Harnack inequalities and quantization properties for the $$n-$$ Liouville equation

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-29 DOI:10.1007/s00526-024-02777-7

Pierpaolo Esposito, Marcello Lucia

引用次数: 0

Abstract

We consider a quasilinear equation involving the $n-$Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known that the exponential nonlinearity converges, up to a subsequence, to a sum of Dirac measures. By performing a precise local asymptotic analysis we complete such a result by showing that the corresponding Dirac masses are quantized as multiples of a given one, related to the mass of limiting profiles after rescaling according to the classification result obtained by the first author in Esposito (Ann. Inst. H. Poincaré Anal. Non Linéaire 35(3), 781–801, 2018). A fundamental tool is provided here by some Harnack inequality of “sup+inf" type, a question of independent interest that we prove in the quasilinear context through a new and simple blow-up approach.

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哈纳克不等式和 $$n-$$ 柳维尔方程的量子化性质

我们考虑的是一个涉及（n-\）拉普拉奇和指数非线性的准线性方程，这个问题包括作为特例的平面中著名的Liouville方程。众所周知，对于一个非紧凑的解序列，指数非线性在一个子序列之前收敛于狄拉克量之和。通过进行精确的局部渐近分析，我们完善了这一结果，表明相应的狄拉克质量被量化为给定质量的倍数，这与第一作者在埃斯波西托（Ann.H. Poincaré Anal.Non Linéaire 35(3), 781-801, 2018）。在这里，"sup+inf "类型的一些哈纳克不等式提供了一个基本工具，我们通过一种新的简单吹胀方法在准线性背景下证明了一个独立关注的问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.