# American Institute of Mathematical Sciences

January  2019, 18(1): 237-253. doi: 10.3934/cpaa.2019013

## Critical system involving fractional Laplacian

 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received  October 2017 Revised  May 2018 Published  August 2018

Fund Project: The authors were supported by NSFC grant 11571125.

In this paper, we study the following critical system with fractional Laplacian:
 $\begin{equation*}\begin{cases}(-Δ)^{s}u = μ_{1}|u|^{2^{*}-2}u+\dfrac{αγ}{2^{*}}|u|^{α-2}u|v|^{β} \ \ \ \text{in} \ \ \mathbb{R}^{n},\\(-Δ)^{s}v = μ_{2}|v|^{2^{*}-2}v+\dfrac{βγ}{2^{*}}|u|^{α}|v|^{β-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n},\\u,v∈ D_{s}(\mathbb{R}^{n}).\end{cases}\end{equation*}$
By using the Nehari manifold, under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.
Citation: Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
##### 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] C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., 42 (2000), 771-787.  doi: 10.1016/S0362-546X(99)00121-2. [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] 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. [5] 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. [6] L. Caffarelli, J. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. Available from: https://hal.archives-ouvertes.fr/hal-00629379v1. doi: 10.4171/JEMS/226. [7] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954. [8] W. Chen and S. Deng, Multiple solutions for a critical fractional elliptic system involving concave-convex nonlinearlities, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 1167-1193.  doi: 10.1017/S0308210516000032. [9] Z. Chen and W. 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. Chen and W. 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. [11] Z. Chen and W. 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. [12] X. 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. [13] 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. [14] 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. [15] M. de Souza and Y. L. Araújo, Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth, Math. Methods Appl. Sci., 40 (2017), 1757-1772.  doi: 10.1002/mma.4095. [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] Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.  doi: 10.1016/j.jmaa.2016.08.069. [18] Y. Guo, Nonexistence and symmetry of solutions to some fractional Laplacian equations in the upper half space, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 836-851.  doi: 10.1016/S0252-9602(17)30040-1. [19] X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, Commun. Pure Appl. Anal., 15 (2016), 1285-1308.  doi: 10.3934/cpaa.2016.15.1285. [20] J. Marcos 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. [21] Q. Li and Z. D. Yang, Multiple positive solution for a fractional Laplacian system with critical nonlinearities, Bull. Malays. Math. Sci. Soc., 2 (2016), 1-27. [22] 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. [23] M. Niu and Z. Tang, Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth, Discrete Contin. Dyn. Syst., 37 (2017), 3963-3987.  doi: 10.3934/dcds.2017168. [24] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26. [25] 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. [26] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. [27] X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13 (2014), 567-584. [28] 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. [29] J. Tan, The Brézis-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. [30] Q. Wang, Positive least energy solutions of fractional Laplacian systems with critical exponent, Electron. J. Differential Equations, 2016 (2016), 1-16. [31] X. Zheng and J. Wang, Symmetry results for systems involving fractional Laplacian, Indian J. Pure Appl. Math., 45 (2014), 39-51.  doi: 10.1007/s13226-014-0050-2.

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##### 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] C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., 42 (2000), 771-787.  doi: 10.1016/S0362-546X(99)00121-2. [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] 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. [5] 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. [6] L. Caffarelli, J. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. Available from: https://hal.archives-ouvertes.fr/hal-00629379v1. doi: 10.4171/JEMS/226. [7] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954. [8] W. Chen and S. Deng, Multiple solutions for a critical fractional elliptic system involving concave-convex nonlinearlities, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 1167-1193.  doi: 10.1017/S0308210516000032. [9] Z. Chen and W. 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. Chen and W. 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. [11] Z. Chen and W. 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. [12] X. 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. [13] 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. [14] 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. [15] M. de Souza and Y. L. Araújo, Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth, Math. Methods Appl. Sci., 40 (2017), 1757-1772.  doi: 10.1002/mma.4095. [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] Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.  doi: 10.1016/j.jmaa.2016.08.069. [18] Y. Guo, Nonexistence and symmetry of solutions to some fractional Laplacian equations in the upper half space, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 836-851.  doi: 10.1016/S0252-9602(17)30040-1. [19] X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, Commun. Pure Appl. Anal., 15 (2016), 1285-1308.  doi: 10.3934/cpaa.2016.15.1285. [20] J. Marcos 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. [21] Q. Li and Z. D. Yang, Multiple positive solution for a fractional Laplacian system with critical nonlinearities, Bull. Malays. Math. Sci. Soc., 2 (2016), 1-27. [22] 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. [23] M. Niu and Z. Tang, Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth, Discrete Contin. Dyn. Syst., 37 (2017), 3963-3987.  doi: 10.3934/dcds.2017168. [24] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26. [25] 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. [26] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. [27] X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13 (2014), 567-584. [28] 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. [29] J. Tan, The Brézis-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. [30] Q. Wang, Positive least energy solutions of fractional Laplacian systems with critical exponent, Electron. J. Differential Equations, 2016 (2016), 1-16. [31] X. Zheng and J. Wang, Symmetry results for systems involving fractional Laplacian, Indian J. Pure Appl. Math., 45 (2014), 39-51.  doi: 10.1007/s13226-014-0050-2.
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