Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials

Xue Zhang, Marco Squassina () and Jianjun Zhang
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Xue Zhang: College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China
Marco Squassina: Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy
Jianjun Zhang: College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

Mathematics, 2024, vol. 12, issue 5, 1-20

Abstract: We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation ( − Δ ) s u + V ( ε x ) u = λ u + h ( ε x ) f ( u ) i n R N , ∫ R N | u | 2 d x = a , , where ( − Δ ) s is the fractional Laplacian, s ∈ ( 0 , 1 ) , a , ε > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, h : R N → [ 0 , + ∞ ) are bounded and continuous, and f is L 2 -subcritical. Under some assumptions on the potential V , we show the existence of normalized solutions depends on the global maximum points of h when ε is small enough.

Keywords: fractional Laplacian; normalized solution; mass critical exponent (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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