×

Symmetry and nonexistence of positive solutions for fractional Choquard equations. (English) Zbl 1444.35152

The authors consider the following fractional and nonlocal equation in \(\mathbb R^{n}\), \(n\geq2\): \[ (-\Delta )^{\alpha/2} u=\left( |x|^{\beta-n}*u^{p}\right)u ^{p-1}, \quad u\geq0, \] where \(0<\alpha\), \(\beta<2\), \(1\leq p<+\infty\).
By means of the maximum principle, moving planes and the Kelvin transform, the authors prove that in the subcritical case (\(n/(n -\alpha )\leq p< (n+\beta)/(n-\alpha)\)) the problem has no positive solutions, while in the critical case \(p=(n+\beta)/(n-\alpha)\), the positive solutions are radially symmetric and monotone decreasing about some point.

MSC:

35R11 Fractional partial differential equations
35B06 Symmetries, invariants, etc. in context of PDEs
35B09 Positive solutions to PDEs
35B50 Maximum principles in context of PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] 10.1007/s00209-004-0663-y · Zbl 1059.35037
[2] 10.1017/S0308210511000175 · Zbl 1290.35304
[3] 10.1016/j.aim.2016.11.038 · Zbl 1362.35320
[4] 10.1142/S0218202515500384 · Zbl 1323.35205
[5] 10.1007/BF01214768 · Zbl 0407.35063
[6] 10.1007/s11425-016-0231-x · Zbl 1383.35010
[7] 10.1002/sapm197757293 · Zbl 0369.35022
[8] 10.1007/BF01609845
[9] 10.1016/0362-546X(80)90016-4 · Zbl 0453.47042
[10] 10.1007/s00205-008-0208-3 · Zbl 1185.35260
[11] ; Pitaevskii, J. Exper. Theor. Phys. (USSR), 40, 646 (1961)
[12] 10.3934/dcds.2016.36.1125 · Zbl 1322.31007
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.