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

Liping Wang; Chunyi Zhao

doi:10.3934/dcds.2017071

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2017, Volume 37, Issue 3: 1707-1731. Doi: 10.3934/dcds.2017071

This issue Previous Article A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition Next Article Conserved quantities, global existence and blow-up for a generalized CH equation

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

Liping Wang^1, and
Chunyi Zhao^1,

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

Received: May 31, 2015

Revised: September 30, 2016

Published: May 2017

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Abstract

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.

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

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[1]	A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.
[2]	A. Ambrosetti, A. 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. Ao, L. 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. Bartsch, N. 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. Berestycki, T.-C. Lin, J. 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. Cao, E. 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 Pino, M. 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. Dancer, J. 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 Pino, J. 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. Gidas, W.-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. Mahmoudi, A. 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. Musso, F. 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. Noris, H. Tavares, S. 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. Wang, J. 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. Wang, J. 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.