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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.   Google Scholar

[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.  Google Scholar

[26]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

Q. Wang, Positive least energy solutions of fractional Laplacian systems with critical exponent, Electron. J. Differential Equations, 2016 (2016), 1-16.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.   Google Scholar

[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.  Google Scholar

[26]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

Q. Wang, Positive least energy solutions of fractional Laplacian systems with critical exponent, Electron. J. Differential Equations, 2016 (2016), 1-16.   Google Scholar

[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.  Google Scholar

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