Variational methods for strongly indefinite problems
Ding Yanheng · Scientia Sinica Mathematica · 2017
This paper reviews some recent development in studying variational methods for strongly indefinite problems under the support of National Natural Science Foundation of China. We firstly describe the ideas for establishing the variational setting, then state a deformation theory in locally convex topological linear spaces and several critical point theorems for dealing with strongly indefinite problems based on the deformation. Some applications of the theory are given including mainly: Existence and multiplicity of solutions to non-autonomous stationary Dirac system, in particular, the existence, concentration phenomena and the exponential decay of semi-classical states; existence and multiplicity of global solutions to the nonlinear (non-autonomous, unbounded Hamilton-type) reaction-diffusion system, especially the existence, concentration phenomena and decay of its ground states of the system with singular perturbation; depth study of homoclinic orbits Hamiltonian system, global Schrödinger equation; other initial work, such as bifurcation of Dirac equation on spin manifolds.