Variational methods for scaled functionals with applications to the Schr\"{o}dinger-Poisson-Slater equation

Carlo Mercuri; Kanishka Perera

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arXiv · 2411.15887

Variational methods for scaled functionals with applications to the Schr\"{o}dinger-Poisson-Slater equation

Carlo Mercuri·Kanishka Perera

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Abstract

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 $\mathbb Z_2$ 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 Schr\"{o}dinger-Poisson-Slater equations are given.

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Carlo Mercuri, Kanishka Perera. 2024-11-24. Variational methods for scaled functionals with applications to the Schr\"{o}dinger-Poisson-Slater equation. https://arxiv.org/abs/2411.15887

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