## Differential and Integral Equations

- Differential Integral Equations
- Volume 25, Number 11/12 (2012), 1189-1202.

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

J. Fleckinger, J.-P. Gossez, and F. de Thélin

#### 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 $\hat \lambda =
\lambda_4 = \lambda_5$.

#### Article information

**Source**

Differential Integral Equations, Volume 25, Number 11/12 (2012), 1189-1202.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012257

**Mathematical Reviews number (MathSciNet)**

MR3013410

**Zentralblatt MATH identifier**

1274.35088

**Subjects**

Primary: 35J25: Boundary value problems for second-order elliptic equations

#### Citation

Fleckinger, J.; Gossez, J.-P.; de Thélin, F. Nodal domains for an elliptic problem with the spectral parameter near the fourth eigenvalue. Differential Integral Equations 25 (2012), no. 11/12, 1189--1202. https://projecteuclid.org/euclid.die/1356012257