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2015 POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS ARISING IN A THEORY OF THERMAL EXPLOSION
Eunkyung Ko, S. Prashanth
Taiwanese J. Math. 19(6): 1759-1775 (2015). DOI: 10.11650/tjm.19.2015.5968

Abstract

In this paper we study a mathematical model of thermal explosion which is described by the boundary value problem \[\begin{cases}-\Delta u = \lambda e^{u^{\alpha}}, &x \in \Omega, \\\mathbf{n} \cdot \nabla u + g(u) u = 0, &x \in \partial \Omega,\end{cases}\]where the constant $\alpha \in (0,2],~ g:[0, \infty)\rightarrow (0, \infty)$ is an nondecreasing $C^1$ function,  $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial \Omega$ and $\lambda \gt 0$ is a bifurcation parameter.Using variational methodswe show that there exists $0\lt \Lambda \lt \infty$ such that the problem has at least two positive  solutions if $0 \lt \lambda \lt \Lambda,$ no solution if $\lambda \gt \Lambda$ and at least one positive solution when $\lambda =\Lambda.$

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Eunkyung Ko. S. Prashanth. "POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS ARISING IN A THEORY OF THERMAL EXPLOSION." Taiwanese J. Math. 19 (6) 1759 - 1775, 2015. https://doi.org/10.11650/tjm.19.2015.5968

Information

Published: 2015
First available in Project Euclid: 4 July 2017

zbMATH: 1357.35263
MathSciNet: MR3434276
Digital Object Identifier: 10.11650/tjm.19.2015.5968

Subjects:
Primary: 35J66, 35K57

Rights: Copyright © 2015 The Mathematical Society of the Republic of China

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Vol.19 • No. 6 • 2015
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