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

Abstract

We study the positive solutions of the logistic elliptic equation with a nonlinear Neumann boundary condition that models coastal fishery harvesting ([18]). An essential role is played by the smallest eigenvalue of the Dirichlet eigenvalue problem, with respect to which a noncritical case is studied in [32]. In this paper, we extend our analysis to the critical case and further study the noncritical case for a more precise description of the positive solution set. Our approach relies on the energy method, sub- and supersolutions, and implicit function analysis.

Figures (3)

• Figure 1: Possible positive solution sets in the case where $\beta_\Omega<1$.
• Figure 2: Suggested positive solution set in the case where $\beta_\Omega =1$ and $pq<1$.
• Figure 3: Suggested positive solution set in the case where $\beta_\Omega = 1$ and $pq=1$, and $\lambda_c\in \Gamma_0$.

Theorems & Definitions (40)

• Theorem 0
• Remark 0
• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• proof