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.

我们在欧几里得空间上的光滑有界域上构造椭圆系统的爆破解族，这是临界Lane-Emden系统的变体，类似于Brezis-Nirenberg问题。我们发现一个控制爆炸点和爆炸速率的函数，观察到它反映了系统的强非线性特性。通过使用它，我们还证明了单个爆炸解决方案存在于一般域中，并构造了允许多个爆炸解决方案存在的可收缩域示例。我们相信，这里提出的各种新思想和论据将有助于分析相关哈密顿型系统中的爆炸现象。