Bounds for the extremal parameter of nonlinear eigenvalue problems and applications

Abstract

We consider the nonlinear eigenvalue problem $\mathit{L}\mathit{u}=\mathit{\lambda }\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}$ , posed in a smooth bounded domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^{\mathit{N}}$ with Dirichlet boundary condition, where L is a uniformly elliptic second-order linear differential operator, $\mathit{\lambda }\mathrm{>}0$ and $\mathit{f}:[0\mathrm{,}{\mathit{a}}_{\mathit{f}}\mathrm{)}\to {\mathbb{R}}_{+}$ ( $0\mathrm{<}{\mathit{a}}_{\mathit{f}}⩽\infty$ ) is a smooth, increasing and convex nonlinearity such that $\mathit{f}\mathrm{(}0\mathrm{)}\mathrm{>}0$ and which blows up at ${\mathit{a}}_{\mathit{f}}$ . First we present some upper and lower bounds for the extremal parameter ${\mathit{\lambda }}^{\ast }$ and the extremal solution ${\mathit{u}}^{\ast }$ . Then we apply the results to the operator ${\mathit{L}}_{\mathit{A}}=-\mathrm{\Delta }+\mathit{A}\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}$ with $\mathit{A}\mathrm{>}0$ and $\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}$ is a divergence-free flow in Ω. We show that, if ${\mathit{\psi }}_{\mathit{A}\mathrm{,}\mathrm{\Omega }}$ is the maximum of the solution ${\mathit{\psi }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}$ of the equation ${\mathit{L}}_{\mathit{A}}\mathit{u}=1$ in Ω with Dirichlet boundary condition, then for any incompressible flow $\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}$ we have, ${\mathit{\psi }}_{\mathit{A}\mathrm{,}\mathrm{\Omega }}⟶0$ as $\mathit{A}⟶\infty$ if and only if $\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}$ has no non-zero first integrals in ${\mathit{H}}_0^1\mathrm{(}\mathrm{\Omega }\mathrm{)}$ . Also, taking $\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}=-\mathit{x}\mathit{\rho }\mathrm{(}|\mathit{x}|\mathrm{)}$ where ρ is a smooth real function on $[0\mathrm{,}1]$ then $\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}$ is never divergence-free in unit ball $\mathit{B}\subset {\mathbb{R}}^{\mathit{N}}$ , but our results completely determine the behaviour of the extremal parameter ${\mathit{\lambda }}_{\mathit{A}}^{\ast }$ as $\mathit{A}⟶\infty$ .