Bounds for the extremal parameter of nonlinear eigenvalue problems and application to the explosion problem in a flow

We consider the nonlinear eigenvalue problem $Lu=\lambda f(u)$, posed in a smooth bounded domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ with Dirichlet boundary condition, where $L$ is a uniformly elliptic second-order linear differential operator, $\lambda >0$ and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $(0<a_f⩽\mathrm{\infty })$ is a smooth, increasing and convex nonlinearity such that $f(0)>0$ and which blows up at $a_f$. First we present some upper and lower bounds for the extremal parameter ${\lambda }^{\ast }$ and the extremal solution $u^{\ast }$. Then we apply the results to the operator $L_A=-\mathrm{\Delta }+Ac(x)$ with $A>0$ and $c(x)$ is a divergence-free flow in $\mathrm{\Omega }$. We show that, if ${\psi }_{A,\mathrm{\Omega }}$ is the maximum of the solution ${\psi }_A(x)$ of the equation $L_{A}u=1$ in $\mathrm{\Omega }$ with Dirichlet boundary condition, then for any incompressible flow $c(x)$ we have, ${\psi }_{A,\mathrm{\Omega }}⟶0$ as $A⟶\mathrm{\infty }$ if and only if $c(x)$ has no non-zero first integrals in $H_0^1(\mathrm{\Omega })$. Also, taking $c(x)=-x\rho (\mathrm{\mid }x\mathrm{\mid })$ where $\rho$ is a smooth real function on $[0,1]$ then $c(x)$ is never divergence-free in unit ball $B\subset {\mathbb{R}}^N$, but our results completely determine the behaviour of the extremal parameter ${\lambda }_A^{\ast }$ as $A⟶\mathrm{\infty }$.