Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities. (29th December 2019)

• Record Type: Journal Article
• Authors: Xie, Weihong; Chen, Haibo; Shi, Hongxia
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 6(2020)
• Page Start: 3602
• Page End: 3616
• Publication Date: 2019-12-29
• Subjects: multiplicity -- normalized solutions -- Schrödinger‐Poisson system -- variational methods; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6140
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we study the existence and multiplicity of standing waves with prescribed L 2 ‐norm Schrödinger‐Poisson equations with general nonlinearities in R 3 : i ∂ t ψ + Δ ψ − κ ( | x | − 1 * | φ | 2 ) ψ + f ( ψ ) = 0, where κ > 0 and f is superlinear and satisfies the monotonicity condition. To this end, we look for critical points of the following functional E κ ( u ) = 1 2 ∫ R 3 | ∇ u | 2 + κ 4 ∫ R 3 ( | x | − 1 * u 2 ) u 2 − ∫ R 3 F ( u ) constrained on the L 2 ‐spheres S ( c ) = u ∈ H 1 ( R 3 ) : | | u | | 2 2 = c, c > 0, where F ( s ) : = ∫ 0 s f ( t ) d t . We consider the case where E κ is unbounded from below on S ( c ) and establish the existence of critical points of E κ on S ( c ) for c > 0 sufficiently small and under some mild assumptions on f . In addition, we show that there are infinitely many radial critical points { u n κ } of E κ on S ( c ) when f is odd and present a convergence property of u n κ as κ ↘ 0 . Our results generalize some recent results in the literature.