Fleckinger, J.; Gossez, J.-P.; de Thélin, F. Maximum and antimaximum principles: beyond the first eigenvalue. (English) Zbl 1240.35128 Differ. Integral Equ. 22, No. 9-10, 815-828 (2009). Summary: We consider the Dirichlet problem \(-\Delta u=\mu u+f\) 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 the above problem has the same number of nodal domains as \(\widehat {\varphi }\), and moreover, the nodal domains of \(u\) appear as small deformations of those of \(\widehat {\varphi }\). Cited in 1 ReviewCited in 2 Documents MSC: 35J25 Boundary value problems for second-order elliptic equations 35B50 Maximum principles in context of PDEs 35P05 General topics in linear spectral theory for PDEs Keywords:Dirichlet problem; nodal domain; maximum principle; antimaximum principle PDFBibTeX XMLCite \textit{J. Fleckinger} et al., Differ. Integral Equ. 22, No. 9--10, 815--828 (2009; Zbl 1240.35128)