Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional \(p\)-Laplacian

Abstract

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 \(a>0\).

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Le, P. Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional \(p\)-Laplacian. Acta Appl Math 170, 387–398 (2020). https://doi.org/10.1007/s10440-020-00338-6

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Mathematics Subject Classification (2010)

  • 35R11
  • 35J92
  • 35B06

Keywords

  • Choquard equations
  • Fractional p-Laplacian
  • Symmetry of solutions