×

Maximum and antimaximum principles: beyond the first eigenvalue. (English) Zbl 1240.35128

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 \(\widehat {\varphi }\).

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
PDFBibTeX XMLCite