A priori estimates and Blow-up behavior for solutions of $-Q_{N}u=Ve^{u}$ in bounded domain in $\mathbb{R}^{N}$

TL;DR

This paper investigates the properties, a priori estimates, and blow-up behavior of solutions to a class of anisotropic Laplacian equations involving exponential nonlinearities in bounded domains.

Contribution

It introduces new properties of the $Q_N$ operator and provides the first comprehensive blow-up analysis for solutions of $-Q_N u=V e^u$ in bounded domains.

Findings

Established weak maximum principle, comparison principle, and mean value property for $Q_N$.

Derived a priori estimates for solutions.

Analyzed the behavior of solutions at blow-up points.

Abstract

Let $Q_{N}$ be $N$-anisotropic Laplacian operator, which contains the ordinary Laplacian operator, $N$-Laplacian operator and anisotropic Laplacian operator. In this paper, we firstly obtain the properties for $Q_{N}$, which contain the weak maximal principle, the comparison principle and the mean value property. Then a priori estimates and blow-up analysis for solutions of $-Q_{N}u=Ve^{u}$ in bounded domain in $\mathbb{R}^{N}$, $N\geq 2$ are established. Finally, the behavior of sole blow-up point is further considered.