Harnack inequalities and quantization properties for the n-Liouville equation

TL;DR

This paper investigates the quantization and asymptotic behavior of solutions to a quasilinear n-Liouville equation involving the n-Laplacian and exponential nonlinearity, extending known results and introducing a new Harnack inequality.

Contribution

It provides a detailed local asymptotic analysis of solution sequences and establishes a new

Findings

Dirac masses are quantized as multiples of a fundamental mass.

A new Harnack inequality of

contribution

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 P. Esposito, A classification result for the quasi-linear Liouville equation. Ann. Inst. H. Poincar\'e Anal. Non Lin\`eaire 35 (2018), no. 3, 781--801. 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.