Bifurcation into spectral gaps.(English)Zbl 0864.47037

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.

MSC:

 47J05 Equations involving nonlinear operators (general) 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces 47B25 Linear symmetric and selfadjoint operators (unbounded) 34C23 Bifurcation theory for ordinary differential equations 37G99 Local and nonlocal bifurcation theory for dynamical systems 35B32 Bifurcations in context of PDEs