×

Bound states for fractional Schrödinger-Poisson system with critical exponent. (English) Zbl 1480.35198

Summary: This paper deals with the fractional Schrödinger-Poisson system \[ \begin{cases} \varepsilon^{2s}(-\Delta )^su+V(x)u+K(x)\phi u = |u|^{2_s^*-2}u, \quad & \text{in } \mathbb{R}^3,\\ (-\Delta)^t\phi = K(x)u^2, & \text{in } \mathbb{R}^3, \end{cases} \] where \( s\in (\frac{3}{4}, 1) \), \( t\in(0, 1) \), \( \varepsilon \) is a positive parameter, \( 2_s^* = \frac{6}{3-2s} \) is the critical Sobolev exponent. \( K(x)\in L^{\frac{6}{2t+4s-3}}(\mathbb{R}^3) \), \( V(x)\in L^{\frac{3}{2s}}(\mathbb{R}^3) \) and \(V(x)\) is assumed to be zero in some region of \(\mathbb{R}^3 \), which means that the problem is of the critical frequency case. In virtue of a global compactness result in fractional Sobolev space and Lusternik-Schnirelman theory of critical points, we succeed in proving the multiplicity of bound states.

MSC:

35J61 Semilinear elliptic equations
35R11 Fractional partial differential equations
35J50 Variational methods for elliptic systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Ambrosetti; D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10, 391-404 (2008) · Zbl 1188.35171
[2] A. Azzollini; P. d’Avenia; A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 779-791 (2010) · Zbl 1187.35231
[3] A. Azzollini; A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345, 90-108 (2008) · Zbl 1147.35091
[4] V. Benci; G. Cerami, Existence of positive solutions of the equation \(-\Delta u+a(x)u = u^{(N+2)/(N-2)}\) in \({\Bbb R}^N\), J. Funct. Anal., 88, 90-117 (1990) · Zbl 0705.35042
[5] V. Benci; D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11, 283-293 (1998) · Zbl 0926.35125
[6] G. Cerami; R. Molle, Positive bound state solutions for some Schrödinger-Poisson systems, Nonlinearity, 29, 3103-3119 (2016) · Zbl 1408.35022
[7] J. Chabrowski; J. Yang, Multiple semiclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugal. Math., 57, 273-284 (2000) · Zbl 0965.35065
[8] W. Chen; C. Li; B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59, 330-343 (2006) · Zbl 1093.45001
[9] T. D’Aprile; D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4, 307-322 (2004) · Zbl 1142.35406
[10] T. D’Aprile; D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134, 893-906 (2004) · Zbl 1064.35182
[11] T. D’Aprile; J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37, 321-342 (2005) · Zbl 1096.35017
[12] E. Di Nezza; G. Palatucci; E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573 (2012) · Zbl 1252.46023
[13] R. L. Frank; E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39, 85-99 (2010) · Zbl 1204.39024
[14] R. L. Frank; E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Oper. Theory Adv. Appl., 219, 55-67 (2012) · Zbl 1297.39023
[15] L. Guo and Q. Li, Multiple bound state solutions for fractional Choquard equation with Hardy-Littlewood-Sobolev critical exponent, J. Math. Phys., 61 (2020), 121501, 20 pp. · Zbl 1454.81254
[16] I. Ianni; G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8, 573-595 (2008) · Zbl 1216.35138
[17] G. Li; S. Peng; S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12, 1069-1092 (2010) · Zbl 1206.35082
[18] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118, 349-374 (1983) · Zbl 0527.42011
[19] Z. Liu; J. Zhang, Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth, ESAIM Control Optim. Calc. Var., 23, 1515-1542 (2017) · Zbl 1516.35467
[20] E. G. Murcia; G. Siciliano, Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30, 231-258 (2017) · Zbl 1413.35018
[21] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674 (2006) · Zbl 1136.35037
[22] D. Ruiz; G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential, Rev. Mat. Iberoam., 27, 253-271 (2011) · Zbl 1216.35024
[23] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261, 3061-3106 (2016) · Zbl 1386.35458
[24] K. Teng; R. P. Agarwal, Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Methods Appl. Sci., 41, 8258-8293 (2018) · Zbl 1405.35252
[25] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. · Zbl 0856.49001
[26] Y. Yu, F. Zhao and L. Zhao, The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 116, 25 pp. · Zbl 1437.35699
[27] J. Zhang; J. M. do Ó.; M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16, 15-30 (2016) · Zbl 1334.35407
[28] H. Zhang, J. Xu and F. Zhang, Multiplicity of semiclassical states for Schrödinger-Poisson systems with critical frequency, Z. Angew. Math. Phys., 71 (2020), Paper No. 5, 15 pp. · Zbl 1433.35065
[29] H. Zhang and F. Zhang, Multiplicity of semiclassical states for fractional Schrödinger equations with critical frequency, Nonlinear Anal., 190 (2020), 111599, 15pp. · Zbl 1430.35072
[30] L. Zhao; F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346, 155-169 (2008) · Zbl 1159.35017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.