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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 & Continuous Dynamical Systems - A, 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. Google Scholar [2] A. Ambrosetti, E. 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. Google Scholar [3] A. Ambrosetti, G. 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. Google Scholar [4] V. Ambrosio, Multiple solutions for a nonlinear scalar field equation involving the fractional Laplacian, arXiv: 1603.09538.Google Scholar [5] B. Barrios, E. Colorado, A. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] W. Choi, S. 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. Google Scholar [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. Google Scholar [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. Google Scholar [15] 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. Google Scholar [16] P. Felmer, A. 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. Google Scholar [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. Google Scholar [18] H. Hajaiej, Some fractional functional inequalities and applications to some constrained minimization problems involving a local non-linearity, arXiv: 1104.1414v1.Google Scholar [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. Google Scholar [20] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar [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. Google Scholar [22] E. H. Lieb and M. Loss, Analysis Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/gsm/014. Google Scholar [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. Google Scholar [24] L. A. Maia, E. 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. Google Scholar [25] E. D. Nezza, G. 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. Google Scholar [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. Google Scholar [27] J. Seok, Spike-layer solutions to nonlinear fractional Schrödinger equation with almost optimal nonlinearities, Electron. J. Differential Equations, 196 (2015), 1-19. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] J. Zhang, J. 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. Google Scholar

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. Google Scholar [2] A. Ambrosetti, E. 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. Google Scholar [3] A. Ambrosetti, G. 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. Google Scholar [4] V. Ambrosio, Multiple solutions for a nonlinear scalar field equation involving the fractional Laplacian, arXiv: 1603.09538.Google Scholar [5] B. Barrios, E. Colorado, A. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] W. Choi, S. 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. Google Scholar [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. Google Scholar [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. Google Scholar [15] 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. Google Scholar [16] P. Felmer, A. 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. Google Scholar [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. Google Scholar [18] H. Hajaiej, Some fractional functional inequalities and applications to some constrained minimization problems involving a local non-linearity, arXiv: 1104.1414v1.Google Scholar [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. Google Scholar [20] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar [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. Google Scholar [22] E. H. Lieb and M. Loss, Analysis Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/gsm/014. Google Scholar [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. Google Scholar [24] L. A. Maia, E. 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. Google Scholar [25] E. D. Nezza, G. 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. Google Scholar [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. Google Scholar [27] J. Seok, Spike-layer solutions to nonlinear fractional Schrödinger equation with almost optimal nonlinearities, Electron. J. Differential Equations, 196 (2015), 1-19. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] J. Zhang, J. 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. Google Scholar
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