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 & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
References:
[1]

G. AlbertiG. 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. AlvesD. 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. 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.

[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. CapellaJ. DávilaL. 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. ColoradoA. 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 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.

[17]

Z. GuoS. 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. HeM. 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. MelletS. 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. ShangJ. 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.

show all references

References:
[1]

G. AlbertiG. 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. AlvesD. 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. 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.

[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. CapellaJ. DávilaL. 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. ColoradoA. 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 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.

[17]

Z. GuoS. 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. HeM. 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. MelletS. 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. ShangJ. 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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