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 $\widehat{\varphi}$.
Citation
J. Fleckinger. J.-P. Gossez. F. de Thélin. "Maximum and antimaximum principles: beyond the first eigenvalue." Differential Integral Equations 22 (9/10) 815 - 828, September/October 2009. https://doi.org/10.57262/die/1356019509
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