Σ-Shaped Bifurcation Curves

A Acharya,N Fonseka,J Quiroa,R Shivaji

doi:10.1515/anona-2020-0180

Abstract

AbstractWe study positive solutions to the steady state reaction diffusion equation of the form:−Δu=λf(u); Ω∂u∂η+λu=0; ∂Ω$$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$whereλ> 0 is a positive parameter,Ωis a bounded domain in ℝNwhenN> 1 (with smooth boundary∂ Ω) orΩ= (0, 1), and∂u∂η$\begin{array}{} \displaystyle \frac{\partial u}{\partial \eta} \end{array}$is the outward normal derivative ofu. Heref(s) =ms+g(s) wherem≥ 0 (constant) andg∈C2[0,r) ∩C[0, ∞) for somer> 0. Further, we assume thatgis increasing, sublinear at infinity,g(0) = 0,g′(0) = 1 andg″(0) > 0. In particular, we discuss the existence of multiple positive solutions for certain ranges ofλleading to the occurrence ofΣ-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.

Highlights

In the recent literature there has been considerable interest in reaction di usion models where a parameter in uences the equation as well as the boundary conditions
We study positive solutions to the steady state reaction di usion equation of the form:
Proof of Remark 1.1: First, we show the non-existence of positive solutions for λ ≈

Summary

Introduction

In the recent literature there has been considerable interest in reaction di usion models where a parameter in uences the equation as well as the boundary conditions. See [1, 2, 3] for recent studies in this direction. We enhance this study to show that for certain classes of such models the bifurcation diagram (λ, u ∞) for positive solutions is at least Σ-shaped. We study boundary value problems of the form:. (constant) and g ∈ C [ , r) ∩ C[ , ∞) for some r >. We assume that g is increasing and satis es:. We recall some results from [3]. For k > , let Ak be the principal eigenvalue of the problem:

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NCN R

We also have

Let k k