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

By Giulio Ciraolo, Alessio Figalli, A. Roncoroni. Geometric and Functional Analysis, 2019

118 citations recorded by the source corpus.

Given n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document} and 1<p<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<n$$\end{document}, we consider the critical p-Laplacian equation Δpu+up∗-1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _p u + u^{p^*-1}=0$$\end{document}, 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9ec2a9fe611437f3c12445cb4bda2230ce5b0f32

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In Applied Mathematics

1. D. Monticelli
2. A. Roncoroni
3. G. Catino
4. Xiuxiong Chen
5. Yuanqi Wang

In Computational Theory and Mathematics

1. A. Roncoroni
2. D. Monticelli
3. G. Catino

In Mathematical Physics

1. A. Roncoroni
2. D. Monticelli

Citations in the corpus

• On the critical p-Laplace equation (2022, background): “We mention that this result has been recently generalized in [7] in the anisotropic setting (see also [11]) and in convex cones of Rn (see also [20]) and in [5, 17, 21] in the Riemannian setting (see Appendix A for a more detailed discussion).”