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.


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
Full Text: DOI arXiv


[1] 10.1007/s00209-004-0663-y · Zbl 1059.35037
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