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Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential

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  • In the paper we prove the multiplicity existence of both nonlinear Schrödinger equation and Schrödinger system with slow decaying rate of electric potential at infinity. Namely, for any $\mathit{\boldsymbol{m}},\mathit{\boldsymbol{n > }}{\bf{0}}$ , the potentials $P, Q$ have the asymptotic behavior

    $\left\{ \begin{array}{l}P(r) = 1 + \frac{a}{{{r^m}}} + O\left( {\frac{1}{{{r^{m + \theta }}}}} \right),{\rm{ }}\;\;\;\;\theta > 0\\Q(r) = 1 + \frac{b}{{{r^n}}} + O\left( {\frac{1}{{{r^{n + \widetilde \theta }}}}} \right),\;\;\;\;\;\widetilde \theta > 0\end{array} \right.$

    then Schrödinger equation and Schrödinger system have infinitely many solutions with arbitrarily large energy, which extends the results of [37] for single Schrödinger equation and [30] for Schrödinger system.

    Mathematics Subject Classification: Primary:35B40, 35B45;Secondary:35J40.


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  • [1] A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.
    [2] A. AmbrosettiA. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅰ, Commun. Math. Phys., 235 (2003), 427-466.  doi: 10.1007/s00220-003-0811-y.
    [3] W. W. AoL. Wang and W. Yao, Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials, Comm. Pure Applied Anal., 15 (2016), 965-989.  doi: 10.3934/cpaa.2016.15.965.
    [4] W. W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Cal Var. PDE, 51 (2014), 761-798.  doi: 10.1007/s00526-013-0694-5.
    [5] T. BartschN. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Diff. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.
    [6] H. BerestyckiT.-C. LinJ. Wei and C. Y. Zhao, On Phase-Separation Models: Asymptotics and Qualitative Properties, Arch. Ration. Mech. Anal., 208 (2013), 163-200.  doi: 10.1007/s00205-012-0595-3.
    [7] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.
    [8] H. Berestycki and J. Wei, On least energy solutions to a semilinear elliptic equation in a strip, Disc. Cont. Dyn. Syst., 28 (2010), 1083-1099.  doi: 10.3934/dcds.2010.28.1083.
    [9] J. Y. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc. 229 (2014), ⅷ+89 pp. ISBN: 978-0-8218-9163-6.
    [10] D. M. CaoE. S. Noussair and S. S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 73-111.  doi: 10.1016/S0294-1449(99)80021-3.
    [11] M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.  doi: 10.1007/s002080200327.
    [12] M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.
    [13] M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146. 
    [14] 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.
    [15] E. N. DancerJ. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst.H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009.
    [16] M. del PinoJ. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Cal.Var. PDE, 53 (2015), 473-523.  doi: 10.1007/s00526-014-0756-3.
    [17] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.
    [18] B. GidasW.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, In Mathematical analysis and applications, Part A. Advances in Mathematical Supplementary Studies, Vol.7A (1981), 369-402. 
    [19] M. K. Kwong, Uniqueness of positive solutions of $-Δ u + u =u^p$ in $\mathbb{R}^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.
    [20] X. S. Kang and J. Wei, On interacting spikes of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928. 
    [21] T.-C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004.
    [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. 
    [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), 223-283. 
    [24] F. MahmoudiA. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve, Commun. Pure Appl. Math., 62 (2009), 1155-1264.  doi: 10.1002/cpa.20290.
    [25] M. MussoF. Pacard and J. Wei, Finite energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953.  doi: 10.4171/JEMS/351.
    [26] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.
    [27] B. NorisH. TavaresS. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.  doi: 10.1002/cpa.20309.
    [28] E. S. Noussair and S. S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.
    [29] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation Oxford, 2003.
    [30] S. J. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal., 208 (2013), 305-339.  doi: 10.1007/s00205-012-0598-0.
    [31] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.
    [32] S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condenstates, Arch. Rational Mech. Anal., 194 (2009), 717-741.  doi: 10.1007/s00205-008-0172-y.
    [33] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.
    [34] L. WangJ. Wei and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture for general hypersurfaces, Comm. Partial Diff. Equ., 36 (2011), 2117-2161.  doi: 10.1080/03605302.2011.580033.
    [35] L. WangJ. Wei and S. S. Yan, A Neumann problem with critical exponent in non-convex domains and Lin-Ni's conjecture, Transcation of Amer. Math. Soc., 362 (2010), 4581-4615.  doi: 10.1090/S0002-9947-10-04955-X.
    [36] J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.
    [37] J. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N$, Calc. Var. PDE, 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.
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