Maximum and antimaximum principles: beyond the first eigenvalue

Abstract

We consider the Dirichlet problem (*)$ -\bigtriangleup u=\mu u+f$ in $\Omega,u=0$ on $\partial\Omega$. Let $\widehat{\lambda}$ be an eigenvalue, with $\widehat{\varphi}$ an associated eigenfunction. Under suitable assumptions on $f$ and on the nodal domains of $\widehat{\varphi}$, we show that, if $\mu$ is sufficiently close to $\widehat{\lambda}$, then the solution $u$ of (*) has the same number of nodal domains as $\widehat{\varphi}$, and moreover the nodal domains of $u$ appear as small deformations of those of $\widehat{\varphi}$.