Taiwanese Journal of Mathematics

POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS ARISING IN A THEORY OF THERMAL EXPLOSION

Eunkyung Ko and S. Prashanth

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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 $\lambda =\Lambda.$

Article information

Source
Taiwanese J. Math., Volume 19, Number 6 (2015), 1759-1775.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133738

Digital Object Identifier
doi:10.11650/tjm.19.2015.5968

Mathematical Reviews number (MathSciNet)
MR3434276

Zentralblatt MATH identifier
1357.35263

Subjects
Primary: 35J66: Nonlinear boundary value problems for nonlinear elliptic equations 35K57: Reaction-diffusion equations

Keywords
combustion theory semilinear elliptic equations exponential nonlinearity

Citation

Ko, Eunkyung; Prashanth, S. POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS ARISING IN A THEORY OF THERMAL EXPLOSION. Taiwanese J. Math. 19 (2015), no. 6, 1759--1775. doi:10.11650/tjm.19.2015.5968. https://projecteuclid.org/euclid.twjm/1499133738


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