Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-24T12:16:21.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials

Part of: Existence theories Miscellaneous topics - Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  27 December 2018

Miao Du ,
Lixin Tian ,
Jun Wang  and
Fubao Zhang
Miao Du
Affiliation:
School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, P.R. China (dumiaomath@163.com)
Lixin Tian
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, P.R. China (tianlx@ujs.edu.cn)
Jun Wang
Affiliation:
Institute of Applied System Analysis, Jiangsu University, Zhenjiang, 212013, P.R. China (wangmath2011@126.com)
Fubao Zhang
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, P.R. China (zhangfubao@seu.edu.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.

Keywords

Fractional Schrödinger equationnormalized solutionsmass concentration

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35R11: Fractional partial differential equations 49J40: Variational methods including variational inequalities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 617 - 653
Copyright
Copyright © Royal Society of Edinburgh 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Cabré, X. and Sire, Y.. Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.Google Scholar
2Caffarelli, L. and Silvestre, L.. An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32 (2007), 1245–1260.Google Scholar
3Chang, X. and Wang, Z. Q.. Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J. Differ. Equ. 256 (2014), 2965–2992.Google Scholar
4Chen, G. Y. and Zheng, Y. Q.. Concentration phenomenon for fractional nonlinear Schrödinger equations. Commun. Pure Appl. Anal. 13 (2014), 2359–2376.Google Scholar
5Cheng, M.. Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53 (2012), 043507.Google Scholar
6Cont, R. and Tankov, P.. Financial modelling with jump processes (Boca Raton, FL: Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC press, 2004).Google Scholar
7Coti Zelati, V. and Nolasco, M.. Existence of ground states for nonlinear pseudo relativistic Schrödinger equations. Rend. Lincei Mat. Appl. 22 (2011), 51–72.Google Scholar
8Crandall, M. G. and Rabinowitz, P. H.. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52 (1973), 161–180.Google Scholar
9Dalfovo, F., Giorgini, S., Pitaevskii, L. P. and Stringari, S.. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71 (1999), 463–512.Google Scholar
10Dávila, J., Del Pino, M. and Wei, J.. Concentration phenomenon for fractional nonlinear Schrödinger equations. J. Differ. Equ. 256 (2014), 858–892.Google Scholar
11del Pino, M. and Felmer, P.. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4 (1996), 121–137.Google Scholar
12Di Nezza, E., Palatucci, G. and Valdinoci, E.. Hitchhiker's guide to the fractional sobolev spaces. Bull. des Sci. Math. 136 (2012), 521–573.Google Scholar
13Dipierro, S., Palatucci, G. and Valdinoci, E.. Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68 (2013), 201–216.Google Scholar
14Fall, M. M. and Valdinoci, E.. Uniqueness and nondegeneracy of positive solutions of (−Δ)s u + u = u p in ℝN when s is close to 1. Commun. Math. Phys. 329 (2014), 383–404.Google Scholar
15Felmer, P., Quaas, A. and Tan, J.. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.Google Scholar
16Foler, A. and Weinstein, A.. Nonspreading wave packets for the cubic Schrödinger equations. J. Funct. Anal. 69 (1986), 397–408.Google Scholar
17Frank, R. and Lenzmann, E.. Uniqueness and nondegeneracy of ground states for (−Δ)s Q + Q − Q α + = 0 in ℝ. Acta Math. 210 (2013), 261–318.Google Scholar
18Frank, R., Lenzmann, E. and Silvestre, L.. Uniqueness of radial solutions for the fractional Laplacian. Communications on Pure and Applied Mathematics 69 (2016), 1671–1726.Google Scholar
19Guo, Y. J. and Seiringer, R.. on the mass concentration for Bose-Einstein condensation with attractive interactions. Lett. Math. Phys. 104 (2014), 141–156.Google Scholar
20Guo, Y. J., Wang, Z. Q., Zeng, X. Y. and Zhou, H. S.. Properties for ground states of attractive Gross-Pitaevskii equations with multi-well potentials, e-print arXiv: 1502.01839.Google Scholar
21Guo, Y. J., Zeng, X. Y. and Zhou, H. S.. Energy estimates and symmetry breaking in attractive Bose-Einstein condensation with ring-shaped potential. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 809–828.Google Scholar
22Jackson, R. K. and Weinstein, M. I.. Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation. J. Stat. Phys. 116 (2004), 881–905.Google Scholar
23Kirr, E. W., Kevrekidis, P. G., Shlizerman, E. and Weinstein, M. I.. Symmetry-breaking bifurcation in the nonlinear Schrödinger/Gross-Pitaevskii equations. SIAM J. Math. Anal. 40 (2008), 566–604.Google Scholar
24Kirr, E. W., Kevrekidis, P. G. and Pelinovsky, D. E.. Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials. Comm. Math. Phys. 308 (2011), 795–844.Google Scholar
25Laskin, N.. Fractional quantum mechanics and lévy path integrals. Phys. Lett. A 268 (2000), 298–305.Google Scholar
26Laskin, N.. Fractional Schrödinger equation. Phys. Rev. E 66 (2002), 56–108.Google Scholar
27Lieb, E. H. and Loss, M.. Analysis, graduate studies in mathematics,vol. 14, 2nd edn (Providence: American Mathematical Society, 2001).Google Scholar
28Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case, part I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145.Google Scholar
29Oh, Y. G.. On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131 (1990), 223–253.Google Scholar
30Pitaevskii, L. P.. Vortex lines in an imperfect Bose gas. Sov. Phys, JETP 13 (1961), 451–454.Google Scholar
31Rabinowitz, P. H.. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992), 270–291.Google Scholar
32Secchi, S.. Ground state solutions for nonlinear fractional Schrödinger equations in ℝN. J. Math. Phys. 54 (2013), 031501.Google Scholar
33Secchi, S.. On fractional Schrödinger equations in ℝN without the Ambrosetti-Rabinowitz condition, e-print arXiv:1210.0755.Google Scholar
34Servadei, R. and Valdinoci, E.. The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367 (2015), 67–102.Google Scholar
35Shang, X. and Zhang, J.. Concentrating solutions of nonlinear fractional Schrödinger equation with potentials. J. Differ. Equ. 258 (2015), 1106–1128.Google Scholar
36Silvestre, L.. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60 (2007), 67–112.Google Scholar
37Tan, J.. The Brezis–Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 42 (2011), 21–41.Google Scholar
38Wang, X.. On concentration of positive bound states of nonlinear Schrödinger equations. comm. Math. Phys. 153 (1993), 229–244.Google Scholar
39Weinstein, M. I.. Nonlinear Schrödinger equations and Sharp interpolation estimates. comm. Math. Phys. 87 (1983), 567–576.Google Scholar
40Willem, M.. Minimax theorems (Boston: Birkhäuser, 1996).Google Scholar