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Symmetry and nonexistence results for a fractional Choquard equation with weights

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  • Let $ u $ be a nonnegative solution to the equation

    $ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * |x|^a u^p \right) |x|^a u^{p-1} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $

    where $ n \ge 2 $, $ 0 < \alpha < 2 $, $ 0 < \beta < n $ and $ a>\max\{ -\alpha, -\frac{\alpha+\beta}{2} \} $. By exploiting the method of scaling spheres and moving planes in integral forms, we show that $ u $ must be zero if $ 1\le p<\frac{n+\beta+2a}{n-\alpha} $ and must be radially symmetric about the origin if $ a<0 $ and $ \frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha} $.

    Mathematics Subject Classification: 35R11, 35B06, 35B53, 45G10.


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