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Proceedings of the Royal Society of Edinburgh Section A: Mathematics Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Article contents

Multiplicity and concentration results for a fractional Schrödinger-Poisson type equation with magnetic field

Part of: Variational problems in infinite-dimensional spaces Miscellaneous topics - Partial differential equations Partial differential equations

Published online by Cambridge University Press:  23 January 2019

Vincenzo Ambrosio
Vincenzo Ambrosio
Affiliation:
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, via Brecce Bianche 12, 60131Ancona, Italy (vincenzo.ambrosio2@unina.it)
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vincenzo.ambrosio2@unina.it
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Abstract

This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type

$$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$
where ε > 0 is a parameter, s, t ∈ (0, 1) are such that 2s+2t>3, A:ℝ3 → ℝ3 is a smooth magnetic potential, (−Δ)As is the fractional magnetic Laplacian, V:ℝ3 → ℝ is a continuous electric potential and f:ℝ → ℝ is a C1 subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for e > 0 small enough.

Keywords

Fractional magnetic operators Nehari manifold Ljusternick-Schnirelmann theory

MSC classification

Primary: 35A15: Variational methods
Secondary: 35R11: Fractional partial differential equations 35S05: Pseudodifferential operators 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 655 - 694
Copyright
Copyright © Royal Society of Edinburgh 2019

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