Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic field. (English) Zbl 1496.35430

Summary: In this paper, we consider the following critical fractional magnetic Choquard equation: \[ \begin{aligned} {\varepsilon }^{2s}{(-\Delta)}_{A/\varepsilon}^su+V(x)u \,= \; &{\varepsilon}^{\alpha-N} \left(\int_{\mathbb{R}^N} \frac{|u(y)|^{2_{s,\alpha}^\ast}}{|x-y|^\alpha}\mathrm{d}y\right) |u|^{2_{s,\alpha}^\ast-2}u \\ & +{\varepsilon }^{\alpha-N}\left(\int_{\mathbb{R}^N}\frac{F(y,| u(y)|^2)}{| x-y|^\alpha} \mathrm{d}y\right) f(x,|u|^2)u \quad\text{in }\mathbb{R}^N, \end{aligned} \] where \(\varepsilon > 0\), \(s\in (0,1)\), \(\alpha \in (0,N)\), \(N> \max \{2 \mu +4s,2s+\alpha /2\}\), \(2_{s,\alpha}^{\ast}=\frac{2N-\alpha}{N-2s}\) is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, \((-\Delta)_A^s\) stands for the fractional Laplacian with periodic magnetic field A of \(C^{0,\mu}\)-class with \(\mu \in (0,1]\) and \(V\) is a continuous potential and allows to be sign-changing. Under some mild assumptions imposed on \(V\) and \(f\), we establish the existence of at least one ground state solution.


35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35J61 Semilinear elliptic equations
35R09 Integro-partial differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: DOI


[1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), 423-443. · Zbl 1059.35037
[2] C. O. Alves, G. M. Figueiredo, and M. Yang, Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field, Asymptot. Anal. 96 (2016), 135-159. · Zbl 1339.35278
[3] C. O. Alves, F. Gao, M. Squassina, and M. Yang, Singularly perturbed critical Choquard equations, J. Differ. Equ. 263 (2017), 3943-3988. · Zbl 1378.35113
[4] C. O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differ. Equ. 257 (2014), 4133-4164. · Zbl 1309.35036
[5] C. O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 23-58. · Zbl 1366.35050
[6] V. Ambrosio, Concentration phenomena for a fractional Choquard equation with magnetic field, Dyn. Partial Differ. Equ. 16 (2019), 125-149. · Zbl 1409.35211
[7] V. Ambrosio, Multiplicity and concentration results for a fractional Choquard equation via penalization method, Potential Anal. 50 (2019), 55-82. · Zbl 1408.35001
[8] V. Ambrosio and P. d’Avenia, Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity, J. Differ. Equ. 264 (2018), 3336-3368. · Zbl 1447.35345
[9] H. Bueno, N. da Hora Lisboa, and L. L. Vieira, Nonlinear perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent, Z. Angew. Math. Phys. 71 (2020), no. 143, 26. · Zbl 1464.35311
[10] B. Buffoni, L. Jeanjean, and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993), 179-186. · Zbl 0789.35052
[11] D. Cassani, J. Van Schaftingen, and J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 1377-1400. · Zbl 1437.35329
[12] S. Chen and L. Xiao, Existence of a nontrivial solution for a strongly indefinite periodic Choquard system, Calc. Var. Partial Differ. Equ. 54 (2015), 599-614. · Zbl 1323.35028
[13] S. Cingolani, S. Secchi, and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 973-1009. · Zbl 1215.35146
[14] R. Clemente, J. C. de Albuquerque, and E. Barboza, Existence of solutions for a fractional Choquard-type equation in R with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), Paper no. 16, 13 pp. · Zbl 1466.35175
[15] A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), 225-236. · Zbl 1084.26009
[16] P. d’Avenia and M. Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var. 24 (2018), 1-24. · Zbl 1400.49059
[17] M. del Pino, and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ. 4 (1996), 121-137. · Zbl 0844.35032
[18] Y. Deng, L. Jin, and S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differ. Equ. 253 (2012), 1376-1398. · Zbl 1248.35058
[19] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. · Zbl 1252.46023
[20] F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math. 61 (2018), 1219-1242. · Zbl 1397.35087
[21] Z. Gao, X. Tang, and S. Chen, On existence and concentration behavior of positive ground state solutions for a class of fractional Schrödinger-Choquard equations, Z. Angew. Math. Phys. 69 (2018), Paper no. 122, 21 pp. · Zbl 1401.35314
[22] N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H Poincaré Anal. Non Linéaire 6 (1989), 321-330. · Zbl 0711.58008
[23] T. Guo and X. Tang, Ground state solutions for nonlinear Choquard equations with inverse-square potentials, Asymptot. Anal. 117 (2020), 141-160. · Zbl 1475.35007
[24] T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, mathematical physics, spectral theory and stochastic analysis, in Operator Theory: Advances and Applications, vol. 232, Birkhäuser/Springer Basel AG, Basel, 2013, pp. 247-297. · Zbl 1266.81079
[25] Q. Li, K. Teng, and J. Zhang, Ground state solutions for fractional Choquard equations involving upper critical exponent, Nonlinear Anal. 197 (2020), 111846, 11. · Zbl 1440.35113
[26] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies Appl. Math. 57 (1976/77), 93-105. · Zbl 0369.35022
[27] E. H. Lieb, M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence, 2001. · Zbl 0966.26002
[28] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire. 1 (1984), 109-145. · Zbl 0541.49009
[29] M. Liu and Z. W. Tang, Pseudoindex theory and Nehari method for a fractional Choquard equation, Pacific J. Math. 304 (2020), 103-142. · Zbl 1437.35304
[30] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455-467. · Zbl 1185.35260
[31] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153-184. · Zbl 1285.35048
[32] V. Moroz, and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ. 52 (2015), 199-235. · Zbl 1309.35029
[33] V. Moroz, and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), 773-813. · Zbl 1360.35252
[34] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. · Zbl 0058.45503
[35] D. Qin, V. D. Rădulescu, and X. Tang, Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations, J. Differ. Equ., 275 (2021), 652-683. · Zbl 1456.35187
[36] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in RN, J. Math. Phys. 54 (2013), Paper no. 031501, 17 pp. · Zbl 1281.81034
[37] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67-102. · Zbl 1323.35202
[38] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
[39] Q. Wu, D. Qin, and J. Chen, Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials, Nonlinear Anal. 197 (2020), Paper no. 111863, 20 pp. · Zbl 1440.35144
[40] Z. Yang and F. Zhao, Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth, Adv. Nonlinear Anal. 10 (2021), 732-774. · Zbl 1466.35304
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.