# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2022007
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## Stability and instability of standing waves for Gross-Pitaevskii equations with double power nonlinearities

 University of Electronic Science and Technology of China, School of Mathematical Sciences, Chengdu 611731, Sichuan Province, China

* Corresponding author: Jian Zhang

Received  September 2021 Early access March 2022

Fund Project: The project is supported by the National Natural Science Foundation of China grant 11871138

In this paper, we investigate Gross-Pitaevskii equations with double power nonlinearities. Firstly, due to the defocusing effect from the lower power order nonlinearity, Gross-Pitaevskii equations still have standing waves when the frequency $\omega$ is the negative of the first eigenvalue of the linear operator $- \Delta + \gamma|x{|^2}$. The existence of this class of standing waves is proved by the variational method, especially the mountain pass lemma. Secondly, by establishing the relationship to the known standing waves of the classical nonlinear Schrödinger equations, we study the instability of standing waves for $q \ge 1 + 4/N$ and $\omega$ sufficiently large. Finally, we use the variational argument to prove the stability of standing waves for $q \le 1 + 4/N$.

Citation: Yue Zhang, Jian Zhang. Stability and instability of standing waves for Gross-Pitaevskii equations with double power nonlinearities. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022007
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] V. Ambrosio, Zero mass case for a fractional Berestycki-Lions-type problem, Adv. Nonlinear Anal., 7 (2018), 365-374.  doi: 10.1515/anona-2016-0153. [3] H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981), 489-492. [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [5] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3. [6] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504. [8] S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2007/08), 1070-1111.  doi: 10.1137/050648389. [9] E. N. Dancer and S. Santra, Singular perturbed problems in the zero mass case: Asymptotic behavior of spikes, Ann. Mat. Pura Appl., 189 (2010), 185-225.  doi: 10.1007/s10231-009-0105-x. [10] W. Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308.  doi: 10.1007/BF00282336. [11] N. Fukaya and M. Hayashi, Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities, Trans. Amer. Math. Soc., 374 (2021), 1421-1447.  doi: 10.1090/tran/8269. [12] N. Fukaya and M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math., 56 (2019), 713-726. [13] R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 691-706. [14] R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 111-128. [15] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I., J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9. [16] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II., J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E. [17] M. Hirose and M. Ohta, Structure of positive radial solutions to scalar field equations with harmonic potential, J. Differential Equations, 178 (2002), 519-540.  doi: 10.1006/jdeq.2000.4010. [18] M. Hirose and M. Ohta, Uniqueness of positive solutions to scalar field equations with harmonic potential, Funkcial. Ekvac., 50 (2007), 67-100.  doi: 10.1619/fesi.50.67. [19] C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25-R62.  doi: 10.1088/0951-7715/14/5/201. [20] H. Kikuchi, Existence of standing waves for the nonlinear Schrödinger equation with double power nonlinearity and harmonic potential, in Asymptotic Analysis and Singularities—Elliptic and Parabolic PDEs and Related Problems, Adv. Stud. Pure Math., 47-2, Math. Soc. Japan, Tokyo, 2007,623–633. doi: 10.2969/aspm/04720623. [21] M. K. Kwong, Uniqueness of positive solutions $\Delta u - u + {u^p} = 0$ in ${\mathbb{R}^n}$, Arch. Rational Mech. Anal., 105 (1989), 234-266.  doi: 10.1007/BF00251502. [22] 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), 109-145.  doi: 10.1016/s0294-1449(16)30428-0. [23] 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), 223-282.  doi: 10.1016/s0294-1449(16)30422-x. [24] Y.-G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274.  doi: 10.1016/0022-0396(89)90123-X. [25] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré Phys. Théor, 62 (1995), 69-80. [26] M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential, Funkcial. Ekvac., 61 (2018), 135-143.  doi: 10.1619/fesi.61.135. [27] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003. [28] S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. [29] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [30] J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190.  doi: 10.1007/BF01212446. [31] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517. [32] L. Véron, Comportement asymptotique des solutions d'équations elliptiques semi-linéaires dans ${\mathbb{R}^n}$, Ann. Mat. Pura Appl., 127 (1981), 25-50.  doi: 10.1007/BF01811717. [33] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103. [34] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. [35] J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443.  doi: 10.1080/03605300500299539. [36] J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.  doi: 10.1023/A:1026437923987.

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] V. Ambrosio, Zero mass case for a fractional Berestycki-Lions-type problem, Adv. Nonlinear Anal., 7 (2018), 365-374.  doi: 10.1515/anona-2016-0153. [3] H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981), 489-492. [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [5] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3. [6] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504. [8] S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2007/08), 1070-1111.  doi: 10.1137/050648389. [9] E. N. Dancer and S. Santra, Singular perturbed problems in the zero mass case: Asymptotic behavior of spikes, Ann. Mat. Pura Appl., 189 (2010), 185-225.  doi: 10.1007/s10231-009-0105-x. [10] W. Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308.  doi: 10.1007/BF00282336. [11] N. Fukaya and M. Hayashi, Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities, Trans. Amer. Math. Soc., 374 (2021), 1421-1447.  doi: 10.1090/tran/8269. [12] N. Fukaya and M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math., 56 (2019), 713-726. [13] R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 691-706. [14] R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 111-128. [15] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I., J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9. [16] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II., J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E. [17] M. Hirose and M. Ohta, Structure of positive radial solutions to scalar field equations with harmonic potential, J. Differential Equations, 178 (2002), 519-540.  doi: 10.1006/jdeq.2000.4010. [18] M. Hirose and M. Ohta, Uniqueness of positive solutions to scalar field equations with harmonic potential, Funkcial. Ekvac., 50 (2007), 67-100.  doi: 10.1619/fesi.50.67. [19] C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25-R62.  doi: 10.1088/0951-7715/14/5/201. [20] H. Kikuchi, Existence of standing waves for the nonlinear Schrödinger equation with double power nonlinearity and harmonic potential, in Asymptotic Analysis and Singularities—Elliptic and Parabolic PDEs and Related Problems, Adv. Stud. Pure Math., 47-2, Math. Soc. Japan, Tokyo, 2007,623–633. doi: 10.2969/aspm/04720623. [21] M. K. Kwong, Uniqueness of positive solutions $\Delta u - u + {u^p} = 0$ in ${\mathbb{R}^n}$, Arch. Rational Mech. Anal., 105 (1989), 234-266.  doi: 10.1007/BF00251502. [22] 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), 109-145.  doi: 10.1016/s0294-1449(16)30428-0. [23] 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), 223-282.  doi: 10.1016/s0294-1449(16)30422-x. [24] Y.-G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274.  doi: 10.1016/0022-0396(89)90123-X. [25] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré Phys. Théor, 62 (1995), 69-80. [26] M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential, Funkcial. Ekvac., 61 (2018), 135-143.  doi: 10.1619/fesi.61.135. [27] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003. [28] S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. [29] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [30] J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190.  doi: 10.1007/BF01212446. [31] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517. [32] L. Véron, Comportement asymptotique des solutions d'équations elliptiques semi-linéaires dans ${\mathbb{R}^n}$, Ann. Mat. Pura Appl., 127 (1981), 25-50.  doi: 10.1007/BF01811717. [33] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103. [34] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. [35] J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443.  doi: 10.1080/03605300500299539. [36] J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.  doi: 10.1023/A:1026437923987.
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