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- Title
The existence and multiplicity of L<sup>2</sup>-normalized solutions to nonlinear Schrödinger equations with variable coefficients.
- Authors
Ikoma, Norihisa; Yamanobe, Mizuki
- Abstract
The existence of L2–normalized solutions is studied for the equation − Δ u + μ u = f (x , u) in R N , ∫ R N u 2 d x = m. Here m > 0 and f(x, s) are given, f(x, s) has the L2-subcritical growth and (μ, u) ∈ R × H1(RN) are unknown. In this paper, we employ the argument in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.
- Subjects
NONLINEAR Schrodinger equation; SCHRODINGER equation; LAGRANGIAN points; LAGRANGIAN functions; MULTIPLICITY (Mathematics); SCALAR field theory
- Publication
Advanced Nonlinear Studies, 2024, Vol 24, Issue 2, p477
- ISSN
1536-1365
- Publication type
Article
- DOI
10.1515/ans-2022-0056