Nodal domains for an elliptic problem with the spectral parameter near the fourth eigenvalue

Abstract

We consider the Dirichlet problem $ - \Delta u = \mu u + f$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb R^N$. Let $\hat \lambda$ be an eigenvalue with $\hat \phi$ an associated eigenfunction. We study the following question $(*)$: Assuming $ \int_{\Omega} f \hat \phi \neq 0$, has $u$ the same number of nodal domains as $\hat \phi$ if $\mu $ is sufficiently close to $\hat \lambda$? The answer to $(*)$ is known to be affirmative in various cases; see [1], [5], and [6]. Here we study a specific situation where, on the contrary, the answer to $(*)$ is not always affirmative: $\Omega=$ the unit disk in $\mathbb R^2$ and $\hat \lambda = \lambda_4 = \lambda_5$.