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June  2017, 37(6): 3327-3352. doi: 10.3934/dcds.2017141

On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling

1. 

School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

2. 

School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000, China

Received  September 2016 Revised  January 2017 Published  February 2017

In this paper, a class of systems of two coupled nonlinear fractional Laplacian equations are investigated. Under very weak assumptions on the nonlinear terms $f$ and $g$, we establish some results about the existence of positive vector solutions and vector ground state solutions for the fractional Laplacian systems by using variational methods. In addition, we also study the asymptotic behavior of these solutions as the coupling parameter $β$ tends to zero.

Citation: Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141
References:
[1]

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp. doi: 10.1007/s00526-016-0983-x.

[2]

A. AmbrosettiE. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.

[3]

A. AmbrosettiG. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb{R}^{n}$, J. Funct. Anal., 254 (2008), 2816-2845.  doi: 10.1016/j.jfa.2007.11.013.

[4]

V. Ambrosio, Multiple solutions for a nonlinear scalar field equation involving the fractional Laplacian, arXiv: 1603.09538.

[5]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations (Ⅰ): Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250555.

[7]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.

[8]

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.

[9]

X. Chang and Z.-Q. Wang, Ground state of scalar field equations involving fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.

[10]

Z. Chen and W. Zou, On linearly coupled Schrödinger systems, Proc. Amer. Math. Soc., 142 (2014), 323-333.  doi: 10.1090/S0002-9939-2013-12000-9.

[11]

Z. Chen and W. Zou, Standing waves for a coupled system of nonlinear Schrödinger equations, Annali di Matematica, 194 (2015), 183-220.  doi: 10.1007/s10231-013-0371-5.

[12]

W. ChoiS. Kim and K. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.  doi: 10.1016/j.jfa.2014.02.029.

[13]

W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153.  doi: 10.1016/j.na.2015.03.007.

[14]

S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119.  doi: 10.1016/j.jde.2013.04.001.

[15]

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

[16]

P. FelmerA. Quaas and J. 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.

[17]

Q. Guo and X. He, Least energy solutions for a weakly coupled fractional Schrödinger system, Nonlinear Anal., 132 (2016), 141-159.  doi: 10.1016/j.na.2015.11.005.

[18]

H. Hajaiej, Some fractional functional inequalities and applications to some constrained minimization problems involving a local non-linearity, arXiv: 1104.1414v1.

[19]

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

[20]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[21]

R. Lehrei and L. A. Maia, Positive solutions of asymptotically linear equations via Pohožaev manifold, J. Funct. Anal., 266 (2014), 213-246.  doi: 10.1016/j.jfa.2013.09.002.

[22]

E. H. Lieb and M. Loss, Analysis Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/gsm/014.

[23]

C. Lin and S. Peng, Segregated vector solutions for linearly coupled nonlinear Schrödinger systems, Indiana Univ. Math. J., 63 (2014), 939-967.  doi: 10.1512/iumj.2014.63.5310.

[24]

L. A. MaiaE. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002.

[25]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[26]

S. Secchi, Ground state solutions for nonlinear fractional Schrodinger equations in $\mathbb{R}^{N}$ ,J. Math. Phys. , 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990.

[27]

J. Seok, Spike-layer solutions to nonlinear fractional Schrödinger equation with almost optimal nonlinearities, Electron. J. Differential Equations, 196 (2015), 1-19. 

[28]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{n}$, Commun. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.

[29] M. Struwe, Variational Methods-Application to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1996.  doi: 10.1007/978-3-662-04194-9.
[30]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.  doi: 10.1007/s00526-010-0378-3.

[31]

Z. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508.  doi: 10.3934/dcds.2016.36.499.

[32]

V. C. Zelati and M. Nolasco, Existence of ground states for nonlinear, pseudo relativistic Schrödinger equations, Rend. Lincei Mat. Appl., 22 (2011), 51-72.  doi: 10.4171/RLM/587.

[33]

J. ZhangJ. Marcos do Ó and M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30.  doi: 10.1515/ans-2015-5024.

show all references

References:
[1]

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp. doi: 10.1007/s00526-016-0983-x.

[2]

A. AmbrosettiE. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.

[3]

A. AmbrosettiG. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb{R}^{n}$, J. Funct. Anal., 254 (2008), 2816-2845.  doi: 10.1016/j.jfa.2007.11.013.

[4]

V. Ambrosio, Multiple solutions for a nonlinear scalar field equation involving the fractional Laplacian, arXiv: 1603.09538.

[5]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations (Ⅰ): Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250555.

[7]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.

[8]

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.

[9]

X. Chang and Z.-Q. Wang, Ground state of scalar field equations involving fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.

[10]

Z. Chen and W. Zou, On linearly coupled Schrödinger systems, Proc. Amer. Math. Soc., 142 (2014), 323-333.  doi: 10.1090/S0002-9939-2013-12000-9.

[11]

Z. Chen and W. Zou, Standing waves for a coupled system of nonlinear Schrödinger equations, Annali di Matematica, 194 (2015), 183-220.  doi: 10.1007/s10231-013-0371-5.

[12]

W. ChoiS. Kim and K. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.  doi: 10.1016/j.jfa.2014.02.029.

[13]

W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153.  doi: 10.1016/j.na.2015.03.007.

[14]

S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119.  doi: 10.1016/j.jde.2013.04.001.

[15]

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

[16]

P. FelmerA. Quaas and J. 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.

[17]

Q. Guo and X. He, Least energy solutions for a weakly coupled fractional Schrödinger system, Nonlinear Anal., 132 (2016), 141-159.  doi: 10.1016/j.na.2015.11.005.

[18]

H. Hajaiej, Some fractional functional inequalities and applications to some constrained minimization problems involving a local non-linearity, arXiv: 1104.1414v1.

[19]

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

[20]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[21]

R. Lehrei and L. A. Maia, Positive solutions of asymptotically linear equations via Pohožaev manifold, J. Funct. Anal., 266 (2014), 213-246.  doi: 10.1016/j.jfa.2013.09.002.

[22]

E. H. Lieb and M. Loss, Analysis Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/gsm/014.

[23]

C. Lin and S. Peng, Segregated vector solutions for linearly coupled nonlinear Schrödinger systems, Indiana Univ. Math. J., 63 (2014), 939-967.  doi: 10.1512/iumj.2014.63.5310.

[24]

L. A. MaiaE. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002.

[25]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[26]

S. Secchi, Ground state solutions for nonlinear fractional Schrodinger equations in $\mathbb{R}^{N}$ ,J. Math. Phys. , 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990.

[27]

J. Seok, Spike-layer solutions to nonlinear fractional Schrödinger equation with almost optimal nonlinearities, Electron. J. Differential Equations, 196 (2015), 1-19. 

[28]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{n}$, Commun. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.

[29] M. Struwe, Variational Methods-Application to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1996.  doi: 10.1007/978-3-662-04194-9.
[30]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.  doi: 10.1007/s00526-010-0378-3.

[31]

Z. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508.  doi: 10.3934/dcds.2016.36.499.

[32]

V. C. Zelati and M. Nolasco, Existence of ground states for nonlinear, pseudo relativistic Schrödinger equations, Rend. Lincei Mat. Appl., 22 (2011), 51-72.  doi: 10.4171/RLM/587.

[33]

J. ZhangJ. Marcos do Ó and M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30.  doi: 10.1515/ans-2015-5024.

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