$ under the Dirichlet boundary condition. An interpretation of our main results from an ecological viewpoint is presented. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2202_09442 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting Umezu, Kenichiro Analysis of PDEs Let $0<q<1<p$. In this study, we investigate positive solutions of the logistic elliptic equation $-Δu = u(1-u^{p-1})$ in a smooth bounded domain $Ω$ of $\mathbb{R}^N$, $N\geq1$, with the nonlinear boundary condition $\frac{\partial u}{\partial ν}=-λu^q$ on $\partialΩ$. This nonlinear boundary condition arises from coastal fishery harvesting. When $p>1$ is subcritical, we prove that in the case of $λ_Ω>1$, there exist at least two positive solutions for $λ>0$ sufficiently small but no positive solutions for $λ>0$ large enough. In the case of $λ_Ω<1$, there exists at least one positive solution for every $λ>0$. Here, $λ_Ω>0$ is the smallest eigenvalue of $-Δ$ under the Dirichlet boundary condition. An interpretation of our main results from an ecological viewpoint is presented. |
| title | Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2202.09442 |