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

Critical system involving fractional Laplacian

Maoding Zhen , Jinchun He and  Haoyun Xu

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.
Keywords: Fractional Laplacian system, Nehari manifold, least energy solution.
Mathematics Subject Classification: Primary: 35J50, 35B33, 35R11.
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. 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]

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. Google Scholar

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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

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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, 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. 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. 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

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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. 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

[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

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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 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]

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. 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. 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. 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. 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

[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. Shang, J. 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. 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]

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. 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]

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. 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. 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. 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

[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 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]

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. 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. 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. 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. 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

[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. Shang, J. 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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