November  2017, 37(11): 5561-5601. doi: 10.3934/dcds.2017242

Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

2. 

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

3. 

Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received  March 2017 Revised  June 2017 Published  July 2017

We consider the fractional nonlinear Schrödinger equation
\begin{equation*}(-Δ)^su+V(x)u=u^p \mbox{ in }\mathbb{R}^N, u→0~\mathrm{as}~|x|→+∞,\end{equation*}
where
$V(x)$
is a uniformly positive potential and $p>1.$ Assuming that
\begin{equation*}V(x)=V_{∞}+\frac{a}{|x|^m}+O\Big(\frac{1}{|x|^{m+σ}}\Big)~\mathrm{as}~|x|→+∞,\end{equation*}
and
$p,m,σ,s$
satisfy certain conditions, we prove the existence of infinitely many positive solutions for
$N=2$
. For
$s=1$
, this corresponds to the multiplicity result given by Del Pino, Wei, and Yao [24] for the classical nonlinear Schrödinger equation.
Citation: Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242
References:
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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

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A. AmbrosettiM. 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

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A. AmbrosettiM. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅱ, Indiana Univ. Math. J., 53 (2004), 297-329.  doi: 10.1512/iumj.2004.53.2400.  Google Scholar

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W. W. Ao and J. C. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differ. Equ., 51 (2014), 761-798.  doi: 10.1007/s00526-013-0694-5.  Google Scholar

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A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.  doi: 10.4171/RMI/92.  Google Scholar

[7]

A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linaire, 14 (1997), 365-413.  doi: 10.1016/S0294-1449(97)80142-4.  Google Scholar

[8]

X. Cabré and J. G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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D. Cao, Positive solution and bifurcation from the essential spectrum of a semilinear elliptic equation on $\mathbb{R}$, Nonlinear Anal., 15 (1990), 1045-1052.  doi: 10.1016/0362-546X(90)90152-7.  Google Scholar

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D. CaoE. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré Anal. Non Lineaire, 15 (1998), 73-111.  doi: 10.1016/S0294-1449(99)80021-3.  Google Scholar

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D. CaoE. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. R. Soc. Edinb. Sect. A, 129 (1999), 235-264.  doi: 10.1017/S030821050002134X.  Google Scholar

[13]

G. CeramiD. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.  Google Scholar

[14]

G. CeramiR. Molle and D. Passaseo, Multiplicity of positive and nodal solutions for scalar field equations, J. Differential Eqn., 257 (2014), 3554-3606.  doi: 10.1016/j.jde.2014.07.002.  Google Scholar

[15]

G. CeramiD. Passaseo and S. Solimini, Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps, Ann. Inst. H. Poincaré Anal. Non Lineaire, 32 (2015), 23-40.  doi: 10.1016/j.anihpc.2013.08.008.  Google Scholar

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G. CeramiG. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.  doi: 10.1007/s00526-004-0293-6.  Google Scholar

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W. Chen, Soft matter and fractional mathematics: Insights into mesoscopic quantum and time-space structures, Preprint. http://arxiv.org/abs/1305.4426. Google Scholar

[18]

J. D$\acute{a}$vilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

[19]

M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[20]

M. del Pino and P. Felmer, Multi-peak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. NonLineaire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[21]

M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.  doi: 10.1006/jfan.1996.3085.  Google Scholar

[22]

M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations: A variational reduction method, Math. Ann., 324 (2002), 1-32.  doi: 10.1007/s002080200327.  Google Scholar

[23]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schródinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.  Google Scholar

[24]

M. Del PinoJ. Wei and W. Yao, Intermediate reduction method 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

[25]

W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91 (1986), 283-308.  doi: 10.1007/BF00282336.  Google Scholar

[26]

P. FelmerA. Quaas and J. G. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[27]

A. Floer and M. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[28]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[29]

R. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure. Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[30]

D. Giulini, That strange procedure called quantisation, Quantum gravity, Lecture Notes in Phys., vol. 631, Springer, Berlin, 2003, 17-40. doi: 10.1007/978-3-540-45230-0_2.  Google Scholar

[31]

X. S. Kang and J. C. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Diff. Eqn., 5 (2000), 899-928.   Google Scholar

[32]

I. Kra and S. R. Sinmanca, On circulant matrices, Notices AMS, 59 (2012), 368-377.  doi: 10.1090/noti804.  Google Scholar

[33]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.  doi: 10.1142/10541.  Google Scholar

[34]

N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[35]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[36]

Y. Li and W. M. Ni, On conformal scalar curvature equations in $\mathbb{R}^n$, Duke Math, J., 57 (1988), 895-924.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

[37]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[38]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[39]

F. MahmoudiA. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve, Comm. Pure Appl. Math., 62 (2009), 1155-1264.  doi: 10.1002/cpa.20290.  Google Scholar

[40]

M. Musso and J. C. Wei, Nondegeneracy of nonlinear nodal solutions to Yamabe problem, Comm. Math. Phy., 340 (2015), 1049-1107.  doi: 10.1007/s00220-015-2462-1.  Google Scholar

[41]

E. S. Noussair and S. S. Yan, On positive multi-peak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.  Google Scholar

[42]

Y. J. Oh, On positive multi-lump bound states nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.  Google Scholar

[43]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.  Google Scholar

[44]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[45]

L. P. WangJ. C. Wei and S. S. Yan, A Neumann problem with critical exponent in non-convex domains and Lin-Ni's conjecture, Tran. American Math. Society, 362 (2010), 4581-4615.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[46]

L. P. Wang and C. Y. Zhao, Infinitely many solutions for the prescribed boundary mean curvature problem in $\mathbb{R}^N$, Canad. J. Math., 65 (2013), 927-960.  doi: 10.4153/CJM-2012-054-2.  Google Scholar

[47]

L. P. Wang and C. Y. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv: 1403.0042v1. Google Scholar

[48]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[49]

L. Wei, S. J. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schródinger equations, arXiv: 1402.1902v1. Google Scholar

[50]

J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

show all references

References:
[1]

L. AbdelouhabJ. L. BonaM. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360-392.  doi: 10.1016/0167-2789(89)90050-X.  Google Scholar

[2]

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

[3]

A. AmbrosettiM. 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

[4]

A. AmbrosettiM. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅱ, Indiana Univ. Math. J., 53 (2004), 297-329.  doi: 10.1512/iumj.2004.53.2400.  Google Scholar

[5]

W. W. Ao and J. C. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differ. Equ., 51 (2014), 761-798.  doi: 10.1007/s00526-013-0694-5.  Google Scholar

[6]

A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.  doi: 10.4171/RMI/92.  Google Scholar

[7]

A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linaire, 14 (1997), 365-413.  doi: 10.1016/S0294-1449(97)80142-4.  Google Scholar

[8]

X. Cabré and J. G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[10]

D. Cao, Positive solution and bifurcation from the essential spectrum of a semilinear elliptic equation on $\mathbb{R}$, Nonlinear Anal., 15 (1990), 1045-1052.  doi: 10.1016/0362-546X(90)90152-7.  Google Scholar

[11]

D. CaoE. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré Anal. Non Lineaire, 15 (1998), 73-111.  doi: 10.1016/S0294-1449(99)80021-3.  Google Scholar

[12]

D. CaoE. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. R. Soc. Edinb. Sect. A, 129 (1999), 235-264.  doi: 10.1017/S030821050002134X.  Google Scholar

[13]

G. CeramiD. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.  Google Scholar

[14]

G. CeramiR. Molle and D. Passaseo, Multiplicity of positive and nodal solutions for scalar field equations, J. Differential Eqn., 257 (2014), 3554-3606.  doi: 10.1016/j.jde.2014.07.002.  Google Scholar

[15]

G. CeramiD. Passaseo and S. Solimini, Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps, Ann. Inst. H. Poincaré Anal. Non Lineaire, 32 (2015), 23-40.  doi: 10.1016/j.anihpc.2013.08.008.  Google Scholar

[16]

G. CeramiG. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.  doi: 10.1007/s00526-004-0293-6.  Google Scholar

[17]

W. Chen, Soft matter and fractional mathematics: Insights into mesoscopic quantum and time-space structures, Preprint. http://arxiv.org/abs/1305.4426. Google Scholar

[18]

J. D$\acute{a}$vilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

[19]

M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[20]

M. del Pino and P. Felmer, Multi-peak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. NonLineaire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[21]

M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.  doi: 10.1006/jfan.1996.3085.  Google Scholar

[22]

M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations: A variational reduction method, Math. Ann., 324 (2002), 1-32.  doi: 10.1007/s002080200327.  Google Scholar

[23]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schródinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.  Google Scholar

[24]

M. Del PinoJ. Wei and W. Yao, Intermediate reduction method 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

[25]

W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91 (1986), 283-308.  doi: 10.1007/BF00282336.  Google Scholar

[26]

P. FelmerA. Quaas and J. G. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[27]

A. Floer and M. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[28]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[29]

R. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure. Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[30]

D. Giulini, That strange procedure called quantisation, Quantum gravity, Lecture Notes in Phys., vol. 631, Springer, Berlin, 2003, 17-40. doi: 10.1007/978-3-540-45230-0_2.  Google Scholar

[31]

X. S. Kang and J. C. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Diff. Eqn., 5 (2000), 899-928.   Google Scholar

[32]

I. Kra and S. R. Sinmanca, On circulant matrices, Notices AMS, 59 (2012), 368-377.  doi: 10.1090/noti804.  Google Scholar

[33]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.  doi: 10.1142/10541.  Google Scholar

[34]

N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[35]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[36]

Y. Li and W. M. Ni, On conformal scalar curvature equations in $\mathbb{R}^n$, Duke Math, J., 57 (1988), 895-924.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

[37]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[38]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[39]

F. MahmoudiA. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve, Comm. Pure Appl. Math., 62 (2009), 1155-1264.  doi: 10.1002/cpa.20290.  Google Scholar

[40]

M. Musso and J. C. Wei, Nondegeneracy of nonlinear nodal solutions to Yamabe problem, Comm. Math. Phy., 340 (2015), 1049-1107.  doi: 10.1007/s00220-015-2462-1.  Google Scholar

[41]

E. S. Noussair and S. S. Yan, On positive multi-peak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.  Google Scholar

[42]

Y. J. Oh, On positive multi-lump bound states nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.  Google Scholar

[43]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.  Google Scholar

[44]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[45]

L. P. WangJ. C. Wei and S. S. Yan, A Neumann problem with critical exponent in non-convex domains and Lin-Ni's conjecture, Tran. American Math. Society, 362 (2010), 4581-4615.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[46]

L. P. Wang and C. Y. Zhao, Infinitely many solutions for the prescribed boundary mean curvature problem in $\mathbb{R}^N$, Canad. J. Math., 65 (2013), 927-960.  doi: 10.4153/CJM-2012-054-2.  Google Scholar

[47]

L. P. Wang and C. Y. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv: 1403.0042v1. Google Scholar

[48]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[49]

L. Wei, S. J. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schródinger equations, arXiv: 1402.1902v1. Google Scholar

[50]

J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

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