Maximum and antimaximum principles near the second eigenvalue. (English) Zbl 1240.35127

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

Differ. Integral Equ. 24, No. 3-4, 389-400 (2011).

Summary: 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 the above problem has also 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.

MSC:

35J25	Boundary value problems for second-order elliptic equations
35B50	Maximum principles in context of PDEs

Keywords:

Dirichlet problem; eigenvalue; nodal domain