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Symmetric properties for Choquard equations involving fully nonlinear nonlocal operators. (English) Zbl 1479.35930

Summary: In this paper we consider the following nonlinear nonlocal Choquard equation \[ \mathcal{F}_{\alpha}\left( u(x)\right) +\omega u(x) = C_{n,2s} \Big( |x|^{2s-n} \ast u^q (x)\Big) u^r (x),\, x\in\mathbb{R}^n, \] where \(0<s<1, \, 0< \alpha < 2, \mathcal{F}_{\alpha}\) is the fully nonlinear nonlocal operator: \[ \mathcal{F}_{\alpha} (u(x)) = C_{n,\alpha} P.V.\int_{\mathbb{R}^n} \frac{F(u(x) - u(y))}{\left| x-y \right|^{n + \alpha}} dy. \] The positive solution to nonlinear nonlocal Choquard equation is shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li [W. Chen et al., Calc. Var. Partial Differ. Equ. 56, No. 2, Paper No. 29, 18 p. (2017; Zbl 1368.35110)]. We first turn single equation into equivalent system of equations. Then the key ingredients are to obtain the “narrow region principle” and “decay at infinity” for the corresponding problems. We also get radial symmetry results of positive solution for the Schrödinger-Maxwell nonlocal equation. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities.

MSC:

35R11 Fractional partial differential equations
35A09 Classical solutions to PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
35B09 Positive solutions to PDEs
35R09 Integro-partial differential equations

Citations:

Zbl 1368.35110
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References:

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