AI 中文总结
本文针对1<p≤2的分数阶p-拉普拉斯方程，在弱形式下发展移动平面法，证明了有界域中弱解的径向对称性等性质，完善了Chen-Li的相关研究成果。
AI 中文摘要
本文针对奇异范围1<p≤2的分数阶p-拉普拉斯方程，完全在弱形式框架下发展了移动平面法。我们首先建立了反对称函数的小区域原理，并将其应用于证明有界域中分数阶p-拉普拉斯方程非负弱解的径向对称性与单调性。我们还考虑了ℝⁿ中的非局部拟线性Lane–Emden方程(-Δ)ᵖₛu=uᵠ。在Sobolev临界情形下，我们证明了有限能量弱解的径向对称性、单调性及无穷远处的精确渐近行为。在适当的衰减条件下，我们还得到了q>p-1全范围的径向对称性。我们的结果完善了Chen-Li（Adv. Math., 2018:735-758）的工作，该工作针对逐点意义下的C¹,¹解得到了类似结果。
英文摘要
In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional $p$-Laplacian in the singular range $1<p\leq2$. We first establish a small region principle for antisymmetric functions and apply it to prove radial symmetry and monotonicity of nonnegative weak solutions of fractional $p$-Laplacian equations in a bounded domain. We also consider the nonlocal quasilinear Lane--Emden equation $(-Δ)^s_pu=u^q$ in $\mathbb R^n$. In the Sobolev critical case, we establish radial symmetry, monotonicity, and precise asymptotic behavior at infinity for finite-energy weak solutions. Under a suitable decay condition, we also obtain radial symmetry for the full range $q>p-1$. Our results complete those of Chen-Li (Adv. Math., 2018: 735-758), where analogous results were obtained for $C^{1,1}$ solutions in the pointwise sense.
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