We study positive solutions to the steady state reaction diffusion equation of the form: − Δ u = λ f ( u ) ; Ω ∂ u ∂ η + λ u = 0 ; ∂ Ω $$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$ where λ > 0 is a positive parameter, Ω is a bounded domain in ℝ N when N > 1 (with smooth boundary ∂ Ω) or Ω = (0, 1), and ∂ u ∂ η $\begin{array}{} \displaystyle \frac{\partial u}{\partial \eta} \end{array}$ is the outward normal derivative of u. Here f(s) = ms + g(s) where m ≥ 0 (constant) and g ∈ C 2[0, r) ∩ C[0, ∞) for some r > 0. Further, we assume that g is increasing, sublinear at infinity, g(0) = 0, g′(0) = 1 and g″(0) > 0. In particular, we discuss the existence of multiple positive solutions for certain ranges of λ leading to the occurrence of Σ-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.
We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely, −∆u = λu(1 − u); x ∈ Ω ∂u ∂η + γ √ λ[(A − u) 2 + ]u = 0; x ∈ ∂Ω where Ω is a bounded domain in R N ; N > 1 with smooth boundary ∂Ω or Ω = (0, 1), ∂u ∂η is the outward normal derivative of u on ∂Ω, λ is a domain scaling parameter, γ is a measure of the exterior matrix (Ω c) hostility, and A ∈ (0, 1) and > 0 are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for u < A and increasing for u > A. We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of λ. When Ω = (0, 1) we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter γ varies. Our results indicate that when γ is large there is no Allee effect for any λ. We employ a method of sub-supersolutions to obtain existence and multiplicity results when N > 1, and the quadrature method to study the case N = 1.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.