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Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger–Poisson systems

Tomas Dutko, Carlo Mercuri, Megan Tyler

Calculus of Variations and Partial Differential Equations, Volume: 60, Issue: 5

Swansea University Authors: Carlo Mercuri, Megan Tyler

Abstract

We study a nonlinear Schrödinger-Poisson system which reduces to a nonlinear andnonlocal PDE set on the whole spatial domain, whose variational formulation requires propertiesof a suitable finite energy space, such as separability and compactness, that we study and use.Our equation involves a weight...

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Published in: Calculus of Variations and Partial Differential Equations
ISSN: 0944-2669 1432-0835
Published: Springer Science and Business Media LLC 2021
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa57454
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Abstract: We study a nonlinear Schrödinger-Poisson system which reduces to a nonlinear andnonlocal PDE set on the whole spatial domain, whose variational formulation requires propertiesof a suitable finite energy space, such as separability and compactness, that we study and use.Our equation involves a weight function which is allowed as particular scenarios, to either 1) vanish on a region and be finite at infinity, or 2) be large at infinity. We find least energy solutionsin both cases, studying the vanishing case by means of a priori integral bounds on sequences ofapproximating solutions and highlighting the role of certain positive universal constants for thesebounds to hold. Within the Ljusternik-Schnirelman theory we show the existence of infinitelymany distinct pairs of high energy solutions, having a min-max characterisation given by meansof the Krasnoselskii genus.
Keywords: Nonlinear Schrödinger–Poisson system; Weighted Sobolev spaces; Palais–Smale sequences; Compactness; Multiple solutions; Nonexistence
College: Faculty of Science and Engineering
Issue: 5