Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one. (English) Zbl 1458.81017
Summary: We investigate the problem of existence and uniqueness of ground states at fixed mass for two families of focusing nonlinear Schrödinger equations on the line. The first family consists of NLS with power nonlinearities concentrated at a point. For such model, we prove existence and uniqueness of ground states at every mass when the nonlinearity power is \(L^2\)-subcritical and at a threshold value of the mass in the \(L^2\)-critical regime. The second family is obtained by adding a standard power nonlinearity to the previous setting. In this case, we prove existence and uniqueness at every mass in the doubly subcritical case, namely when both the powers related to the pointwise and the standard nonlinearity are subcritical. If only one power is critical, then existence and uniqueness hold only at masses lower than the critical mass associated to the critical nonlinearity. Finally, in the doubly critical case ground states exist only at critical mass, whose value results from a non-trivial interplay between the two nonlinearities.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 35G55 | Initial value problems for systems of nonlinear higher-order PDEs |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral 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 |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
Keywords:
nonlinear Schrödinger; pointwise nonlinearity; fixed mass ground states; threshold phenomena; minimizationReferences:
| [1] | Adami, R.; Carlone, R.; Correggi, M.; Tentarelli, L., Blow up for the pointwise NLS in dimension two: absence of critical power, J. Differ. Equ., 269, 1, 1-37 (2020) · Zbl 1436.35051 |
| [2] | Adami, R.; Carlone, R.; Correggi, M.; Tentarelli, L., Stability of the standing waves of the concentrated NLSE in dimension two, Math. Eng., 3, 2, 1-15 (2021) · Zbl 1494.35143 |
| [3] | Adami, R.; Dell’Antonio, G.; Figari, R.; Teta, A., The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 20, 3, 477-500 (2003) · Zbl 1028.35137 |
| [4] | Adami, R.; Dell’Antonio, G.; Figari, R.; Teta, A., Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21, 1, 121-137 (2004) · Zbl 1042.35070 |
| [5] | Adami, R.; Teta, A., A simple model of concentrated nonlinearity, Oper. Theory, Adv. Appl., 108, 183-189 (1999) · Zbl 0967.81010 |
| [6] | Adami, R.; Teta, A., A class of nonlinear Schrödinger equations with concentrated nonlinearity, J. Funct. Anal., 180, 1, 148-175 (2001) · Zbl 0979.35130 |
| [7] | Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H., Solvable Models in Quantum Mechanics (2005), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI, xiv+488 pp · Zbl 1078.81003 |
| [8] | Banica, V.; Visciglia, N., Scattering for NLS with a delta potential, J. Differ. Equ., 260, 5, 4410-4439 (2016) · Zbl 1342.35200 |
| [9] | Berestycki, H.; Lions, P. L., Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029 |
| [10] | Borrelli, W.; Carlone, R.; Tentarelli, L., An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity, Symmetry, 11, 2, Article 169 pp. (2019) · Zbl 1416.35287 |
| [11] | Borrelli, W.; Carlone, R.; Tentarelli, L., Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit, SIAM J. Math. Anal., 51, 2, 1046-1081 (2019) · Zbl 1411.35263 |
| [12] | Borrelli, W.; Carlone, R.; Tentarelli, L., On the nonlinear Dirac equation on noncompact metric graphs (2019) |
| [13] | Bulashenko, O. M.; Kochelap, V. A.; Bonilla, L. L., Coherent patterns and self-induced diffraction of electrons on a thin nonlinear layer, Phys. Rev. B, 54, 3, 1537 (1996) |
| [14] | Cacciapuoti, C.; Carlone, R.; Noja, D.; Posilicano, A., The 1-D Dirac equation with concentrated nonlinearity, SIAM J. Math. Anal., 49, 3, 2246-2268 (2017) · Zbl 1371.35236 |
| [15] | Cacciapuoti, C.; Finco, D.; Noja, D.; Teta, A., The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit, Lett. Math. Phys., 104, 1557-1570 (2014) · Zbl 1303.81072 |
| [16] | Cacciapuoti, C.; Finco, D.; Noja, D.; Teta, A., The point-like limit for a NLS equation with concentrated nonlinearity in dimension three, J. Funct. Anal., 273, 5, 1762-1809 (2017) · Zbl 1378.35274 |
| [17] | Cardanobile, S.; Mugnolo, D., Analysis of a FitzHugh-Nagumo-Rall model of a neuronal network, Math. Methods Appl. Sci., 30, 18, 2281-2308 (2007) · Zbl 1195.92007 |
| [18] | Carlone, R.; Correggi, M.; Tentarelli, L., Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 36, 1, 257-294 (2019) · Zbl 1410.35196 |
| [19] | Carlone, R.; Figari, R.; Negulescu, C., The quantum beating and its numerical simulation, J. Math. Anal. Appl., 450, 2, 1294-1316 (2017) · Zbl 1376.81077 |
| [20] | Carlone, R.; Finco, D.; Tentarelli, L., Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one, Nonlinearity, 32, 8, 3112-3143 (2019) · Zbl 1421.35299 |
| [21] | Carlone, R.; Fiorenza, A.; Tentarelli, L., The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions, J. Funct. Anal., 273, 3, 1258-1294 (2017) · Zbl 1530.47060 |
| [22] | Cazenave, T., Semilinear Schrödinger Equations, Courant Lecture Notes, vol. 10 (2003), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1055.35003 |
| [23] | Cazenave, T.; Lions, P. L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85, 4, 549-561 (1982) · Zbl 0513.35007 |
| [24] | Coclite, G. M.; Holden, H., The Schrödinger-Maxwell system with Dirac mass, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 24, 5, 773-793 (2007) · Zbl 1132.35024 |
| [25] | Coclite, G. M.; Holden, H., Erratum to: The Schrödinger-Maxwell system with Dirac mass, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 25, 4, 833-836 (2008) · Zbl 1157.35334 |
| [26] | Coclite, G. M.; Holden, H., Ground states of the Schrödinger-Maxwell system with Dirac mass: existence and asymptotics, Discrete Contin. Dyn. Syst., 27, 1, 117-132 (2010) · Zbl 1190.35189 |
| [27] | Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 3, 463-512 (1999) |
| [28] | Dolbeault, J.; Esteban, M. J.; Laptev, A.; Michael, L., One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows, J. Lond. Math. Soc., 90, 2, 525-550 (2014) · Zbl 1320.26017 |
| [29] | Dovetta, S.; Tentarelli, L., \( L^2\)-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features, Calc. Var. Partial Differ. Equ., 58, 3, Article 108 pp. (2019) · Zbl 1432.35212 |
| [30] | Dovetta, S.; Tentarelli, L., Ground states of the \(L^2\)-critical NLS equation with localized nonlinearity on a tadpole graph, Oper. Theory, Adv. Appl., 281, 113-125 (2020) · Zbl 1473.35503 |
| [31] | Gnutzman, S.; Smilansky, U.; Derevyanko, S., Stationary scattering from a nonlinear network, Phys. Rev. A, 83, Article 033831 pp. (2001) |
| [32] | Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal., 74, 160-197 (1987) · Zbl 0656.35122 |
| [33] | Holmer, J.; Liu, C., Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: basic theory, J. Math. Anal. Appl., 483, 1, Article 123522 pp. (2020) · Zbl 1436.35290 |
| [34] | Holmer, J.; Liu, C., Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: supercritical blow-up profiles (2017) |
| [35] | Jona-Lasinio, G.; Presilla, C.; Sjöstrand, J., On Schrödinger equations with concentrated nonlinearities, Ann. Phys., 240, 1-21 (1995) · Zbl 0820.34050 |
| [36] | Lannes, D., The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, vol. 188 (2013), AMS · Zbl 1410.35003 |
| [37] | Malomed, B.; Azbel, B., Modulational instability of a wave scattered by a nonlinear center, Phys. Rev. B, 47, 16 (1993) |
| [38] | Molina, M. I.; Bustamante, C. A., The attractive nonlinear delta-function potential, Am. J. Phys., 70, 1, 67-70 (2002) |
| [39] | Nier, F., The dynamics of some quantum open system with short-range nonlinearities, Nonlinearity, 11, 1127 (1998) · Zbl 0909.34052 |
| [40] | Presilla, C.; Jona-Lasinio, G.; Capasso, F., Nonlinear feedback oscillations in resonant tunneling through double barriers, Phys. Rev. B, 43, 6, 5200-5203 (1991) |
| [41] | Pelinovsky, D.; Ponomarev, D., Justification of a nonlinear Schrödinger model for laser beams in photopolymers, Z. Angew. Math. Phys., 65, 3, 405-433 (2014) · Zbl 1296.35175 |
| [42] | Serra, E.; Tentarelli, L., Bound states of the NLS equation on metric graphs with localized nonlinearities, J. Differ. Equ., 260, 7, 5627-5644 (2016) · Zbl 1408.35206 |
| [43] | Serra, E.; Tentarelli, L., On the lack of bound states for certain NLS equations on metric graphs, Nonlinear Anal., 145, 68-82 (2016) · Zbl 1347.35226 |
| [44] | Sukhorukov, A. A.; Kivshar, Y. S.; Bang, O., Two-color nonlinear localized photonic modes, Phys. Rev. E, 60, 1, R41-R44 (1999) |
| [45] | Sukhorukov, A. A.; Kivshar, Y. S.; Bang, O.; Rasmussen, J. J.; Christiansen, P. L., Nonlinearity and disorder: classification and stability of nonlinear impurity modes, Phys. Rev. E, 63, 3-II, Article 036601 pp. (2001) |
| [46] | Tentarelli, L., NLS ground states on metric graphs with localized nonlinearities, J. Math. Anal. Appl., 433, 1, 291-304 (2016) · Zbl 1321.05278 |
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