Given \(n \ge 2\) and \(1<p<n\), we consider the critical p-Laplacian equation \(\Delta _p u + u^{p^*-1}=0\), 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.

中文翻译：

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凸锥中临界各向异性p -Laplacian方程的对称结果

给定\（n \ ge 2 \）和\（1 <p <n \），我们考虑临界p-拉普拉斯方程\（\ Delta _p u + u ^ {p ^ *-1} = 0 \），对应于Sobolev不等式的临界点。利用移动平面方法，最近已显示出对整个空间中的正解进行了分类。由于移动平面方法强烈依赖于方程和域的对称性，因此在本文中，我们为这一Liouville型问题提供了一种新方法，使我们能够在各向异性的情况下给出溶液的完整分类。更确切地说，我们描述的解决方案，关键p-由任何凸锥内的光滑范数引起的拉普拉斯方程。此外，使用最优输运，我们证明了任意凸锥内的一类（加权）各向异性Sobolev不等式。