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 & 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,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[3]

X. J. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity,, \emph{Nonlinearity}, 26 (2013), 479.  doi: 10.1088/0951-7715/26/2/479.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian,, \emph{Le Matematiche}, 68 (2013), 201.   Google Scholar

[9]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Critical and subcritical fractional problems with vanishing potentials,, \arXiv{1410.0843v3}., ().   Google Scholar

[10]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth,, \arXiv{1411.0233v1}., ().   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

N. Laskin, Fractional Schrödinger equation,, \emph{Physical Review}, 66 (2002), 56.  doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[16]

Y. Q. Li, Z. Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 23 (2006), 829.  doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, I, II},, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.   Google Scholar

[18]

S. A. Molchanov and E. Ostrovskij, Symmetric stable processes as traces of degenerate diffusion processes,, \emph{Theor. Probab. Appl.}, 14 (1969), 128.   Google Scholar

[19]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. A. Math. Phy.}, 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[20]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^N$,, \arXiv{1208.2545}., ().   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

X. D. Shang, J. H. Zhang and Y. Yang, On fractional Schrödinger equation in $\mathbbR^N$ with critical growth,, \emph{J. Math. Phys.}, 54 (2013).   Google Scholar

[25]

X. D. Shang, J. H. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 567.   Google Scholar

[26]

K. M. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\mathbbR^N$,, \emph{Nonlinear Anal. Real World Appl.}, 21 (2015), 76.  doi: 10.1016/j.nonrwa.2014.06.008.  Google Scholar

[27]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: A tutorial, in T. Bountis, Order and Chaos, vol. 10, (2008).   Google Scholar

[28]

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

[29]

M. Willem, Minimax Theorems,, Birkh\, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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,, \emph{Acta Math. Sci. (English Ed.)}, 7 (1987), 341.   Google Scholar

[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}., ().   Google Scholar

[32]

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

show all references

References:
[1]

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

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[3]

X. J. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity,, \emph{Nonlinearity}, 26 (2013), 479.  doi: 10.1088/0951-7715/26/2/479.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian,, \emph{Le Matematiche}, 68 (2013), 201.   Google Scholar

[9]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Critical and subcritical fractional problems with vanishing potentials,, \arXiv{1410.0843v3}., ().   Google Scholar

[10]

J. M. Do ó, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth,, \arXiv{1411.0233v1}., ().   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

N. Laskin, Fractional Schrödinger equation,, \emph{Physical Review}, 66 (2002), 56.  doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[16]

Y. Q. Li, Z. Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 23 (2006), 829.  doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, I, II},, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.   Google Scholar

[18]

S. A. Molchanov and E. Ostrovskij, Symmetric stable processes as traces of degenerate diffusion processes,, \emph{Theor. Probab. Appl.}, 14 (1969), 128.   Google Scholar

[19]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. A. Math. Phy.}, 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[20]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^N$,, \arXiv{1208.2545}., ().   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

X. D. Shang, J. H. Zhang and Y. Yang, On fractional Schrödinger equation in $\mathbbR^N$ with critical growth,, \emph{J. Math. Phys.}, 54 (2013).   Google Scholar

[25]

X. D. Shang, J. H. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 567.   Google Scholar

[26]

K. M. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\mathbbR^N$,, \emph{Nonlinear Anal. Real World Appl.}, 21 (2015), 76.  doi: 10.1016/j.nonrwa.2014.06.008.  Google Scholar

[27]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: A tutorial, in T. Bountis, Order and Chaos, vol. 10, (2008).   Google Scholar

[28]

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

[29]

M. Willem, Minimax Theorems,, Birkh\, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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,, \emph{Acta Math. Sci. (English Ed.)}, 7 (1987), 341.   Google Scholar

[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}., ().   Google Scholar

[32]

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

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