# 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 and Continuous Dynamical Systems, 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] W. Rudin, Real and Complex Analysis, 3nd edition, McGraw-Hill Book Co., New York, 1987. [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. [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. [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. [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. [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. [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. [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. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] W. Rudin, Real and Complex Analysis, 3nd edition, McGraw-Hill Book Co., New York, 1987. [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. [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. [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. [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. [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. [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. [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.  [1] 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 [2] Yu Su, Zhaosheng Feng. Ground state solutions for the fractional problems with dipole-type potential and critical exponent. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1953-1968. doi: 10.3934/cpaa.2021111 [3] Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179 [4] Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143 [5] Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure and Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567 [6] Yu Su, Zhaosheng Feng. Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022112 [7] Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131 [8] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [9] Qilin Xie, Jianshe Yu. Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. Communications on Pure and Applied Analysis, 2019, 18 (1) : 129-158. doi: 10.3934/cpaa.2019008 [10] Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} \$dinger-Poisson system involving critical exponent. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104 [11] Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038 [12] Juncheng Wei, Ke Wu. Local behavior of solutions to a fractional equation with isolated singularity and critical Serrin exponent. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4031-4050. doi: 10.3934/dcds.2022044 [13] Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108 [14] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [15] Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357 [16] Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645 [17] Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165 [18] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [19] Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231 [20] Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351

2021 Impact Factor: 1.588