Variational methods for scaled functionals with applications to the Schrödinger–Poisson–Slater equation

We develop novel variational methods for solving scaled equations that do not have the mountain pass geometry, classical linking geometry based on linear subspaces, or Z2 symmetry, and therefore cannot be solved using classical variational arguments. Our contributions here include new critical group estimates for scaled functionals, nonlinear saddle point and linking geometries based on scaling, a notion of local linking based on scaling, and scaling-based multiplicity results for symmetric functionals. We develop these methods in an abstract setting involving scaled operators and scaled eigenvalue problems. Applications to subcritical and critical Schrödinger–Poisson–Slater equations are given.