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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation Oxford, 2003. Google Scholar

[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. Google Scholar

[31]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation Oxford, 2003. Google Scholar

[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. Google Scholar

[31]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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