Maximum and antimaximum principles: beyond the first eigenvalue
Maximum and antimaximum principles: beyond the first eigenvalue
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 …