# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242
##### References:
 [1] L. Abdelouhab, J. L. Bona, M. 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. [2] 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. [3] A. Ambrosetti, M. 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. [4] A. Ambrosetti, M. 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. [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. [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. [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. [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. [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. [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. [11] D. Cao, E. 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. [12] D. Cao, E. 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. [13] G. Cerami, D. 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. [14] G. Cerami, R. 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. [15] G. Cerami, D. 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. [16] G. Cerami, G. 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. [17] W. Chen, Soft matter and fractional mathematics: Insights into mesoscopic quantum and time-space structures, Preprint. http://arxiv.org/abs/1305.4426. [18] J. D$\acute{a}$vila, M. 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. [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. [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. [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. [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. [23] M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schródinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135. [24] M. Del Pino, J. 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. [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. [26] P. Felmer, A. 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. [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. [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. [29] R. Frank, E. 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. [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. [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. [32] I. Kra and S. R. Sinmanca, On circulant matrices, Notices AMS, 59 (2012), 368-377.  doi: 10.1090/noti804. [33] N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.  doi: 10.1142/10541. [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. [35] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. [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. [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. [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. [39] F. Mahmoudi, A. 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. [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. [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. [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. [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. [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. [45] L. P. Wang, J. 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. [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. [47] L. P. Wang and C. Y. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv: 1403.0042v1. [48] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642. [49] L. Wei, S. J. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schródinger equations, arXiv: 1402.1902v1. [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.

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##### References:
 [1] L. Abdelouhab, J. L. Bona, M. 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. [2] 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. [3] A. Ambrosetti, M. 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. [4] A. Ambrosetti, M. 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. [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. [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. [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. [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. [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. [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. [11] D. Cao, E. 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. [12] D. Cao, E. 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. [13] G. Cerami, D. 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. [14] G. Cerami, R. 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. [15] G. Cerami, D. 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. [16] G. Cerami, G. 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. [17] W. Chen, Soft matter and fractional mathematics: Insights into mesoscopic quantum and time-space structures, Preprint. http://arxiv.org/abs/1305.4426. [18] J. D$\acute{a}$vila, M. 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. [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. [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. [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. [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. [23] M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schródinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135. [24] M. Del Pino, J. 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. [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. [26] P. Felmer, A. 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. [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. [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. [29] R. Frank, E. 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. [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. [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. [32] I. Kra and S. R. Sinmanca, On circulant matrices, Notices AMS, 59 (2012), 368-377.  doi: 10.1090/noti804. [33] N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.  doi: 10.1142/10541. [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. [35] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. [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. [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. [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. [39] F. Mahmoudi, A. 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. [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. [41] E. S. Noussair and S. S. 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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. [47] L. P. Wang and C. Y. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv: 1403.0042v1. [48] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642. [49] L. Wei, S. J. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schródinger equations, arXiv: 1402.1902v1. [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.
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