Back to Search
Start Over
The existence and multiplicity of L 2-normalized solutions to nonlinear Schrödinger equations with variable coefficients
- Source :
- Advanced Nonlinear Studies, Vol 24, Iss 2, Pp 477-509 (2024)
- Publication Year :
- 2024
- Publisher :
- De Gruyter, 2024.
-
Abstract
- The existence of L 2–normalized solutions is studied for the equation −Δu+μu=f(x,u) inRN,∫RNu2dx=m. $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 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 L 2 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.
Details
- Language :
- English
- ISSN :
- 21690375
- Volume :
- 24
- Issue :
- 2
- Database :
- Directory of Open Access Journals
- Journal :
- Advanced Nonlinear Studies
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.0e14fc40f41e4166a601d90df19659bb
- Document Type :
- article
- Full Text :
- https://doi.org/10.1515/ans-2022-0056