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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 |
\begin{equation*}(-Δ)^su+V(x)u=u^p \mbox{ in }\mathbb{R}^N, u→0~\mathrm{as}~|x|→+∞,\end{equation*} |
$V(x)$ |
\begin{equation*}V(x)=V_{∞}+\frac{a}{|x|^m}+O\Big(\frac{1}{|x|^{m+σ}}\Big)~\mathrm{as}~|x|→+∞,\end{equation*} |
$p,m,σ,s$ |
$N=2$ |
$s=1$ |
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. |
show all references
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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