# American Institute of Mathematical Sciences

November  2019, 39(11): 6523-6539. doi: 10.3934/dcds.2019283

## Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

 School of Mathematics and Statistics and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author: Haoyuan Xu

Received  December 2018 Revised  May 2019 Published  August 2019

Fund Project: The authors are supported by the NSFC grant 11571125.

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent
 $\begin{cases} (-\Delta)^{s}u+\mu u = |u|^{p-1}u+\lambda v,& x\in\mathbb{R}^{N},\\ (-\Delta)^{s}v+\nu v = |v|^{2^{\ast}-2}v+\lambda u,& x\in\mathbb{R}^{N},\\ \end{cases}$
where
 $(-\Delta)^{s}$
is the fractional Laplacian,
 $02s, \ \lambda <\sqrt{\mu\nu },\ 1 is the Sobolev critical exponent. By using the Nehari\ manifold, we show that there exists a $ \mu_{0}\in(0,1) $, such that when $ 0<\mu\leq\mu_{0} $, the system has a positive ground state solution. When $ \mu>\mu_{0} $, there exists a $ \lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu}) $such that if $ \lambda>\lambda_{\mu,\nu} $, the system has a positive ground state solution, if $ \lambda<\lambda_{\mu,\nu} $, the system has no ground state solution. Citation: Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283 ##### References:  [1] G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect, Arch. Rational Mech. Anal., 144 (1998), 1-46. doi: 10.1007/s002050050111. Google Scholar [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [3] 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 [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. (I): Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. Google Scholar [5] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3. Google Scholar [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar [7] L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226. Google Scholar [8] 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 [9] Z. J. Chen and W. M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations, 48 (2013), 695-711. doi: 10.1007/s00526-012-0568-2. Google Scholar [10] Z. J. Chen and W. M. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107. doi: 10.1016/j.jfa.2012.01.001. Google Scholar [11] Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551. doi: 10.1007/s00205-012-0513-8. Google Scholar [12] Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467. doi: 10.1007/s00526-014-0717-x. Google Scholar [13] X. Y. Cheng and S. Ma, Existence of three nontrivial solutions for elliptic systems with critical exponents and weights, Nonlinear Anal., 69 (2008), 3537-3548. doi: 10.1016/j.na.2007.09.040. Google Scholar [14] E. Colorado, A. de Pablo and U. Sánchez, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65. Google Scholar [15] A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034. Google Scholar [16] E. Di 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 [17] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in$\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. Google Scholar [18] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar [19] Q. Guo and X. M. 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 [20] Z. Y. Guo, S. P. Luo and W. M. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. doi: 10.1016/j.jmaa.2016.08.069. Google Scholar [21] D. F. Lü and S. J. Peng, On the positive vector solutions for nonlinear fractional Laplacian system with linear coupling, Discrete Contin. Dys. Syst., 37 (2017), 3327-3352. doi: 10.3934/dcds.2017141. Google Scholar [22] J. Marcos do Ó and D. Ferraz, Concentration-compactness principle for nonlocal scalar field equations with critical growth, J. Math. Anal. Appl., 449 (2017), 1189-1228. doi: 10.1016/j.jmaa.2016.12.053. Google Scholar [23] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2. Google Scholar [24] S. J. Peng, Y.-F. Peng and Z.-Q. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp. doi: 10.1007/s00526-016-1091-7. Google Scholar [25] S. J. Peng, W. Shuai and Q. F. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differential Equations, 263 (2017), 709-731. doi: 10.1016/j.jde.2017.02.053. Google Scholar [26] W. Rudin, Real and Complex Analysis, 3nd edition, McGraw-Hill Book Co., New York, 1987. Google Scholar [27] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. Google Scholar [28] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [29] 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. doi: 10.3934/cpaa.2014.13.567. Google Scholar [30] 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. Google Scholar [31] Z. P. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 449-508. doi: 10.3934/dcds.2016.36.499. Google Scholar [32] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc. Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [33] M. D. Zhen, J. C. He and H. Y. Xu, Critical system involving fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 237-253. doi: 10.3934/cpaa.2019013. Google Scholar show all references ##### References:  [1] G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect, Arch. Rational Mech. Anal., 144 (1998), 1-46. doi: 10.1007/s002050050111. Google Scholar [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [3] 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 [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. (I): Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. Google Scholar [5] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3. Google Scholar [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar [7] L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226. Google Scholar [8] 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 [9] Z. J. Chen and W. M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations, 48 (2013), 695-711. doi: 10.1007/s00526-012-0568-2. Google Scholar [10] Z. J. Chen and W. M. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107. doi: 10.1016/j.jfa.2012.01.001. Google Scholar [11] Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551. doi: 10.1007/s00205-012-0513-8. Google Scholar [12] Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467. doi: 10.1007/s00526-014-0717-x. Google Scholar [13] X. Y. Cheng and S. Ma, Existence of three nontrivial solutions for elliptic systems with critical exponents and weights, Nonlinear Anal., 69 (2008), 3537-3548. doi: 10.1016/j.na.2007.09.040. Google Scholar [14] E. Colorado, A. de Pablo and U. Sánchez, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65. Google Scholar [15] A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034. Google Scholar [16] E. Di 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 [17] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in$\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. Google Scholar [18] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar [19] Q. Guo and X. M. 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 [20] Z. Y. Guo, S. P. Luo and W. M. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. doi: 10.1016/j.jmaa.2016.08.069. Google Scholar [21] D. F. Lü and S. J. Peng, On the positive vector solutions for nonlinear fractional Laplacian system with linear coupling, Discrete Contin. Dys. Syst., 37 (2017), 3327-3352. doi: 10.3934/dcds.2017141. Google Scholar [22] J. Marcos do Ó and D. Ferraz, Concentration-compactness principle for nonlocal scalar field equations with critical growth, J. Math. Anal. Appl., 449 (2017), 1189-1228. doi: 10.1016/j.jmaa.2016.12.053. Google Scholar [23] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2. Google Scholar [24] S. J. Peng, Y.-F. Peng and Z.-Q. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp. doi: 10.1007/s00526-016-1091-7. Google Scholar [25] S. J. Peng, W. Shuai and Q. F. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differential Equations, 263 (2017), 709-731. doi: 10.1016/j.jde.2017.02.053. Google Scholar [26] W. Rudin, Real and Complex Analysis, 3nd edition, McGraw-Hill Book Co., New York, 1987. Google Scholar [27] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. Google Scholar [28] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [29] 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. doi: 10.3934/cpaa.2014.13.567. Google Scholar [30] 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. Google Scholar [31] Z. P. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 449-508. doi: 10.3934/dcds.2016.36.499. Google Scholar [32] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc. Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [33] M. D. Zhen, J. C. He and H. Y. Xu, Critical system involving fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 237-253. doi: 10.3934/cpaa.2019013. Google Scholar  [1] 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 [2] Yinbin Deng, Wentao Huang. 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