论文标题
在具有关键Sobolev指数的哈密顿系统上
On Hamiltonian systems with critical Sobolev exponents
论文作者
论文摘要
在本文中，我们考虑了在有限的光滑域$ω\ subset \ mathbb {r}^n $上构成的关键车道填充系统的低阶扰动，并带有$ n \ geq3 $，灵感来自brezis和nirenberg \ cite {brezisnirenbergg1983}的经典结果。我们解决了为所有维度找到积极解决方案$ n \ geq 4 $的问题。对于临界维度$ n = 3 $，我们显示出一种新现象，而没有观察到标量问题。也就是说，在关键的双曲线上有一些部分，其中所有$ 1 $均匀或亚临界的超级线性扰动存在解决方案，而某些扰动中没有解决方案的部分。
In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain $Ω\subset \mathbb{R}^N$, with $N \geq3$, inspired by the classical results of Brezis and Nirenberg \cite{BrezisNirenberg1983}. We solve the problem of finding a positive solution for all dimensions $N \geq 4$. For the critical dimension $N=3$ we show a new phenomenon, not observed for scalar problems. Namely, there are parts on the critical hyperbola where solutions exist for all $1$-homogeneous or subcritical superlinear perturbations and parts where there are no solutions for some of those perturbations.
