March  2017, 37(3): 1707-1731. doi: 10.3934/dcds.2017071

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

Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

Received  June 2015 Revised  October 2016 Published  December 2016

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.
Citation: Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
References:
[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.

show all references

References:
[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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