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 [
Citation: |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
W. Chen, Soft matter and fractional mathematics: Insights into mesoscopic quantum and time-space structures, Preprint. http://arxiv.org/abs/1305.4426.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
I. Kra
and S. R. Sinmanca
, On circulant matrices, Notices AMS, 59 (2012)
, 368-377.
doi: 10.1090/noti804.![]() ![]() ![]() |
|
N. Laskin
, Fractional quantum mechanics, Phys. Rev. E, 62 (2000)
, 31-35.
doi: 10.1142/10541.![]() ![]() |
|
N. Laskin
, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000)
, 298-305.
doi: 10.1016/S0375-9601(00)00201-2.![]() ![]() ![]() |
|
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
L. P. Wang and C. Y. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv: 1403.0042v1.
![]() |
|
X. Wang
, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993)
, 229-244.
doi: 10.1007/BF02096642.![]() ![]() ![]() |
|
L. Wei, S. J. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schródinger equations, arXiv: 1402.1902v1.
![]() |
|
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.![]() ![]() ![]() |