| Abstract: |
| In this talk we consider a mathematical model of thermal explosion which is described by the boundary value problem
$$\,\,\,\,\,\,\,\,\,\,\,\,\left\{
\begin{array}{rlll}
-div(|\nabla u|^{N-2}\nabla u) &=&\lambda e^{u^\alpha}, & x \in\Omega,\\
\\
|\nabla u|^{N-2}\frac{\partial u}{\partial \nu}+g(u) &=&0, & x \in \partial\Omega,
\end{array}\right.
$$
where $\Omega$ is a bounded domain in $ \mathbb{R}^N, N\geq 2,$
$\partial \Omega$ is the smooth boundary of $\Omega$
with outward normal $\nu,$ $\alpha \in (0, \frac{N}{N-1}]$ and
$\lambda$ is a positive parameter.
Using variational methods
we show that there exists $0< \Lambda < \infty$ such that the problem has at least two positive solutions if $0 < \lambda < \Lambda,$ no solution if $\lambda > \Lambda$ and at least one positive solution when $\lambda =\Lambda.$ |
|