On critical fractional systems with Hardy-Littlewood-Sobolev nonlinearities. (English) Zbl 07274827

Hong, Qianyu; Yang, Yang

Rocky Mt. J. Math. 50, No. 5, 1661-1683 (2020).

The authors study the following critical fractional Laplacian system with Choquard type nonlinearities: \[ \left\{ \begin{array} [c]{lll} (-\Delta)^s u=k_1\displaystyle\left(\int_{\mathbb{R}^N}\frac{|u(y)|^{2_\mu^*}}{|x-y|^\mu}\right)|u|^{2_\mu^*-2}u+\frac{\alpha\gamma}{^{2_\mu^*}}\left(\int_{\mathbb{R}^N}\frac{|v(y)|^{\beta}}{|x-y|^\mu}\right)|u|^{\alpha-2}u & \mathrm{in} & \mathbb{R}^N,\\ (-\Delta)^s v=k_2\displaystyle\left(\int_{\mathbb{R}^N}\frac{|v(y)|^{2_\mu^*}}{|x-y|^\mu}\right)|v|^{2_\mu^*-2}v+\frac{\beta\gamma}{^{2_\mu^*}}\left(\int_{\mathbb{R}^N}\frac{|u(y)|^{\alpha}}{|x-y|^\mu}\right)|v|^{\beta-2}v & \mathrm{in} & \mathbb{R}^N, \end{array} \right. \] where \(s\in(0,1)\), \(N>2s\), \(k_1,k_2>0\), \(\gamma\neq 0\), \(\mu\in(0,N)\), \(\alpha,\beta>1\) and \(2_\mu^*=(2N-\mu)/(N-2s)\) is the fractional upper critical exponent in the Hardy-Littlewood-Sobolev inequality. By using minimization techniques over the Nehari set they provide existence and non-existence of solutions, for certain regions of parameters \((\alpha,\beta)\).

Reviewer: Kaye Silva (Goiânia)

MSC:

35A15	Variational methods applied to PDEs
35R09	Integro-partial differential equations
35R11	Fractional partial differential equations

Keywords:

Nehari manifold; fractional Laplacian; Choquard equation; Hardy-Littlewood-Sobolev critical exponent

References:

[1]	D. Applebaum, “Lévy processes - from probability to finance and quantum groups”, Notices Amer. Math. Soc. 51:11 (2004), 1336-1347. Mathematical Reviews (MathSciNet): MR2105239 Zentralblatt MATH: 1053.60046 · Zbl 1053.60046
[2]	J. Bertoin, Lévy processes, Cambridge Tracts in Mathematics 121, Cambridge University Press, 1996. Mathematical Reviews (MathSciNet): MR1406564 Zentralblatt MATH: 0861.60003 · Zbl 0861.60003
[3]	X. Cabré and Y. Sire, “Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates”, Ann. Inst. H. Poincaré Anal. Non Linéaire 31:1 (2014), 23-53. Mathematical Reviews (MathSciNet): MR3165278 Zentralblatt MATH: 1286.35248 Digital Object Identifier: doi:10.1016/j.anihpc.2013.02.001 · Zbl 1286.35248
[4]	Y.-H. Chen and C. Liu, “Ground state solutions for non-autonomous fractional Choquard equations”, Nonlinearity 29:6 (2016), 1827-1842. Mathematical Reviews (MathSciNet): MR3502230 Zentralblatt MATH: 1381.35213 Digital Object Identifier: doi:10.1088/0951-7715/29/6/1827 · Zbl 1381.35213
[5]	Z. Chen and W. Zou, “Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent”, Arch. Ration. Mech. Anal. 205:2 (2012), 515-551. Mathematical Reviews (MathSciNet): MR2947540 Zentralblatt MATH: 1256.35132 Digital Object Identifier: doi:10.1007/s00205-012-0513-8 · Zbl 1256.35132
[6]	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 52:1-2 (2015), 423-467. Mathematical Reviews (MathSciNet): MR3299187 Zentralblatt MATH: 1312.35158 Digital Object Identifier: doi:10.1007/s00526-014-0717-x · Zbl 1312.35158
[7]	Z. Chen, C.-S. Lin, and W. Zou, “Sign-changing solutions and phase separation for an elliptic system with critical exponent”, Comm. Partial Differential Equations 39:10 (2014), 1827-1859. Mathematical Reviews (MathSciNet): MR3250976 Zentralblatt MATH: 1308.35084 Digital Object Identifier: doi:10.1080/03605302.2014.908391 · Zbl 1308.35084
[8]	P. d’Avenia, G. Siciliano, and M. Squassina, “Existence results for a doubly nonlocal equation”, São Paulo J. Math. Sci. 9:2 (2015), 311-324. Mathematical Reviews (MathSciNet): MR3457463 Zentralblatt MATH: 1369.35109 Digital Object Identifier: doi:10.1007/s40863-015-0023-3 · Zbl 1369.35109
[9]	P. d’Avenia, G. Siciliano, and M. Squassina, “On fractional Choquard equations”, Math. Models Methods Appl. Sci. 25:8 (2015), 1447-1476. Mathematical Reviews (MathSciNet): MR3340706 Zentralblatt MATH: 1323.35205 Digital Object Identifier: doi:10.1142/S0218202515500384 · Zbl 1323.35205
[10]	Z. Deng and Y. Huang, “Positive symmetric results for a weighted quasilinear elliptic system with multiple critical exponents in \(\smash{\mathbb R^N} \)”, Bound. Value Probl. (2017), article no. 28. Mathematical Reviews (MathSciNet): MR3613640 Zentralblatt MATH: 1362.35122 · Zbl 1362.35122
[11]	F. Gao and M. Yang, “On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents”, J. Math. Anal. Appl. 448:2 (2017), 1006-1041. Mathematical Reviews (MathSciNet): MR3582271 Zentralblatt MATH: 1357.35106 Digital Object Identifier: doi:10.1016/j.jmaa.2016.11.015 · Zbl 1357.35106
[12]	F. Gao and M. Yang, “The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation”, Sci. China Math. 61:7 (2018), 1219-1242. Mathematical Reviews (MathSciNet): MR3817173 Zentralblatt MATH: 1397.35087 Digital Object Identifier: doi:10.1007/s11425-016-9067-5 · Zbl 1397.35087
[13]	A. Garroni and S. Müller, “\( \Gamma \)-limit of a phase-field model of dislocations”, SIAM J. Math. Anal. 36:6 (2005), 1943-1964. Mathematical Reviews (MathSciNet): MR2178227 Zentralblatt MATH: 1094.82008 Digital Object Identifier: doi:10.1137/S003614100343768X · Zbl 1094.82008
[14]	J. Giacomoni, T. Mukherjee, and K. Sreenadh, “Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity”, J. Math. Anal. Appl. 467:1 (2018), 638-672. Mathematical Reviews (MathSciNet): MR3834826 Zentralblatt MATH: 1396.35068 Digital Object Identifier: doi:10.1016/j.jmaa.2018.07.035 · Zbl 1396.35068
[15]	Z. Guo and W. Zou, “On a class of coupled Schrödinger systems with critical Sobolev exponent growth”, Math. Methods Appl. Sci. 39:7 (2016), 1730-1746. Mathematical Reviews (MathSciNet): MR3499041 Zentralblatt MATH: 1341.35040 Digital Object Identifier: doi:10.1002/mma.3598 · Zbl 1341.35040
[16]	Z. Guo, S. Luo, and W. Zou, “On critical systems involving fractional Laplacian”, J. Math. Anal. Appl. 446:1 (2017), 681-706. Mathematical Reviews (MathSciNet): MR3554751 Zentralblatt MATH: 1409.35217 Digital Object Identifier: doi:10.1016/j.jmaa.2016.08.069 · Zbl 1409.35217
[17]	Z. Guo, M. Melgaard, and W. Zou, “Schrödinger equations with magnetic fields and Hardy-Sobolev critical exponents”, Electron. J. Differential Equations (2017), article no. 199. Mathematical Reviews (MathSciNet): MR3690226 Zentralblatt MATH: 1375.35484 · Zbl 1375.35484
[18]	Z. Guo, K. Perera, and W. Zou, “On critical \(p\)-Laplacian systems”, Adv. Nonlinear Stud. 17:4 (2017), 641-659. Mathematical Reviews (MathSciNet): MR3709034 Zentralblatt MATH: 1372.35034 Digital Object Identifier: doi:10.1515/ans-2017-6029 · Zbl 1372.35034
[19]	E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001. Mathematical Reviews (MathSciNet): MR1817225 Zentralblatt MATH: 0966.26002 · Zbl 0966.26002
[20]	D. Lü and G. Xu, “On nonlinear fractional Schrödinger equations with Hartree-type nonlinearity”, Appl. Anal. 97:2 (2018), 255-273. Mathematical Reviews (MathSciNet): MR3750483 Zentralblatt MATH: 1393.35224 Digital Object Identifier: doi:10.1080/00036811.2016.1260708 · Zbl 1393.35224
[21]	C. Mou, “Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space”, Commun. Pure Appl. Anal. 14:6 (2015), 2335-2362. Mathematical Reviews (MathSciNet): MR3411110 Zentralblatt MATH: 1326.35125 Digital Object Identifier: doi:10.3934/cpaa.2015.14.2335 · Zbl 1326.35125
[22]	T. Mukherjee and K. Sreenadh, “Fractional Choquard equation with critical nonlinearities”, NoDEA Nonlinear Differential Equations Appl. 24:6 (2017), article no. 63. Mathematical Reviews (MathSciNet): MR3720946 Zentralblatt MATH: 1387.35608 · Zbl 1387.35608
[23]	Y. J. Park, “Fractional Polya-Szegő inequality”, J. Chungcheong Math. Soc. 24:2 (2011), 267-271.
[24]	S. Peng, Y. Peng, and Z. Wang, “On elliptic systems with Sobolev critical growth”, Calc. Var. Partial Differential Equations 55:6 (2016), article no. 142. Mathematical Reviews (MathSciNet): MR3567641 Zentralblatt MATH: 1364.35091 Digital Object Identifier: doi:10.1007/s00526-016-1091-7 · Zbl 1364.35091
[25]	X. Ros-Oton and J. Serra, “The Dirichlet problem for the fractional Laplacian: regularity up to the boundary”, J. Math. Pures Appl. \((9) 101\):3 (2014), 275-302. Mathematical Reviews (MathSciNet): MR3168912 Zentralblatt MATH: 1285.35020 Digital Object Identifier: doi:10.1016/j.matpur.2013.06.003 · Zbl 1285.35020
[26]	R. Servadei and E. Valdinoci, “The Brezis-Nirenberg result for the fractional Laplacian”, Trans. Amer. Math. Soc. 367:1 (2015), 67-102. Mathematical Reviews (MathSciNet): MR3271254 Zentralblatt MATH: 1323.35202 Digital Object Identifier: doi:10.1090/S0002-9947-2014-05884-4 · Zbl 1323.35202
[27]	L. Silvestre, “Regularity of the obstacle problem for a fractional power of the Laplace operator”, Comm. Pure Appl. Math. 60:1 (2007), 67-112. Mathematical Reviews (MathSciNet): MR2270163 Zentralblatt MATH: 1141.49035 Digital Object Identifier: doi:10.1002/cpa.20153 · Zbl 1141.49035
[28]	Y. Wu and W. Zou, “Spikes of the two-component elliptic system in \(\mathbb{R}^4\) with the critical Sobolev exponent”, Calc. Var. Partial Differential Equations 58:1 (2019), article no. 24. Mathematical Reviews (MathSciNet): MR3895387 Zentralblatt MATH: 1406.35028 · Zbl 1406.35028
[29]	M. Yang, J. C. de Albuquerque, E. D. Silva, and M. L. Silva, “On the critical cases of linearly coupled Choquard systems”, Appl. Math. Lett. 91 (2019), 1-8. Mathematical Reviews (MathSciNet): MR3886985 Zentralblatt MATH: 1411.35106 Digital Object Identifier: doi:10.1016/j.aml.2018.11.005 · Zbl 1411.35106
[30]	Y. Yang, Q. Y. Hong, and X. Shang, “Existence of solutions for non-local elliptic systems with Hardy-Littlewood-Sobolev critical nonlinearities”, Electron. J. Differential Equations (2019), article no. 90. Mathematical Reviews (MathSciNet): MR3992301 Zentralblatt MATH: 1418.35373 · Zbl 1418.35373
[31]	S. You, P. Zhao, and Q. Wang, “Positive ground states for coupled nonlinear Choquard equations involving Hardy-Littlewood-Sobolev critical exponent”, Nonlinear Anal. Real World Appl. 48 (2019), 182-211. Mathematical Reviews (MathSciNet): MR3909887 Zentralblatt MATH: 1428.35543 Digital Object Identifier: doi:10.1016/j.nonrwa.2019.01.015 · Zbl 1428.35543
[32]	M. · Zbl 1401.35071