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Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting

Kenichiro Umezu

Abstract

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.

Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting

Abstract

Let . In this study, we investigate positive solutions of the logistic elliptic equation in a smooth bounded domain of , , with the nonlinear boundary condition on . This nonlinear boundary condition arises from coastal fishery harvesting. When is subcritical, we prove that in the case of , there exist at least two positive solutions for sufficiently small but no positive solutions for large enough. In the case of , there exists at least one positive solution for every . Here, is the smallest eigenvalue of under the Dirichlet boundary condition. An interpretation of our main results from an ecological viewpoint is presented.
Paper Structure (10 sections, 44 theorems, 151 equations, 3 figures)

This paper contains 10 sections, 44 theorems, 151 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Existence of positive solutions via the implicit function theorem and sub and supersolutions
  3. Multiplicity of positive solutions via Nehari manifold
  4. Nehari manifold and fibering map
  5. Existence of a global minimizer on $\mathcal{N}_{\lambda}^{+}$
  6. Existence of a global minimizer on $\mathcal{N}_{\lambda}^{-}$
  7. Asymptotic profiles of positive solutions
  8. Asymptotic profiles of positive solutions as $\lambda\to\infty$
  9. Asymptotic profiles of positive solutions as $\lambda\to 0^{+}$
  10. Existence of bounded, closed, and connected subsets of positive solutions

Key Result

Theorem 1.1

Let $u$ be a positive solution of p for $\lambda>0$. Then, $u<1$ in $\overline{\Omega}$ and $u>0$ on $\Gamma$ with some $\Gamma\subset \partial\Omega$ satisfying that $|\Gamma|>0$. Conversely, problem p has a smooth positive solution curve $\{ (\lambda, u_{1,\lambda}) : 0\leq \lambda<\overline{\lam

Figures (3)

  • Figure 1: Possible positive solution set of \ref{['p']} in the case of $\lambda_{\Omega}<1$.
  • Figure 2: Possible positive solution set of \ref{['p']} in the case of $\lambda_{\Omega}>1$.
  • Figure 3: Component $\mathcal{C}_{\varepsilon}$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 73 more