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arXiv 1906.00622math.AP
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临界各向异性$p$ -Laplacian方程在凸锥上的对称性结果

Symmetry results for critical anisotropic $p$-Laplacian equations in convex cones

Giulio Ciraolo, Alessio Figalli, Alberto Roncoroni

摘要:

给定$n \geq 2$和$1<p<n$,我们考虑临界$p$ -Laplacian方程$Δ_p u + u^{p^*-1}=0$,它对应于Sobolev不等式的临界点.利用移动平面方法,最近已经证明了在整个空间中的正解是可分类的。由于移动平面法强烈依赖于方程和域的对称性,在本文中,我们提供了一种新的方法,这个刘维尔型问题,使我们能够在各向异性设置的解决方案给出一个完整的分类。更准确地说,我们描述了由任何凸锥内部的光滑范数诱导的临界$p$ -拉普拉斯方程的解。此外,利用最优输运,我们证明了任意凸锥内的一类(加权)各向异性Sobolev不等式。

英文摘要:

Given $n \geq 2$ and $1<p<n$, we consider the critical $p$-Laplacian equation $Δ_p u + u^{p^*-1}=0$, which corresponds to critical points of the Sobolev inequality. Exploiting the moving planes method, it has been recently shown that positive solutions in the whole space are classified. Since the moving plane method strongly relies on the symmetries of the equation and the domain, in this paper we provide a new approach to this Liouville-type problem that allows us to give a complete classification of solutions in an anisotropic setting. More precisely, we characterize solutions to the critical $p$-Laplacian equation induced by a smooth norm inside any convex cone. In addition, using optimal transport, we prove a general class of (weighted) anisotropic Sobolev inequalities inside arbitrary convex cones.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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