## Existence and stability of standing waves for supercritical NLS with a partial confinement.(English)Zbl 1367.35150

Summary: We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $$L^{2}$$-supercritical; in particular, we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs
Full Text:

### References:

 [1] Anderson, M.H.; Ensher, J.R.; Matthews, M.R.; Wieman, C.E.; Cornell, E.A., Bose-Einstein condensation in a dilute atomic vapor, Science, 269, 14, (1995) [2] Antonelli, P.; Carles, R.; Drumond Silva, J., Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Commun. Math. Phys., 334, 367-396, (2015) · Zbl 1309.35124 [3] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1-135, (2013) · Zbl 1266.82009 [4] Bao, W.; Jaksch, D.; Markowich, P.A., Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187, 318-342, (2003) · Zbl 1028.82501 [5] Bellazzini, J.; Jeanjean, L., On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48, 2028-2058, (2016) · Zbl 1352.35157 [6] Bellazzini, J.; Jeanjean, L.; Luo, T., Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc. (3), 107, 303-339, (2013) · Zbl 1284.35391 [7] Benci, V.; Visciglia, N., Solitary waves with non-vanishing angular momentum, Adv. Nonlinear Stud., 3, 151-160, (2003) · Zbl 1030.35051 [8] Berestycki, H.; Cazenave, T., Instabilité des états stationnaires dans LES équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293, 489-492, (1981) · Zbl 0492.35010 [9] Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88, 486-490, (1983) · Zbl 0526.46037 [10] Brock, F.; Solynin, A.Y., An approach to symmetrization via polarization, Trans. Am. Math. Soc., 352, 1759-1796, (2000) · Zbl 0965.49001 [11] Brothers, J.E.; Ziemer, W.P., Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384, 153-179, (1988) · Zbl 0633.46030 [12] Byeon, J.; Jeanjean, L.; Mariş, M., Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differ. Equ., 36, 481-492, (2009) · Zbl 1226.35041 [13] Carles, R., Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci., 12, 1513-1523, (2002) · Zbl 1029.35208 [14] Cazenave, T.: Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003) · Zbl 1226.35041 [15] Cazenave, T.; Lions, P.-L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85, 549-561, (1982) · Zbl 0513.35007 [16] Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 463, (1999) [17] Erdős, L.; Schlein, B.; Yau, H.-T., Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Commun. Pure Appl. Math., 59, 1659-1741, (2006) · Zbl 1122.82018 [18] Erdős, L.; Schlein, B.; Yau, H.-T., Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Am. Math. Soc., 22, 1099-1156, (2009) · Zbl 1207.82031 [19] Fukuizumi, R., Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential, Discrete Contin. Dyn. Syst., 7, 525-544, (2001) · Zbl 0992.35094 [20] Fukuizumi, R.; Hadj Selem, F.; Kikuchi, H., Stationary problem related to the nonlinear Schrödinger equation on the unit ball, Nonlinearity, 25, 2271-2301, (2012) · Zbl 1254.35207 [21] Fukuizumi, R.; Ohta, M., Instability of standing waves for nonlinear Schrödinger equations with potentials, Differ. Integral Equ., 16, 691-706, (2003) · Zbl 1031.35131 [22] Fukuizumi, R.; Ohta, M., Stability of standing waves for nonlinear Schrödinger equations with potentials, Differ. Integral Equ., 16, 111-128, (2003) · Zbl 1031.35132 [23] Griffin A., Snoke D., Stringari S.: Bose-Einstein Condensation. Cambridge University Press, Cambridge (1996) [24] Hajaiej, H.; Stuart, C.A., Symmetrization inequalities for composition operators of Carathéodory type, Proc. Lond. Math. Soc. (3), 87, 396-418, (2003) · Zbl 1052.26020 [25] Hajaiej, H.; Stuart, C.A., On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation, Adv. Nonlinear Stud., 4, 469-501, (2004) · Zbl 1068.35155 [26] Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 1633-1659, (1997) · Zbl 0877.35091 [27] Kavian O.: Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, vol. 13 of Mathématiques and Applications (Berlin) [Mathematics and Applications]. Springer, Paris (1993) · Zbl 0797.58005 [28] Kenig, C.E.: Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pp. 948-960. Am. Math. Soc., Providence, RI (1987) · Zbl 1030.35051 [29] Li, Y.; Ni, W.-M., Radial symmetry of positive solutions of nonlinear elliptic equations in R\^{}{$$n$$}, Commun. Partial Differ. Equ., 18, 1043-1054, (1993) · Zbl 0788.35042 [30] Lieb E., Loss M.: Analysis, Volume 14 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2001) [31] Lieb, E.; Seiringer, R., Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett., 88, 170409, (2002) [32] Lieb E., Seiringer R., Solovej J.P., Yngvason J.: The Mathematics of the Bose Gas and Its Condensation, Volume 34 of Oberwolfach Seminars. Birkhäuser Verlag, Basel (2005) · Zbl 1104.82012 [33] Lopes, O., Radial symmetry of minimizers for some translation and rotation invariant functionals, J. Differ. Equ., 124, 378-388, (1996) · Zbl 0842.49004 [34] Noris, B.; Tavares, H.; Verzini, G., Existence and orbital stability of the ground states with prescribed mass for the $$L$$\^{}{2}-critical and supercritical NLS on bounded domains, Anal. PDE, 7, 1807-1838, (2014) · Zbl 1314.35168 [35] Rousset, F.; Tzvetkov, N., Stability and instability of the KdV solitary wave under the KP-I flow, Commun. Math. Phys., 313, 155-173, (2012) · Zbl 1252.35052 [36] Stuart, C.A.: Private Communication. (2016) · Zbl 1314.35168 [37] Terracini, S.; Tzvetkov, N.; Visciglia, N., The nonlinear Schrödinger equation ground states on product spaces, Anal. PDE, 7, 73-96, (2014) · Zbl 1294.35148 [38] Weinstein, M.I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., 39, 51-67, (1986) · Zbl 0594.35005 [39] Willem M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston (1996) · Zbl 0856.49001 [40] Zhang, J., Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51, 498-503, (2000) · Zbl 0985.35085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.