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On fractional Choquard equations. (English) Zbl 1323.35205

This article deals with non-local problems of the form \[ (-\Delta)^s u + \omega u = (\mathcal{K}_\alpha \ast| u|^p)| u|^{p-2}u, \quad u \in H^s(\mathbb R^N), \] where \(\omega >0\), \(N\geq 3\), \(\alpha \in (0,N)\), \(p>1\) and \(s\in (0,1)\). The kernel is \(\mathcal{K}_\alpha (x) = | x|^{\alpha-N}\) and the Hilbert space \( H^s(\mathbb R^N)\) is defined by \[ H^s(\mathbb R^N) = \left\{ u \in L^2(\mathbb R^N):(-\Delta)^{s/2} u \in L^2(\mathbb R^N) \right\} \] with inner product \[ (u,v) = \int (-\Delta)^{s/2} u (-\Delta)^{s/2} v +\omega\int uv \] and \[ (-\Delta)^s u (x) = -\frac{C(N,s)}{2}\int\frac{u(x+y)-u(x-y)-2u(x)}{| y|^{N+2s}}\, dy. \] The authors prove cases of existence, multiplicity, and nonexistence of solutions. In the existence cases, the regularity, symmetry and asymptotic behavior of solutions are also discussed.

MSC:

35R11 Fractional partial differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B06 Symmetries, invariants, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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References:

[1] A. Baernstein, Partial Differential Equations of Elliptic Type, Symposia Mathematica XXXV (Cambridge Univ. Press, 1994) pp. 47–91.
[2] Berestycki H., Arch. Ration. Mech. Anal. 82 pp 347– (1983)
[3] DOI: 10.1006/jfan.1993.1133 · Zbl 0790.35021
[4] DOI: 10.1016/j.jmaa.2014.02.063 · Zbl 1332.35066
[5] Bongers A., Z. Angew. Math. Mech. 60 pp 240– (1980)
[6] DOI: 10.1080/03605300600987306 · Zbl 1143.26002
[7] Cazenave T., Textos de Métodos Matemáticos 26, in: An Introduction to Nonlinear Schrödinger Equations (1996)
[8] DOI: 10.1088/0951-7715/26/2/479 · Zbl 1276.35080
[9] DOI: 10.1002/cpa.20116 · Zbl 1093.45001
[10] DOI: 10.1017/S0308210509000584 · Zbl 1215.35146
[11] DOI: 10.1016/j.bulsci.2011.12.004 · Zbl 1252.46023
[12] DOI: 10.1002/cpa.20134 · Zbl 1113.81032
[13] DOI: 10.1016/j.jfa.2012.06.018 · Zbl 1260.35050
[14] DOI: 10.1017/S0308210511000746 · Zbl 1290.35308
[15] DOI: 10.1007/s00220-007-0272-9 · Zbl 1126.35064
[16] DOI: 10.1088/0951-7715/20/5/001 · Zbl 1124.35084
[17] DOI: 10.1103/PhysRevE.66.056108
[18] Lenzmann E., Anal. Partial Differential Equations (Basel) 2 pp 1– (2009)
[19] DOI: 10.4171/JEMS/6 · Zbl 1075.45006
[20] DOI: 10.1002/sapm197757293 · Zbl 0369.35022
[21] DOI: 10.1016/0022-1236(82)90072-6 · Zbl 0501.46032
[22] DOI: 10.1016/0362-546X(80)90016-4 · Zbl 0453.47042
[23] DOI: 10.1016/S0370-1573(00)00070-3 · Zbl 0984.82032
[24] DOI: 10.1088/0305-4470/37/31/R01 · Zbl 1075.82018
[25] Miao C., Forum Math. 27 pp 373– (2015)
[26] DOI: 10.1016/j.jfa.2013.04.007 · Zbl 1285.35048
[27] Pekar S., Untersuchung uber die Elektronentheorie der Kristalle (1954)
[28] Penrose R., Philos. Trans. Roy. Soc. 356 pp 1– (1998)
[29] DOI: 10.1007/s00205-014-0740-2 · Zbl 1361.35199
[30] DOI: 10.1080/03605302.2014.918144 · Zbl 1315.35098
[31] Stein E. M., Princeton Mathematical Series 30, in: Singular Integrals and Differentiability Properties of Functions (1970) · Zbl 0207.13501
[32] DOI: 10.1016/j.jmaa.2013.09.054 · Zbl 1332.35343
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