## Differential and Integral Equations

### Maximum and antimaximum principles near the second eigenvalue

#### Abstract

We consider the Dirichlet problem $(*)$ $-\Delta u = \mu u + f$ in $\Omega$, $u=0$ on $\partial \Omega$, with $\Omega$ either a bounded smooth convex domain in $\mathbb R^2$, or a ball or an annulus in $\mathbb R^N$. Let $\lambda_2$ be the second eigenvalue, with $\varphi_2$ an associated eigenfunction. Although the two nodal domains of $\varphi_2$ do not satisfy the interior ball condition, we are able to prove under suitable assumptions that, if $\mu$ is sufficiently close to $\lambda_2$, then the solution $u$ of $(*)$ also has two nodal domains which appear as small deformations of the nodal domains of $\varphi_2$. For $N=2$, use is made in the proof of several results relative to the Payne conjecture.

#### Article information

Source
Differential Integral Equations, Volume 24, Number 3/4 (2011), 389-400.

Dates
First available in Project Euclid: 20 December 2012