论文标题
沿海渔业收获引起的非线性边界条件的逻辑椭圆方程
Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting
论文作者
论文摘要
令$ 0 <q <1 <p $。在这项研究中，我们研究了逻辑椭圆方程$-ΔU= u（1-u^{p-1}）$的阳性解决方案，以平滑的有限域$ \ Mathbb {r}^n $，$ n \ geq1 $的$ \ Mathbb {r} $ \partialΩ$。这种非线性边界条件来自沿海渔业收获。当$ p> 1 $是亚批判性时，我们证明，在$λ_Ω> 1 $的情况下，至少有两个$λ> 0 $的正解决方案足够小，但没有$λ> 0 $的正面解决方案。在$λ_Ω<1 $的情况下，每个$λ> 0 $都至少存在一个积极的解决方案。在这里，在Dirichlet边界条件下，$λ_Ω> 0 $是$-δ$的最小特征值。从生态学角度提出了对我们主要结果的解释。
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.
