Perturbation methods and semilinear elliptic problems on Rn

Více o knize

Several important problems in Physics, Differential Geometry, and related fields lead to the consideration of semilinear variational elliptic equations on R, which have been the focus of extensive study. Mathematically, the primary interest lies in the limitations of Nonlinear Functional Analysis tools, particularly those based on compactness arguments, necessitating the development of new techniques. Additionally, many elliptic problems on R exhibit a perturbative nature. In certain cases, a natural perturbation parameter arises, such as in bifurcation from the essential spectrum, singularly perturbed equations, or the examination of semiclassical standing waves for NLS. In other instances, perturbations serve as a preliminary step toward achieving global results or offer a valuable perspective for further global studies. A specific approach that leverages this perturbative context appears most suitable. Perturbation methods in critical point theory provide the necessary abstract tools, proving applicable to a wide variety of equations typically regarded as distinct. This monograph aims to discuss these abstract methods alongside their applications to various perturbation problems, all of which share the common feature of involving semilinear Elliptic Partial Differential Equations on R with a variational structure.