## 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
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### References:

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