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

We consider the nonlinear eigenvalue problem $L u = \lambda f(u)$, posed in a smooth bounded domain $\Omega \subseteq \Bbb{R}^{N}$ with Dirichlet boundary condition, where $L$ is a uniformly elliptic second-order linear differential operator, $\lambda > 0$ and $f:[0,a_{f}) \rightarrow \Bbb{R}_{+}$ $(0 < a_{f} \leqslant \infty)$ 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 $\lambda^{*}$ and the extremal solution $u^{*}$. Then we apply the results to the operator $L_A = - \Delta + A c(x)$ with $A>0$ and $c(x)$ is a divergence-free flow in $\Omega$. We show that, if $\psi_{A,\Omega}$ is the maximum of the solution $\psi_{A}(x)$ of the equation $L_A u = 1$ in $\Omega$ with Dirichlet boundary condition, then for any incompressible flow $c(x)$ we have, $\psi_{A,\Omega} \longrightarrow 0$ as $A \longrightarrow \infty$ if and only if $c(x)$ has no non-zero first integrals in $H_{0}^{1}(\Omega)$. Also, taking $c(x)=-x\rho(|x|)$ where $\rho$ is a smooth real function on $[0,1]$ then $c(x)$ is never divergence-free in unit ball $B\subset \Bbb{R}^{N}$, but our results completely determine the behaviour of the extremal parameter $\lambda^{*}_{A}$ as $A \longrightarrow \infty$.