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

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 $\lambda =\Lambda.$