Multiple blowing-up solutions to critical elliptic systems in bounded domains

TL;DR

This paper constructs and analyzes solutions that blow up in critical elliptic systems within bounded domains, revealing the influence of nonlinear characteristics and domain geometry on solution behavior.

Contribution

It introduces a function governing blow-up points and rates, proving the existence of single and multiple blowing-up solutions, and provides new methods for studying such phenomena.

Findings

Existence of single blowing-up solutions in general domains

Construction of domains allowing multiple blowing-up solutions

Identification of a governing function reflecting nonlinear characteristics

Abstract

We construct families of blowing-up solutions to elliptic systems on smooth bounded domains in the Euclidean space, which are variants of the critical Lane-Emden system and analogous to the Brezis-Nirenberg problem. We find a function which governs blowing-up points and rates, observing that it reflects the strong nonlinear characteristic of the system. By using it, we also prove that a single blowing-up solution exists in general domains, and construct examples of contractible domains where multiple blowing-up solutions are allowed to exist. We believe that a variety of new ideas and arguments developed here will help to analyze blowing-up phenomena in related Hamiltonian-type systems.