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

Ma, Pei; Shang, Xudong; Zhang, Jihui

Pac. J. Math. 304, No. 1, 143-167 (2020).

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.

Reviewer: Gaetano Siciliano (São Paulo)

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

Keywords:

method of moving planes; fractional Laplacian; Choquard equation

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