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2021 Singular reaction diffusion equations where a parameter infuluences the reaction term and the boundary conditions
Nalin Fonseka, Amila Muthunayake, Ratnasingham Shivaji, Byungjae Son
Topol. Methods Nonlinear Anal. 57(1): 221-242 (2021). DOI: 10.12775/TMNA.2020.022

Abstract

We analyse positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} -u''=\lambda h(t) f(u) \quad \text{in } (0,1), \\ -du'(0)+\mu(\lambda) u(0)=0,\\ u'(1)+\mu(\lambda) u(1)=0, \end{cases} \end{equation*} where $\lambda>0$ is a parameter, $d\geq 0$ is a constant, $f \in C^2([0,\infty),\mathbb{R}) $ is an increasing function which is sublinear at infinity $\Big (\lim\limits_{s \rightarrow \infty}{f(s)}/{s}=0\Big)$, $h \in C^1((0,1],(0,\infty))$ is a nonincreasing function with $h_1:=h(1)>0$ and there exist constants $d_0>0$, $\alpha \in [0,1)$ such that $h(t)\leq {d_0}/{t^\alpha}$ for all $t \in (0,1]$, and $\mu \in C([0,\infty),[0,\infty))$ is an increasing function such that $\mu(0)\geq 0$. We consider three cases of $f$, namely, $f(0)=0$, $f(0)> 0$ and $f(0) <0$. We will discuss existence and multiplicity results via the method of sub-supersolutions. Further, we will establish uniqueness results for $\lambda\approx 0$ and $\lambda\gg 1$.

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Nalin Fonseka. Amila Muthunayake. Ratnasingham Shivaji. Byungjae Son. "Singular reaction diffusion equations where a parameter infuluences the reaction term and the boundary conditions." Topol. Methods Nonlinear Anal. 57 (1) 221 - 242, 2021. https://doi.org/10.12775/TMNA.2020.022

Information

Published: 2021
First available in Project Euclid: 5 March 2021

Digital Object Identifier: 10.12775/TMNA.2020.022

Rights: Copyright © 2021 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.57 • No. 1 • 2021
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