The Gelfand problem on annular domains of double revolution with monotonicity

Abstract:

We consider the following Gelfand problem \begin{equation*} (P)_\lambda \qquad \left \{\begin {array}{ll} -\Delta u = \lambda a(x) f(u) & \text { in } \Omega , \\ u>0 & \text { in } \Omega , \\ u= 0 & \text { on } \partial \Omega , \end{array}\right . \end{equation*} where $\lambda >0$ is a parameter and $f(u)=e^u$ or $f(u)=(u+1)^p$ where $p>1$ and $a(x)$ is a nonnegative function with certain monotonicity (we allow $a(x)=1$). Here $\Omega$ is an annular domain which is also a double domain of revolution. Our interest will be in the question of the regularity of the extremal solution. We obtain improved compactness because of the annular nature of the domain and we obtain further compactness under some monotonicity assumptions on the domain.