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Ground states for coupled NLS equations with double power nonlinearities

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Abstract

We study the existence of ground states (minimizers of an energy under a fixed masses constraint) for a system of coupled NLS equations with double power nonlinearities (classical and point). We prove that the presence of at least one subcritical power parameter guarantees existence of a ground state for masses below specific values. Moreover, we show that each ground state is given by a pair of strictly positive functions (up to rotation). Using the concentration-compactness approach, under certain restrictions we prove the compactness of each minimizing sequence. One of the principal peculiarities of the model is that in presence of critical power parameters existence of ground states depends on concrete mass parameters related with the optimal function for the Gagliardo–Nirenberg inequality.

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References

  1. Esry, B.D., Greene, C.H., Burke, J.P., Jr., Bohn, J.L.: Hartree–Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997). https://doi.org/10.1103/PhysRevLett.78.3594

    Article  Google Scholar 

  2. Malomed, B., Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-González, R. (eds.) Multi-Component Bose-Einstein Condensates: Theory, pp. 287–305. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-73591-5_15

  3. Bhattarai, S.: Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities. Adv. Nonlinear Anal. 4(2), 73–90 (2015). https://doi.org/10.1515/anona-2014-0058

    Article  MathSciNet  Google Scholar 

  4. Gou, T., Jeanjean, L.: Existence and orbital stability of standing waves for nonlinear Schrödinger systems. Nonlinear Anal. 144, 10–22 (2016). https://doi.org/10.1016/j.na.2016.05.016

    Article  MathSciNet  Google Scholar 

  5. Bartsch, T., Wang, Z.-Q.: Note on ground states of nonlinear Schrödinger systems. J. Partial Differ. Equ. 19(3), 200–207 (2006)

  6. Cipolatti, R., Zumpichiatti, W.: Orbitally stable standing waves for a system of coupled nonlinear Schrödinger equations. Nonlinear Anal. 42(3, Ser. A: Theory Methods), 445–461 (2000). https://doi.org/10.1016/S0362-546X(98)00357-5

  7. Lopes, O.: Stability of solitary waves of some coupled systems. Nonlinearity 19(1), 95–113 (2006). https://doi.org/10.1088/0951-7715/19/1/006

    Article  MathSciNet  Google Scholar 

  8. Lopes, O.: Stability of solitary waves for a generalized nonlinear coupled Schrödinger systems. Sao Paulo J. Math. Sci. 5(2), 175–184 (2011). https://doi.org/10.11606/issn.2316-9028.v5i2p175-184

  9. Maia, A., Montefusco, E., Pellacci, B.: Orbital stability property for coupled nonlinear Schrödinger equations. Adv. Nonlinear Stud. 10(3), 681–705 (2010). https://doi.org/10.1515/ans-2010-0309

    Article  MathSciNet  Google Scholar 

  10. Nguyen, N.V., Wang, Z.-Q.: Orbital stability of solitary waves for a nonlinear Schrödinger system. Adv. Differ. Equ. 16(9–10), 977–1000 (2011)

  11. Pelinovsky, D.E., Yang, J.: Instabilities of multihump vector solitons in coupled nonlinear Schrödinger equations. Stud. Appl. Math. 115(1), 109–137 (2005). https://doi.org/10.1111/j.1467-9590.2005.01565

    Article  MathSciNet  Google Scholar 

  12. Song, X.: Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities. J. Math. Anal. Appl. 366(1), 345–359 (2010). https://doi.org/10.1016/j.jmaa.2009.12.011

    Article  MathSciNet  Google Scholar 

  13. Wei, J., Yao, W.: Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Commun. Pure Appl. Anal. 11(3), 1003–1011 (2012). https://doi.org/10.3934/cpaa.2012.11.1003

    Article  MathSciNet  Google Scholar 

  14. Garrisi, D., Gou, T.: Correction: Orbital stability and concentration of standing-wave solutions to a nonlinear Schrödinger system with mass critical exponents. NoDEA Nonlinear Differ. Equ. Appl. (2023). https://doi.org/10.1007/s00030-023-00843-1

    Article  Google Scholar 

  15. L., K.R.D., George, P.W.: Quantum mechanics of electrons in crystal lattices. Proc. R. Soc. Lond. A 130, 499–513 (1931). https://doi.org/10.3934/cpaa.2012.11.1003

  16. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, Providence (2005)

  17. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness. Academic Press, New York (1975)

  18. Tentarelli, L.: A general review on the NLS equation with point-concentrated nonlinearity. Commun. Appl. Ind. Math. 14(1), 62–84 (2023). https://doi.org/10.2478/caim-2023-0004

  19. Cazenave, T.: Semilinear Schrödinger Equations. New York University, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, RI, New York (2003)

  20. Goloshchapova, N.: Dynamical and variational properties of the NLS-\(\delta ^{\prime }_s\) equation on the star graph. J. Differ. Equ. 310, 1–44 (2022). https://doi.org/10.1016/j.jde.2021.11.047

    Article  MathSciNet  Google Scholar 

  21. Le Coz, S., Fukuizumi, R., Fibich, G., Ksherim, B., Sivan, Y.: Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Physica D 237(8), 1103–1128 (2008). https://doi.org/10.1016/j.physd.2007.12.004

    Article  MathSciNet  Google Scholar 

  22. Fukuizumi, R., Ohta, M., Ozawa, T.: Nonlinear Schrödinger equation with a point defect. Ann. Inst. H. Poincaré C Anal. Non Linéaire 25(5), 837–845 (2008). https://doi.org/10.1016/j.anihpc.2007.03.004

  23. Fukuizumi, R., Jeanjean, L.: Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential. Discret. Contin. Dyn. Syst. 21(1), 121–136 (2008). https://doi.org/10.3934/dcds.2008.21.121

    Article  Google Scholar 

  24. Cely, L., Goloshchapova, N.: Variational and stability properties of coupled NLS equations on the star graph. Nonlinear Anal. 224, 113056–35 (2022). https://doi.org/10.1016/j.na.2022.113056

    Article  MathSciNet  Google Scholar 

  25. Adami, R., Teta, A.: A class of nonlinear Schrödinger equations with concentrated nonlinearity. J. Funct. Anal. 180(1), 148–175 (2001). https://doi.org/10.1006/jfan.2000.3697

    Article  MathSciNet  Google Scholar 

  26. Holmer, J., Liu, C.: Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: basic theory. J. Math. Anal. Appl. (2020). https://doi.org/10.1016/j.jmaa.2019.123522

    Article  Google Scholar 

  27. Boni, F., Dovetta, S.: Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one. J. Math. Anal. Appl. (2021). https://doi.org/10.1016/j.jmaa.2020.124797

    Article  Google Scholar 

  28. Adami, R., Boni, F., Carlone, R., Tentarelli, L.: Ground states for the planar NLSE with a point defect as minimizers of the constrained energy. Calc. Var. Partial Differ. Equ. (2022). https://doi.org/10.1007/s00526-022-02310-8

    Article  MathSciNet  Google Scholar 

  29. Finco, D., Noja, D.: Blow-up and instability of standing waves for the NLS with a point interaction in dimension two. Z. Angew. Math. Phys. 74, 162 (2023). https://doi.org/10.1007/s00033-023-02056-z

  30. Fukaya, N., Georgiev, V., Ikeda, M.: On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction. J. Differ. Equ. 321, 258–295 (2022). https://doi.org/10.1016/j.jde.2022.03.008

    Article  Google Scholar 

  31. Adami, R., Boni, F., Dovetta, S.: Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs. J. Funct. Anal. 283(1), 109483–34 (2022). https://doi.org/10.1016/j.jfa.2022.109483

    Article  MathSciNet  Google Scholar 

  32. Boni, F., Carlone, R.: NLS ground states on the half-line with point interactions. NoDEA Nonlinear Differ. Equ. Appl. (2023). https://doi.org/10.1007/s00030-023-00856-w

    Article  MathSciNet  Google Scholar 

  33. Boni, F., Dovetta, S.: Doubly nonlinear Schrödinger ground states on metric graphs. Nonlinearity 35(7), 3283–3323 (2022). https://doi.org/10.1088/1361-6544/ac7505

    Article  MathSciNet  Google Scholar 

  34. Dolbeault, J., Esteban, M.J., Laptev, A., Loss, M.: One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: remarks on duality and flows. J. Lond. Math. Soc. (2) 90(2), 525–550 (2014). https://doi.org/10.1112/jlms/jdu040

    Article  MathSciNet  Google Scholar 

  35. Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14, p. 346. American Mathematical Society, Providence (2001). https://doi.org/10.1090/gsm/014

  36. Albert, J., Bhattarai, S.: Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system. Adv. Differ. Equ. 18(11–12), 1129–1164 (2013)

  37. Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(3), 191–202 (1984). https://doi.org/10.1007/BF01449041

    Article  MathSciNet  Google Scholar 

  38. Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann. Inst. H. Poincaré C Anal. Non Linéaire 31(6), 1289–1310 (2014). https://doi.org/10.1016/j.anihpc.2013.09.003

  39. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987). https://doi.org/10.1016/0022-1236(87)90044-9

    Article  MathSciNet  Google Scholar 

  40. Garrisi, D., Gou, T.: Orbital stability and concentration of standing-wave solutions to a nonlinear Schrödinger system with mass critical exponents. NoDEA Nonlinear Differential Equations Appl. 30(3), 35–1 (2023). https://doi.org/10.1007/s00030-022-00813-z

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Authors and Affiliations

  1. Mathematics Department, University of São Paulo, Rua do Matão 1010, São Paulo, SP, 05508-090, Brazil

    Nataliia Goloshchapova

  2. São Paulo, SP, Brazil

    Liliana Cely

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  1. Nataliia Goloshchapova
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Contributions

L.C. initially drafted the manuscript for the coupled NLS with a single point nonlinearity, focusing on specific power parameters. Subsequently, N.G. reformulated the problem for the double power nonlinearity, adjusted the manuscript accordingly, and distinguished between cases regarding the existence of the ground state and the compactness of the minimizing sequence.

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Correspondence to Nataliia Goloshchapova.

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Goloshchapova, N., Cely, L. Ground states for coupled NLS equations with double power nonlinearities. Nonlinear Differ. Equ. Appl. 31, 74 (2024). https://doi.org/10.1007/s00030-024-00956-1

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