Bifurcation into spectral gaps
The author gives a unified treatment and some refinements of recent results on the existence and bifurcation of non-trivial solutions of equations having the form \(Su-R(u)= \lambda u\), where \(S\) is an unbounded selfadjoint operator in a real Hilbert space, \(R\) is a potential operator with \(R(u)= o(u)\) for \(u\) near 0 and \(\lambda\not\in \sigma(S)\). The approach is a variational one and is based on the study of an associated equation of the form \((A-\lambda L)u- N(u)=0\), where \(A\) and \(L\) are bounded selfadjoint operators on a Hilbert space \(H\), \(N\in C^1(H,H)\), \(N(0)=0\) and \(N'(0)=0\). Applications are given to bound states for nonlinear Schrödinger equations and homoclinic solutions of Hamiltonian systems.
scientific article; zbMATH DE number 833466