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**On nonlinear Schrödinger equations with attractive inverse-power potentials.**
*(English)*
Zbl 1477.35237

Summary: We study the Cauchy problem for nonlinear Schrödinger equations with attractive inverse-power potentials. By using variational arguments, we first determine a sharp threshold of global well-posedness and blow-up for the equation in the mass-supercritical case. We next study the existence and orbital stability of standing waves for the problem in the mass-subcritical and mass-critical cases. In the mass-critical case, we give a detailed description of the blow-up behavior of standing waves when the mass tends to a critical value.

### MSC:

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35A15 | Variational methods applied to PDEs |

35J35 | Variational methods for higher-order elliptic equations |

35B44 | Blow-up in context of PDEs |

35B35 | Stability in context of PDEs |

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

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

### Keywords:

nonlinear Schrödinger equation; inverse-power potential; standing waves; global well-posedness; blow-up
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\textit{v. D. Dinh}, Topol. Methods Nonlinear Anal. 57, No. 2, 489--523 (2021; Zbl 1477.35237)

### References:

[1] | A.H. Ardila and V.D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys. 71 (2020), 79. · Zbl 1437.35615 |

[2] | J. Bellazzini, N. Boussaid, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys. 353 (2017), 229-339. · Zbl 1367.35150 |

[3] | R. Benguria and L. Jeanneret, Existence and uniqueness of positive solutions of semilinear elliptic equations with coulomb potentials on \(\mathbb R^3\), Commun. Math. Phys. 104 (1986), 291-306. · Zbl 0596.35014 |

[4] | A. Bensouilah, V.D. Dinh and S. Zhu, On stability and instability of standing waves for the nonlinear Schrodinger equation with an inverse-square potential, J. Math. Phys. 59 (2018), 101505. · Zbl 1402.35255 |

[5] | H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I: Existence of ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-345. · Zbl 0533.35029 |

[6] | N. Burq, F. Planchon, J. Stalker and A.S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrodinger equations with inverse-square pontential, J. Funct. Anal. 203 (2003), 519-549. · Zbl 1030.35024 |

[7] | H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490. · Zbl 0526.46037 |

[8] | J. M. Chadam and R.T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Math. Phys. 16 (1975), 1122-1130. · Zbl 0299.35084 |

[9] | E. Csobo and F. Genoud, Minimal mass blow-up solutions for the \(L^2\) critical NLS with inverse-square potential, Nonlinear Anal. 168 (2018), 110-129. · Zbl 1383.35207 |

[10] | C.R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics, Progress in Mathematical Physics, vol. 54, Birkhauser, Berlin, 2009. · Zbl 1165.47001 |

[11] | V.D. Dinh, Global existence and blowup for a class of the focusing nonlinear Schrodinger equation with inverse-square potential, J. Math. Anal. Appl. 468 (2018), 270-303. · Zbl 1403.35273 |

[12] | V.D. Dinh, On instability of radial standing waves for the nonlinear Schrodinger equation with inverse-square potential, Complex Var. Elliptic Equ. 2020 (in press). |

[13] | V.D. Dinh, On nonlinear Schrodinger equations with repulsive inverse-power potentials, Acta Appl. Math. 171 (2021), article no. 14. · Zbl 1472.35353 |

[14] | N. Fukaya and M. Ohta, Strong instability of standing waves for nonlinear Schrodinger equations with attractive inverse power potential, Osaka J. Math. 56 (2019), 713-726. · Zbl 1431.35172 |

[15] | R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrodinger equations with potentials, Differential Integral Equations 16 (2003), 691-706. · Zbl 1031.35131 |

[16] | R.T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations, J. Math. Phys. 18 (1977), 1794-1797. · Zbl 0372.35009 |

[17] | D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, third edition, Springer-Verlag, Berlin, 2001. · Zbl 1042.35002 |

[18] | Y. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), 141-156. · Zbl 1311.35241 |

[19] | Y.J. Guo, X.Y. Zeng and H.S. Zou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. Henri Poincare Nonlinear Anal. 33 (2016), 809-828. · Zbl 1341.35053 |

[20] | Q. Guo, H. Wang and X. Yao, Dynamics of the focusing 3D cubic NLS with slowly decaying potential, 2018, arxiv:1811.07578, preprint. |

[21] | H. Hajaiej, Cases of equality and strict inequality in the extended Hardy-Littlewood inequalities, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 643-661. · Zbl 1102.26013 |

[22] | N. Hayashi and T. Ozawa, Time decay of solutions to the Cauchy problem for timedependent Schrodinger-Hartree equations, Commun. Math. Phys. 110 (1987), 467-478. · Zbl 0648.35078 |

[23] | R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst. 37 (2017), 3831-3866. · Zbl 1366.35168 |

[24] | R. Killip, J. Murphy, M. Visan and J. Zheng, The focusing cubic NLS with inversesquare potential in three space dimension, Differential Integral Equations 30 (2017), 161-206. · Zbl 1413.35407 |

[25] | R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Sobolev spaces adapted to the Schrodinger operator with inverse-square potential, Math. Z. 288 (2018), 1273-1298. · Zbl 1391.35123 |

[26] | E.H. Lieb and M. Loss, Analysis, second edition, Graduate Studies in Mathematics, vol. 14, AMS, Providence, 2001. · Zbl 0966.26002 |

[27] | X. Li and J. Zhao, Orbital stability of standing waves for Schrodinger type equations with slowly decaying linear potential, Comput. Math. Appl. 79 (2020), 303-316. · Zbl 1443.35146 |

[28] | P.L. Lions, Some remarks on Hartree equation, Nonlinear Anal. 5 (1981), 1245-1256. · Zbl 0472.35074 |

[29] | P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. Henri Poincare 1 (1984), 109-145. · Zbl 0541.49009 |

[30] | J. Lu, C. Miao and J. Murphy, Scattering in \(H^1\) for the intercritical NLS with an inverse-square potential, J. Differential Equations 264 (2018), 3174-3211. · Zbl 1387.35554 |

[31] | C. Miao, J. Zhang and J. Zheng, Nonlinear Schrodinger equation with coulomb potential, preprint arxiv:1809.06685, 2018. |

[32] | H. Mizutani, Strichartz estimates for Schrodinger equations with slowly decaying potentials, J. Func. Anal. 279 (2020), no. 12, 108789. · Zbl 1448.35427 |

[33] | N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrodinger equations, Evol. Equ. Control Theory 1, 337-354. · Zbl 1283.35128 |

[34] | T.V. Phan, Blow-up profile of Bose-Einstein condensate with singular potentials, J. Math. Phys. 58 (2017), 072301. · Zbl 1370.82019 |

[35] | N. Shioji and K. Watanabe, A generalized Pohozaev identity and uniqueness of positive radial solutions of \(\Delta u + g(r)u + h(r)u^p = 0\), J. Differential Equations 255 (2013), 4448-4475. · Zbl 1286.35007 |

[36] | Q. Wang and D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condenstates with periodic potentials, J. Differential Equations 262 (2017), 2684-2704. · Zbl 1378.35098 |

[37] | M.I. Weinstein, Nonlinear Schrodinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983), 567-576. · Zbl 0527.35023 |

[38] | X. Zhang, On the Cauchy problem of 3-D energy-critical Schrodinger equations with subcritical perturbations, J. Differential Equations 230 (2006), 422-445. · Zbl 1106.35108 |

[39] | J. Zhang and J. Zheng, Scattering theory for nonlinear Schrodinger equation with inverse-square potential, J. Funct. Anal. 267 (2014), 2907-2932. · Zbl 1298.35130 |

[40] | J. Zheng, Focusing NLS with inverse square potential, J. Math. Phys. 59 (2018), 111502. · Zbl 1408.35178 |

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