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Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term
1. | Center for Applied Mathematics, Tianjin University, Tianjin 300072, China |
2. | Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China |
In this paper we are concerned with the fractional Schrödinger equation $ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $, $ x\in {{\mathbb{R}}^{N}} $, where $ f $ is superlinear, subcritical growth and $ u\mapsto\frac{f(x, u)}{\vert u\vert} $ is nondecreasing. When $ V $ and $ f $ are periodic in $ x_{1},\ldots, x_{N} $, we show the existence of ground states and the infinitely many solutions if $ f $ is odd in $ u $. When $ V $ is coercive or $ V $ has a bounded potential well and $ f(x, u) = f(u) $, the ground states are obtained. When $ V $ and $ f $ are asymptotically periodic in $ x $, we also obtain the ground states solutions. In the previous research, $ u\mapsto\frac{f(x, u)}{\vert u\vert} $ was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.
References:
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G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in ${{\mathbb{R}}^{N}}$, J. Differential Equations, 255 (2013), 2340-2362.
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L. Caffarelli and L. Silvestre,
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K. C. Chang,
Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.
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[5] |
X. J. Chang and Z. Q. Wang,
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.
doi: 10.1088/0951-7715/26/2/479. |
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R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman Hall/CRC Financial Mathematics Series, Boca Raton, 2004. |
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E. Di Nezza, G. Palatucci and E. Valdinaci,
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Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
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On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.
doi: 10.1007/s00526-011-0447-2. |
[11] |
R. Metzler and J. Klafter,
The random walls guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[12] |
G. Molica Bisci and V. Rădulescu,
Ground state solutions of scalar field fractional for Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.
doi: 10.1007/s00526-015-0891-5. |
[13] |
G. Molica Bisci, V. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316282397.![]() ![]() ![]() |
[14] |
A. Pankov,
Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
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F. O. de Pavia, W. Kryszewski and A. Szulkin,
Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term, Proc. Amer. Math. Soc., 145 (2017), 4783-4794.
doi: 10.1090/proc/13609. |
[16] |
P. Pucci, M. Q. Xia and B. L. Zhang,
Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[17] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 54 (2013), 031501, 17pp.
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|
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L. Silvestre,
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doi: 10.1002/cpa.20153. |
[20] |
M. Struwe, Variational Methods, 2$^{nd}$ edition, 34. Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-662-03212-1. |
[21] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[22] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[23] |
H. Zhang, J. X. Xu and F. B. Zhao, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 56 (2015), 091502, 13pp.
doi: 10.1063/1.4929660. |
[24] |
X. Zhong and W. Zou,
Ground state and multiple solutions via generalized Nehari mandifold, Nonlinear Anal., 102 (2014), 251-263.
doi: 10.1016/j.na.2014.02.018. |
show all references
References:
[1] |
G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in ${{\mathbb{R}}^{N}}$, J. Differential Equations, 255 (2013), 2340-2362.
doi: 10.1016/j.jde.2013.06.016. |
[2] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. in Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[3] |
J. Chabrowski, Variational Methods for Potential Operator Equations, de Gruyter, Berlin, 1997.
doi: 10.1515/9783110809374. |
[4] |
K. C. Chang,
Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.
doi: 10.1016/0022-247X(81)90095-0. |
[5] |
X. J. Chang and Z. Q. Wang,
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.
doi: 10.1088/0951-7715/26/2/479. |
[6] |
R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman Hall/CRC Financial Mathematics Series, Boca Raton, 2004. |
[7] |
E. Di Nezza, G. Palatucci and E. Valdinaci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[8] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[9] |
N. Laskin, Fractional Schrödinger equations, Phys. Rev., 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[10] |
S. B. Liu,
On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.
doi: 10.1007/s00526-011-0447-2. |
[11] |
R. Metzler and J. Klafter,
The random walls guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[12] |
G. Molica Bisci and V. Rădulescu,
Ground state solutions of scalar field fractional for Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.
doi: 10.1007/s00526-015-0891-5. |
[13] |
G. Molica Bisci, V. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316282397.![]() ![]() ![]() |
[14] |
A. Pankov,
Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
doi: 10.1007/s00032-005-0047-8. |
[15] |
F. O. de Pavia, W. Kryszewski and A. Szulkin,
Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term, Proc. Amer. Math. Soc., 145 (2017), 4783-4794.
doi: 10.1090/proc/13609. |
[16] |
P. Pucci, M. Q. Xia and B. L. Zhang,
Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[17] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[18] |
S. Secchi,
On fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$ without the Ambrosetti-Rabinowitz condition, Topol. Methods Nonlinear Anal., 47 (2016), 19-41.
|
[19] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[20] |
M. Struwe, Variational Methods, 2$^{nd}$ edition, 34. Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-662-03212-1. |
[21] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[22] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[23] |
H. Zhang, J. X. Xu and F. B. Zhao, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 56 (2015), 091502, 13pp.
doi: 10.1063/1.4929660. |
[24] |
X. Zhong and W. Zou,
Ground state and multiple solutions via generalized Nehari mandifold, Nonlinear Anal., 102 (2014), 251-263.
doi: 10.1016/j.na.2014.02.018. |
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