Abstract
In this paper, we study the normalized solutions for the following fractional Schrödinger equation with combined nonlinearities
where
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12371121
Funding statement: The authors were supported by National Natural Science Foundation of China, Grant No. 12371121.
References
[1] F. J. Almgren, Jr. and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), no. 4, 683–773. 10.1090/S0894-0347-1989-1002633-4Search in Google Scholar
[2] L. Appolloni and S. Secchi, Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations 286 (2021), 248–283. 10.1016/j.jde.2021.03.016Search in Google Scholar
[3] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. 10.1090/S0002-9939-1983-0699419-3Search in Google Scholar
[4] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. 10.1002/cpa.3160360405Search in Google Scholar
[5] A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), no. 1, 225–236. 10.1016/j.jmaa.2004.03.034Search in Google Scholar
[6] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar
[7] 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
[8] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 6, 1237–1262. 10.1017/S0308210511000746Search in Google Scholar
[9]
B. Feng, J. Ren and Q. Wang,
Existence and instability of normalized standing waves for the fractional Schrödinger equations in the
[10] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. 10.1002/cpa.21591Search in Google Scholar
[11] 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
[12] L. Jeanjean and T. 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
[13] G. Li, X. Luo and T. Yang, Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation, Math. Methods Appl. Sci. 44 (2021), no. 13, 10331–10360. 10.1002/mma.7411Search in Google Scholar
[14] M. Li, J. He, H. Xu and M. Yang, Normalized solutions for a coupled fractional Schrödinger system in low dimensions, Bound. Value Probl. 2020 (2020), Paper No. 166. 10.1186/s13661-020-01463-9Search in Google Scholar
[15]
Q. Li and W. Zou,
The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the
[16] X. Li, Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities, Calc. Var. Partial Differential Equations 60 (2021), no. 5, Paper No. 169. 10.1007/s00526-021-02020-7Search in Google Scholar
[17] 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
[18] M. Liu and W. Zou, Normalized solutions for a system of fractional Schrödinger equations with linear coupling, Minimax Theory Appl. 7 (2022), no. 2, 303–320. Search in Google Scholar
[19] H. Luo and Z. Zhang, Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differential Equations 59 (2020), no. 4, Paper No. 143. 10.1007/s00526-020-01814-5Search in Google Scholar
[20] S. Peng and A. Xia, Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential, Commun. Pure Appl. Anal. 20 (2021), no. 11, 3723–3744. 10.3934/cpaa.2021128Search in Google Scholar
[21]
S. Secchi,
Ground state solutions for nonlinear fractional Schrödinger equations in
[22]
M. Shibata,
A new rearrangement inequality and its application for
[23] 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
[24] 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
[25] J. Wei and Y. Wu, Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal. 283 (2022), no. 6, Article ID 109574. 10.1016/j.jfa.2022.109574Search in Google Scholar
[26]
J. Yang,
The existence of normalized solutions for a nonlocal problem in
[27]
T. Yang,
Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal
[28] M. Zhen and B. Zhang, Normalized ground states for the critical fractional NLS equation with a perturbation, Rev. Mat. Complut. 35 (2022), no. 1, 89–132. 10.1007/s13163-021-00388-wSearch in Google Scholar
[29] M. Zhen, B. Zhang and V. D. Rădulescu, Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst. 41 (2021), no. 6, 2653–2676. 10.3934/dcds.2020379Search in Google Scholar
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