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July  2018, 17(4): 1671-1680. doi: 10.3934/cpaa.2018080

Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement

 Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  June 2017 Revised  December 2017 Published  April 2018

Fund Project: The author is supported by JSPS KAKENHI Grant Number 15K04968.

We study the instability of standing wave solutions for nonlinear Schrödinger equations with a one-dimensional harmonic potential in dimension $N≥2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension $N-1$, then any ground states are strongly unstable by blowup.

Citation: Masahito Ohta. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1671-1680. doi: 10.3934/cpaa.2018080
References:
 [1] P. Antonelli, R. Carles and J. Drumond Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.   Google Scholar [2] J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.   Google Scholar [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.   Google Scholar [4] H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.   Google Scholar [5] R. Carles and C. Gallo, Scattering for the nonlinear Schr¨odinger equation with a general one-dimensional confinement, J. Math. Phys., 56 (2015), 101503, 15 pp. Google Scholar [6] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes in Math., 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. Google Scholar [7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.   Google Scholar [8] R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearity and potentials, Adv. Differential Equations, 10 (2005), 259-276.   Google Scholar [9] R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 111-128.   Google Scholar [10] R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 691-706.   Google Scholar [11] M. Hirose and M. Ohta, Structure of positive radial solutions to scalar field equations with harmonic potential, J. Differential Equations, 178 (2002), 519-540.   Google Scholar [12] M. Hirose and M. Ohta, Uniqueness of positive solutions to scalar field equations with harmonic potential, Funkcial. Ekvac., 50 (2007), 67-100.   Google Scholar [13] M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p = 0$ in $\mathbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.   Google Scholar [14] S. Le Coz, A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud., 8 (2008), 455-463.   Google Scholar [15] E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.   Google Scholar [16] Y. Martel, Blow-up for the nonlinear Schrödinger equation in nonisotropic spaces, Nonlinear Anal., 28 (1997), 1903-1908.   Google Scholar [17] M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential, Funkcial. Ekvac., 61 (2018), 135-143.   Google Scholar [18] M. Ohta and T. Yamaguchi, Strong instability of standing waves for nonlinear Schrödinger equations with double power nonlinearity, SUT J. Math., 51 (2015), 49-58.   Google Scholar [19] M. Ohta and T. Yamaguchi, Strong instability of standing waves for nonlinear Schrödinger equations with a delta potential, in Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu, Kyoto University, B56 (2016), 79-92. Google Scholar [20] S. Terracini, N. Tzvetkov and N. Visciglia, The nonlinear Schrödinger equation ground states on product spaces, Anal. PDE, 7 (2014), 73-96.   Google Scholar [21] J. Zhang, Cross-constrained variational problem and nonlinear Schrödinger equation, in Foundations of Computational Mathematics, World Scientific Publishing, River Edge, NJ, (2002), 457-469. Google Scholar

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References:
 [1] P. Antonelli, R. Carles and J. Drumond Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.   Google Scholar [2] J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.   Google Scholar [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.   Google Scholar [4] H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.   Google Scholar [5] R. Carles and C. Gallo, Scattering for the nonlinear Schr¨odinger equation with a general one-dimensional confinement, J. Math. Phys., 56 (2015), 101503, 15 pp. Google Scholar [6] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes in Math., 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. Google Scholar [7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.   Google Scholar [8] R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearity and potentials, Adv. Differential Equations, 10 (2005), 259-276.   Google Scholar [9] R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 111-128.   Google Scholar [10] R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 691-706.   Google Scholar [11] M. Hirose and M. Ohta, Structure of positive radial solutions to scalar field equations with harmonic potential, J. Differential Equations, 178 (2002), 519-540.   Google Scholar [12] M. Hirose and M. Ohta, Uniqueness of positive solutions to scalar field equations with harmonic potential, Funkcial. Ekvac., 50 (2007), 67-100.   Google Scholar [13] M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p = 0$ in $\mathbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.   Google Scholar [14] S. Le Coz, A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud., 8 (2008), 455-463.   Google Scholar [15] E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.   Google Scholar [16] Y. Martel, Blow-up for the nonlinear Schrödinger equation in nonisotropic spaces, Nonlinear Anal., 28 (1997), 1903-1908.   Google Scholar [17] M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential, Funkcial. Ekvac., 61 (2018), 135-143.   Google Scholar [18] M. Ohta and T. Yamaguchi, Strong instability of standing waves for nonlinear Schrödinger equations with double power nonlinearity, SUT J. Math., 51 (2015), 49-58.   Google Scholar [19] M. Ohta and T. Yamaguchi, Strong instability of standing waves for nonlinear Schrödinger equations with a delta potential, in Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu, Kyoto University, B56 (2016), 79-92. Google Scholar [20] S. Terracini, N. Tzvetkov and N. Visciglia, The nonlinear Schrödinger equation ground states on product spaces, Anal. PDE, 7 (2014), 73-96.   Google Scholar [21] J. Zhang, Cross-constrained variational problem and nonlinear Schrödinger equation, in Foundations of Computational Mathematics, World Scientific Publishing, River Edge, NJ, (2002), 457-469. Google Scholar
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