Moving planes for nonlinear fractional Laplacian equation with negative powers. (English) Zbl 1397.35330
Summary: In this paper, we study symmetry properties of positive solutions to the fractional Laplace equation with negative powers on the whole space. We can use the direct method of moving planes introduced by S. Jarohs and T. Weth [Ann. Mat. Pura Appl. (4) 195, No. 1, 273–291 (2016; Zbl 1346.35010)] and W. Chen et al. [Adv. Math. 308, 404–437 (2017; Zbl 1362.35320)]to prove one particular result below. If \(u\in C^{1, 1}_{\mathrm{loc}}(\mathbb{R}^{n})\cap L_{\alpha}\) satisfies
\[
(-\Delta)^{\alpha/2}u(x)+u^{-\beta}(x) = 0,\;\, x\in \mathbb{R}^n,
\]
with the growth/decay property
\[
u(x) = a|x|^{m}+o(1),\;\, \text{as}\; |x|\to \infty,
\]
where \(\frac{\alpha}{\beta+1}<m<1\), \(a>0\) is a constant, then the positive solution \(u(x)\) must be radially symmetric about some point in \(\mathbb{R}^{n}\). Similar result is also true for Hénon type nonlinear fractional Laplace equation with negative powers.
MSC:
| 35R11 | Fractional partial differential equations |
| 35J60 | Nonlinear elliptic equations |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 58J05 | Elliptic equations on manifolds, general theory |
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