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**Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations.**
*(English)*
Zbl 1435.35344

Summary: We consider the Cauchy problem for the nonlinear Schrödinger (NLS) equation with double nonlinearities with opposite sign, with one term mass-critical and the other term mass-supercritical and energy-subcritical, which includes the well-known two-dimensional cubic-quintic NLS equation arising in the study of the boson gas with 2- and 3-body interactions. We prove global well-posedness and scattering in \(H^1(\mathbb{R}^d)\) below the threshold for nonradial data when \(1\le d\le 4\).

### MSC:

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q41 | Time-dependent Schrödinger equations and Dirac equations |

35L70 | Second-order nonlinear hyperbolic equations |

35P25 | Scattering theory for PDEs |

35A01 | Existence problems for PDEs: global existence, local existence, non-existence |

35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |

### References:

[1] | T. Akahori, S. Ibrahim, H. Kikuchi, and H. Nawa, Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth, Differential Integral Equations 25 (2012), nos. 3-4, 383-402. · Zbl 1265.35329 |

[2] | T. Akahori, S. Ibrahim, H. Kikuchi, and H. Nawa, Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth, Selecta Math. (N.S.) 19 (2013), no. 2, 545-609. · Zbl 1379.35284 |

[3] | T. Akahori, S. Ibrahim, H. Kikuchi, and H. Nawa, Global dynamics above the ground state energy for the combined power type nonlinear Schrödinger equations with energy critical growth at low frequencies, preprint, arXiv:1510.08034 [math.AP]. |

[4] | I. V. Barashenkov, A. D. Gocheva, V. G. Makhankov, and I. V. Puzynin, Stability of the soliton-like bubbles, Phys. D 34 (1989), nos. 1-2, 240-254. · Zbl 0697.35127 |

[5] | P. Bégout and A. Vargas, Mass concentration phenomena for the \(L^2\)-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5257-5282. · Zbl 1171.35109 |

[6] | 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. · Zbl 0541.35029 |

[7] | R. Carles and S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equations II: The \(L^2\)-critical case, Trans. Amer. Math. Soc. 359 (2007), no. 1, 33-62. · Zbl 1115.35119 |

[8] | T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Courant Inst. Math. Sci, New York, 2003. · Zbl 1055.35003 |

[9] | T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^s\), Nonlinear Anal. 14 (1990), no. 10, 807-836. · Zbl 0706.35127 |

[10] | X. Cheng, Z. Guo, K. Yang, and L. Zhao, On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide \(\mathbb{R}^2\times \mathbb{T} \), preprint, arXiv:1705.00954 [math.AP]. |

[11] | X. Cheng, Z. Guo, and Z. Zhao, On scattering for the quintic defocusing nonlinear Schrödinger equation on the two-dimensional cylinder, preprint, arXiv:1809.01527 [math.AP]. |

[12] | X. Cheng, C. Miao, and L. Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations 261 (2016), no. 6, 2881-2934. · Zbl 1350.35180 |

[13] | M. Coles and S. Gustafson, Solitary waves and dynamics for subcritical perturbations of energy critical NLS, preprint, arXiv:1707.07219 [math.AP]. |

[14] | J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in \(\mathbb{R}^3 \), Ann. of Math. (2) 167 (2008), no. 3, 767-865. · Zbl 1178.35345 |

[15] | B. Dodson, Global well-posedness and scattering for the defocusing, \(L^2\)-critical, nonlinear Schrödinger equation when \(d\geq 3\), J. Amer. Math. Soc. 25 (2012), no. 2, 429-463. · Zbl 1236.35163 |

[16] | B. Dodson, Global well-posedness and scattering for the defocusing, \(L^2\)-critical, nonlinear Schrödinger equation when \(d=2\), Duke Math. J. 165 (2016), no. 18, 3435-3516. · Zbl 1361.35164 |

[17] | B. Dodson, Global well-posedness and scattering for the defocusing, \(L^2\)-critical, nonlinear Schrödinger equation when \(d=1\), Amer. J. Math. 138 (2016), no. 2, 531-569. · Zbl 1341.35149 |

[18] | B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285 (2015), 1589-1618. · Zbl 1331.35316 |

[19] | B. Dodson, Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension \(d=4\), Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 1, 139-180. · Zbl 1421.35333 |

[20] | T. Duyckaerts, J. Holmer, and S. Roudenko, Scattering for the non-radial 3d cubic nonlinear Schrödinger equation, Math. Res. Lett. 15: 6 (2008), 1233-1250. · Zbl 1171.35472 |

[21] | G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse, Appl. Math. Sci. 192, Springer, Cham, 2015. · Zbl 1351.35001 |

[22] | D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 (2005), no. 1, 1-24. · Zbl 1071.35025 |

[23] | N. Fukaya and M. Ohta, Strong instability of standing waves with negative energy for double power nonlinear Schrödinger equations, SUT J. Math. 54 (2018), no. 2, 131-143. · Zbl 1428.35509 |

[24] | J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3d cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282 (2008), no. 2, 435-467. · Zbl 1155.35094 |

[25] | S. Ibrahim, N. Masmoudi, and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE 4 (2011), no. 3, 405-460. · Zbl 1270.35132 |

[26] | C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645-675. · Zbl 1115.35125 |

[27] | C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), no. 2, 147-212. · Zbl 1183.35202 |

[28] | S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), no. 2, 353-392. · Zbl 1038.35119 |

[29] | S. Keraani, On the blow up phenomenon of the critical Schrödinger equation, J. Funct. Anal. 235 (2006), no. 1, 171-192. · Zbl 1099.35132 |

[30] | R. Killip, T. Oh, O. Pocovnicu, and M. Visan, Solitions and scattering for the cubic-quintic nonlinear Schrödinger equation on \(\mathbb{R}^3 \), Arch. Ration. Mech. Anal. 225 (2017), no. 1, 469-548. · Zbl 1367.35158 |

[31] | R. Killip, T. Tao, and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203-1258. · Zbl 1187.35237 |

[32] | R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math. 132 (2010), no. 2, 361-424. · Zbl 1208.35138 |

[33] | R. Killip and M. Visan, “Nonlinear Schrödinger equations at critical regularity” in Evolution Equations, Clay Math. Proc. 17, Amer. Math. Soc., Providence, 2013, 325-437. · Zbl 1298.35195 |

[34] | R. Killip, M. Visan, and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE 1 (2008), no. 2, 229-266. · Zbl 1171.35111 |

[35] | S. Le Coz and T.-P. Tsai, Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations, Nonlinearity 27 (2014), no. 11, 2689-2709. · Zbl 1304.35646 |

[36] | M. Maeda, Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity, Kodai Math. J. 31 (2008), no. 2, 263-271. · Zbl 1180.35483 |

[37] | F. Merle and L. Vega, Compactness at blow-up time for \(L^2\) solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not. IMRN 8 (1998), 399-425. · Zbl 0913.35126 |

[38] | C. Miao, G. Xu, and L. Zhao, The dynamics of the \(3D\) radial NLS with the combined terms, Comm. Math. Phys. 318 (2013), no. 3, 767-808. · Zbl 1260.35209 |

[39] | C. Miao, G. Xu, and L. Zhao, “The dynamics of the NLS with the combined terms in five and higher dimensions” in Some Topics in Harmonic Analysis and Applications, Adv. Lec. Math. (ALM) 34, 265-298, Int. Press, Somerville, MA, 2016. · Zbl 1350.35183 |

[40] | C. Miao, T. Zhao, and J. Zheng, On the \(4D\) nonlinear Schrödinger equation with combined terms under the energy threshold, Calc. Var. Partial Differential Equations 56 (2017), no. 6, art. 179, 39 pp. · Zbl 1384.35121 |

[41] | K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zur. Lect. Adv. Math, Eur. Math. Soc., Zürich, 2011. · Zbl 1235.37002 |

[42] | K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations 44 (2012), nos. 1-2, 1-45. · Zbl 1237.35148 |

[43] | M. Ohta, Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Math. J. 18 (1995), no. 1, 68-74. · Zbl 0868.35111 |

[44] | M. Ohta, and T. Yamaguchi, Strong instability of standing waves for nonlinear Schrödinger equations with double power nonlinearity, SUT J. Math. 51 (2015), no. 1, 49-58. · Zbl 1337.35138 |

[45] | D. E. Pelinovsky, V. V. Afanasjev, and Y. S. Kivshar, Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation, Phys. Rev. E 53 (1996), no. 2, 1940-1953. |

[46] | A. Stefanov, On the normalized ground states of second order PDE’s with mixed power non-linearities, Comm. Math. Phys. 369 (2019), no. 3, 929-971. · Zbl 1421.35067 |

[47] | T. Tao, M. Visan, and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32 (2007), nos. 7-9, 1281-1343. · Zbl 1187.35245 |

[48] | T. Tao, M. Visan, and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math. 20 (2008), no. 5, 881-919. · Zbl 1154.35085 |

[49] | T. Tao, M. Visan, and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J. 140 (2007), no. 1, 165-202. · Zbl 1187.35246 |

[50] | M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2123-2136. · Zbl 1196.35074 |

[51] | M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567-576. · Zbl 0527.35023 |

[52] | J. Xie, Scattering for focusing combined power-type NLS, Acta Math. Sin. (Engl. Ser). 30 (2014), no. 5, 805-826. · Zbl 1290.35215 |

[53] | G. Xu and J. Yang, Long time dynamics of the \(3D\) radial NLS with the combined terms, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 5, 521-540. · Zbl 1342.35359 |

[54] | X. Zhang, On Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations, J. Differential Equations 230 (2006), no. 2, 422-445. · Zbl 1106.35108 |

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