Symmetry of positive solutions to Choquard type equations involving the fractional \(p\)-Laplacian. (English) Zbl 1462.35020
Summary: We study symmetric properties of positive solutions to the Choquard type equation
\[
(-\Delta)^s_pu+|x|^au=\left(\frac{1}{|x|^{n-\alpha}}*u^q \right)u^r\quad\text{in }\mathbb{R}^n,
\]
where \(0< s< 1,0< \alpha< n,p\geq 2,q >1, r > 0,a\geq 0\) 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 \(a> 0\).
MSC:
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35B09 | Positive solutions to PDEs |
| 35R11 | Fractional partial differential equations |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
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