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| We consider a model of thermal explosion which is described by positive solutions to the boundary value problem
\begin{eqnarray*}
\left\{\begin{array}{ll}
-\Delta u=\lambda f(u), & x\in \Omega, \\
{\bf n} \cdot \nabla u +c(u)u=0, & x \in \partial \Omega,
\end{array}\right.
\end{eqnarray*}
where $f,c: [0,\infty) \rightarrow (0,\infty)$ are $C^1$ and $C^{1,\gamma}$ non decreasing functions satisfying $\lim_{u\rightarrow \infty}\frac{f(u)}{u}=0$, $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega$ and $\lambda> 0$ is a parameter. Using method of sub and super-solutions we show that solution of this problem is unique for large and small vales of parameter $\lambda$, whereas for intermediate values of $\lambda$ solutions are multiple provided nonlinearity
$f$ satisfies some natural assumptions. An example of such nonlinearity which is most relevant to applications and satisfies all our hypotheses is $f(u)=\exp[\frac{\alpha u}{\alpha+u}]$ for $ \alpha \gg 1.$
Also, we extend our results to the problem on a exterior domain
\begin{eqnarray*}
\left\{\begin{array}{ll}
-\Delta u=\lambda K(|x|) f(u), & x\in \Omega_E, \\
{\bf n} \cdot \nabla u +c(u)u=0, & |x| = r_0,\\
u(x)\rightarrow 0, & |x| \rightarrow \infty
\end{array}\right.
\end{eqnarray*}
where $\Omega_E = \{ x \in \mathbb{R^N}: |x| >r_0, N>2, r_0 >0 \}$ and $K:[r_0, \infty) \rightarrow (0, \infty)$ is a continuous function such that $\displaystyle\lim_{r\rightarrow \infty}K(r) =0.$ |
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