Critical blowup exponent to a class of semilinear elliptic equations with constraints in higher dimension - local properties

TL;DR

This paper investigates the blowup behavior and solution classification of a class of constrained semilinear elliptic equations in higher dimensions, revealing parallels to Liouville equations in two dimensions.

Contribution

It introduces a classification of entire solutions, establishes a $ ext{sup}+ ext{inf}$ inequality, and describes the quantized blowup mechanism for these equations.

Findings

Classification of entire solutions in higher dimensions

Establishment of a $ ext{sup}+ ext{inf}$ inequality

Description of the quantized blowup mechanism

Abstract

We study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we establish a classification of entire solutions, the $\sup + \inf$ type inequality and the quantized blowup mechanism.