Abstract
This paper concerns the scalar field equation
under the normalized constraint
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12271313
Award Identifier / Grant number: 12071266
Award Identifier / Grant number: 12101376
Award Identifier / Grant number: 12171204
Funding statement: Xiaojing Feng is supported by NSFC (12271313, 12071266, 12101376), Fundamental Research Program of Shanxi Province (202203021211300, 202203021211309, 20210302124528) and Shanxi Scholarship Council of China (2020–005). Haidong Liu is supported by NSFC (12171204).
References
[1]
C. Alves, C. Ji and O. Miyagaki,
Multiplicity of normalized solutions for a Schrödinger equation with critical growth in
[2] T. Bartsch and S. de Valeriola, Normalized solutions of nonlinear Schrödinger equations, Arch. Math. (Basel) 100 (2013), no. 1, 75–83. 10.1007/s00013-012-0468-xSearch in Google Scholar
[3] T. Bartsch, Y. Liu and Z. Liu, Normalized solutions for a class of nonlinear Choquard equations, Partial Differ. Equ. Appl. 1 (2020), no. 5, Paper No. 34. 10.1007/s42985-020-00036-wSearch in Google Scholar
[4] T. Bartsch, R. Molle, M. Rizzi and G. Verzini, Normalized solutions of mass supercritical Schrödinger equations with potential, Comm. Partial Differential Equations 46 (2021), no. 9, 1729–1756. 10.1080/03605302.2021.1893747Search in Google Scholar
[5] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal. 272 (2017), no. 12, 4998–5037. 10.1016/j.jfa.2017.01.025Search in Google Scholar
[6] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345. 10.1007/BF00250555Search in Google Scholar
[7] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347–375. 10.1007/BF00250556Search in Google Scholar
[8] B. Bieganowski and J. Mederski, Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth, J. Funct. Anal. 280 (2021), no. 11, Paper No. 108989. 10.1016/j.jfa.2021.108989Search in Google Scholar
[9] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549–561. 10.1007/BF01403504Search in Google Scholar
[10] S. Cingolani, M. Gallo and K. Tanaka, Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities, Calc. Var. Partial Differential Equations 61 (2022), no. 2, Paper No. 68. 10.1007/s00526-021-02182-4Search in Google Scholar
[11] S. Cingolani and K. Tanaka, Ground state solutions for the nonlinear Choquard equation with prescribed mass, Geometric Properties for Parabolic and Elliptic PDEs, Springer INdAM Ser. 47, Springer, Cham (2021), 23–41. 10.1007/978-3-030-73363-6_2Search in Google Scholar
[12]
V. Coti Zelati and P. Rabinowitz,
Homoclinic type solutions for a semilinear elliptic PDE on
[13] Y. Ding and X. Zhong, Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case, J. Differential Equations 334 (2022), 194–215. 10.1016/j.jde.2022.06.013Search in Google Scholar
[14]
J. Hirata and K. Tanaka,
Nonlinear scalar field equations with
[15]
N. Ikoma and K. Tanaka,
A note on deformation argument for
[16] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997), no. 10, 1633–1659. 10.1016/S0362-546X(96)00021-1Search in Google Scholar
[17] L. Jeanjean and T. Le, Multiple normalized solutions for a Sobolev critical Schrödinger equation, Math. Ann. 384 (2022), no. 1–2, 101–134. 10.1007/s00208-021-02228-0Search in Google Scholar
[18] L. Jeanjean and S. Lu, Nonradial normalized solutions for nonlinear scalar field equations, Nonlinearity 32 (2019), no. 12, 4942–4966. 10.1088/1361-6544/ab435eSearch in Google Scholar
[19] L. Jeanjean and S. Lu, A mass supercritical problem revisited, Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 174. 10.1007/s00526-020-01828-zSearch in Google Scholar
[20] F. Li, Y. Li and J. Shi, Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent, Commun. Contemp. Math. 16 (2014), no. 6, Article ID 1450036. 10.1142/S0219199714500369Search in Google Scholar
[21]
G. Li and H. Ye,
The existence of positive solutions with prescribed
[22] X. Li and S. Ma, Choquard equations with critical nonlinearities, Commun. Contemp. Math. 22 (2020), no. 4, Article ID 1950023. 10.1142/S0219199719500238Search in Google Scholar
[23] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. 10.1016/s0294-1449(16)30428-0Search in Google Scholar
[24] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283. 10.1016/s0294-1449(16)30422-xSearch in Google Scholar
[25] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam. 1 (1985), no. 1, 145–201. 10.4171/rmi/6Search in Google Scholar
[26] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoam. 1 (1985), no. 2, 45–121. 10.4171/rmi/12Search in Google Scholar
[27] X. Luo, Normalized standing waves for the Hartree equations, J. Differential Equations 267 (2019), no. 7, 4493–4524. 10.1016/j.jde.2019.05.009Search in Google Scholar
[28] R. Molle, G. Riey and G. Verzini, Normalized solutions to mass supercritical Schrödinger equations with negative potential, J. Differential Equations 333 (2022), 302–331. 10.1016/j.jde.2022.06.012Search in Google Scholar
[29] M. Shibata, Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term, Manuscripta Math. 143 (2014), no. 1–2, 221–237. 10.1007/s00229-013-0627-9Search in Google Scholar
[30] N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations 269 (2020), no. 9, 6941–6987. 10.1016/j.jde.2020.05.016Search in Google Scholar
[31] N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), no. 6, Article ID 108610. 10.1016/j.jfa.2020.108610Search in Google Scholar
[32] W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. 10.1007/BF01626517Search in Google Scholar
[33]
C. Stuart,
Bifurcation in
[34] C. Stuart, Bifurcation from the essential spectrum for some noncompact nonlinearities, Math. Methods Appl. Sci. 11 (1989), no. 4, 525–542. 10.1002/mma.1670110408Search in Google Scholar
[35] N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. 10.1002/cpa.3160200406Search in Google Scholar
[36] J. Wei and Y. Wu, Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal. 283 (2022), no. 6, Paper No. 109574. 10.1016/j.jfa.2022.109574Search in Google Scholar
[37] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996. 10.1007/978-1-4612-4146-1Search in Google Scholar
[38] J. Xia and X. Zhang, Normalized saddle solutions for a mass supercritical Choquard equation, J. Differential Equations 364 (2023), 471–497. 10.1016/j.jde.2023.03.049Search in Google Scholar
[39] S. Yao, H. Chen, V. Rădulescu and J. Sun, Normalized solutions for lower critical Choquard equations with critical Sobolev perturbation, SIAM J. Math. Anal. 54 (2022), no. 3, 3696–3723. 10.1137/21M1463136Search in Google Scholar
[40]
H. Ye,
Mass minimizers and concentration for nonlinear Choquard equations in
[41] W. Ye, Z. Shen and M. Yang, Normalized solutions for a critical Hartree equation with perturbation, J. Geom. Anal. 32 (2022), no. 9, Paper No. 242. 10.1007/s12220-022-00986-0Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Open orbits and primitive zero ideals for solvable Lie algebras
- On the Pauli group on 2-qubits in dynamical systems with pseudofermions
- Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
- Perturbation of domain for the linear parabolic equation
- K-theory of flag Bott manifolds
- Some results on Seshadri constants of vector bundles
- Strichartz inequality for orthonormal functions associated with special Hermite operator
- The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds
- On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems
- Boundedness of commutators of rough Hardy operators on grand variable Herz spaces
- Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets
- An alternative proof of Tataru’s dispersive estimates
- The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions
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- Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case
- A class of quaternionic Fourier orthonormal bases
- Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields
- Normalized solutions for scalar field equation involving multiple critical nonlinearities