We study symmetric properties of positive solutions to the Choquard type equation

$$ (-\Delta )^{s}_{p} u + |x|^{a} u = \left (\frac{1}{|x|^{n-\alpha }}*u^{q} \right ) u^{r} \quad \text{in}\ \mathbb{R}^{n}, $$

where \(0< s<1\), \(0<\alpha <n\), \(p\ge 2\), \(q>1\), \(r>0\), \(a\ge 0\) and \((-\Delta )^{s}_{p}\) is the fractional \(p\)-Laplacian. Via a direct method of moving planes, we prove that every positive solution \(u\) which has an appropriate decay property must be radially symmetric and monotone decreasing about some point, which is the origin if \(a>0\).

中文翻译：

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涉及分数p $ p $ -Laplacian的Choquard型方程正解的对称性

我们研究Choquard型方程正解的对称性质

$$（-\ Delta）^ {s} _ {p} u + | x | ^ {a} u = \ left（\ frac {1} {| x | ^ {n- \ alpha}} * u ^ { q} \ right）u ^ {r} \ quad \ text {in} \ \ mathbb {R} ^ {n}，$$

其中\（0 <s <1 \），\（0 <\ alpha <n \），\（p \ ge 2 \），\（q> 1 \），\（r> 0 \），\（a \ ge 0 \）和\（（-\ Delta）^ {s} _ {p} \）是小数\（p \）- Laplacian。通过直接移动平面的方法，我们证明了每个具有适当衰减特性的正解\（u \）必须是径向对称的，并且单调在某个点上减小，如果\（a> 0 \）则为原点。