May  2016, 15(3): 991-1008. doi: 10.3934/cpaa.2016.15.991

Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent

1. 

Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, 030024

2. 

Department of Mathematics, Kunming University, Kunming, Yunnan 650214, China

Received  October 2015 Revised  December 2015 Published  February 2016

In this paper, we establish the existence of ground state solutions for fractional Schrödinger equations with a critical exponent. The methods used here are based on the $s-$harmonic extension technique of Caffarelli and Silvestre, the concentration-compactness principle of Lions and methods of Brezis and Nirenberg.
Citation: Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
References:
[1]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[2]

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.

[3]

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.

[4]

W. X. Chen, C. M. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[5]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bulletin des Sciences Mathematiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[6]

J. Davila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. doi: 10.2140/apde.2015.8.1165.

[7]

J. Davila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equations, 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.

[8]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.

[9]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Critical and subcritical fractional problems with vanishing potentials, arXiv:1410.0843v3.

[10]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, arXiv:1411.0233v1.

[11]

E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.

[12]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961. doi: 10.1088/0951-7715/28/6/1937.

[13]

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

[14]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[15]

N. Laskin, Fractional Schrödinger equation, Physical Review, 66 (2002), 56-108. doi: 10.1103/PhysRevE.66.056108.

[16]

Y. Q. Li, Z. Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837. doi: 10.1016/j.anihpc.2006.01.003.

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, I, II}, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.

[18]

S. A. Molchanov and E. Ostrovskij, Symmetric stable processes as traces of degenerate diffusion processes, Theor. Probab. Appl., 14 (1969), 128-131.

[19]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. A. Math. Phy., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[20]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N2$, arXiv:1208.2545.

[21]

R. Servadei and and E. Valdinoci, A Brezis-Nirenberg result for nonlocal critical equations in low dimension, Commun. Pure App. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[22]

R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Rev. Mat. Complut., 28 (2015), 655-676. doi: 10.1007/s13163-015-0170-1.

[23]

X. D. Shang and J. H. Zhang, Ground states for fractional Schrödinger equations with critical growt, Nonlinearity, 27 (2014), 187-207. doi: 10.1088/0951-7715/27/2/187.

[24]

X. D. Shang, J. H. Zhang and Y. Yang, On fractional Schrödinger equation in $\mathbb{R}^N2$ with critical growth, J. Math. Phys., 54 (2013), 121502.

[25]

X. D. Shang, J. H. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13 (2014), 567-584.

[26]

K. M. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86. doi: 10.1016/j.nonrwa.2014.06.008.

[27]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: A tutorial, in T. Bountis, Order and Chaos vol. 10, Patras University Press, 2008.

[28]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281. doi: 10.1016/S1007-5704(03)00049-2.

[29]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[30]

J. F. Yang and X. P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains, I. Positive mass case, Acta Math. Sci. (English Ed.), 7 (1987), 341-359.

[31]

J. G. Zhang, X. C. Liu and H. Y. Jiao, Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity, arXiv:1502.02222v1.

[32]

L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116.

show all references

References:
[1]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[2]

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.

[3]

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.

[4]

W. X. Chen, C. M. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[5]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bulletin des Sciences Mathematiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[6]

J. Davila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. doi: 10.2140/apde.2015.8.1165.

[7]

J. Davila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equations, 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.

[8]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.

[9]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Critical and subcritical fractional problems with vanishing potentials, arXiv:1410.0843v3.

[10]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, arXiv:1411.0233v1.

[11]

E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.

[12]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961. doi: 10.1088/0951-7715/28/6/1937.

[13]

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

[14]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[15]

N. Laskin, Fractional Schrödinger equation, Physical Review, 66 (2002), 56-108. doi: 10.1103/PhysRevE.66.056108.

[16]

Y. Q. Li, Z. Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837. doi: 10.1016/j.anihpc.2006.01.003.

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, I, II}, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.

[18]

S. A. Molchanov and E. Ostrovskij, Symmetric stable processes as traces of degenerate diffusion processes, Theor. Probab. Appl., 14 (1969), 128-131.

[19]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. A. Math. Phy., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[20]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N2$, arXiv:1208.2545.

[21]

R. Servadei and and E. Valdinoci, A Brezis-Nirenberg result for nonlocal critical equations in low dimension, Commun. Pure App. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[22]

R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Rev. Mat. Complut., 28 (2015), 655-676. doi: 10.1007/s13163-015-0170-1.

[23]

X. D. Shang and J. H. Zhang, Ground states for fractional Schrödinger equations with critical growt, Nonlinearity, 27 (2014), 187-207. doi: 10.1088/0951-7715/27/2/187.

[24]

X. D. Shang, J. H. Zhang and Y. Yang, On fractional Schrödinger equation in $\mathbb{R}^N2$ with critical growth, J. Math. Phys., 54 (2013), 121502.

[25]

X. D. Shang, J. H. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13 (2014), 567-584.

[26]

K. M. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86. doi: 10.1016/j.nonrwa.2014.06.008.

[27]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: A tutorial, in T. Bountis, Order and Chaos vol. 10, Patras University Press, 2008.

[28]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281. doi: 10.1016/S1007-5704(03)00049-2.

[29]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[30]

J. F. Yang and X. P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains, I. Positive mass case, Acta Math. Sci. (English Ed.), 7 (1987), 341-359.

[31]

J. G. Zhang, X. C. Liu and H. Y. Jiao, Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity, arXiv:1502.02222v1.

[32]

L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116.

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