MAXIMUM AND ANTIMAXIMUM PRINCIPLES: BEYOND THE FIRST EIGENVALUE

A Jean et a ̀ Patrick, avec toute notre amitié We consider the Dirichlet problem (*)−4 u = µu + f in Ω, u = 0 on ∂Ω. Let λ ̂ be an eigenvalue, with ϕ ̂ an associated eigenfunction. Under suitable assumptions on f and on the nodal domains of ϕ̂, we show that if µ is sufficiently close to λ̂, then the solution u of (*) has the same number of nodal domains as ϕ̂, and moreover the nodal domains of u appear as small deformations of those of ϕ̂