论文标题
对哈密顿类型的关键椭圆体系的积极多润滑解决方案的存在和非成分
Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type
论文作者
论文摘要
本文讨论了哈密顿式类型的以下关键椭圆形系统，这些系统是关键车道填充系统的变体，并且类似于规定的曲率问题：\ begin {equination*} \ begin {cases}}-ΔU_1= k_1 = k_1 = k_1（y） -qu_2 = k_2（y）u_1^{q}，\ y \ in \ sathbb {r}^n，\\ u_1，u_2> 0，\ end {cases} \ end {cases} \ end ext {quequation*}其中$ n \ geq 5，p，p，p，q \ in（1，q \ in（1，q \ in（1，q \ in） $ \ frac1 {p+1}+\ frac1 {q+1} = \ frac {n-2} n $，$ k_1（y）$和$ k_2（y）$是积极的径向电势。首先，在$ k_1，k_2 $和指数$ p，q $的一定范围的适当条件下，我们构建了一个无限的非radial阳性矢量解决方案的序列，它们的能量可以任意大。此外，我们通过使用各种Pohozaev身份来证明一种非分类的类型，这是独立兴趣的。无限期的线性操作员和强烈耦合的非线性使哈密顿型系统与梯度类型的系统和单个关键椭圆方程式在规定的曲率问题的研究中都形成鲜明对比。值得注意的是，在高维情况下，$（n \ geq5）$，对于哈密顿式或梯度类型的关键椭圆形系统的许多无限泡沫解决方案都没有结果。
This paper deals with the following critical elliptic systems of Hamiltonian type, which are variants of the critical Lane-Emden systems and analogous to the prescribed curvature problem: \begin{equation*} \begin{cases} -Δu_1=K_1(y)u_2^{p},\ y\in \mathbb{R}^N,\\ -Δu_2=K_2(y)u_1^{q}, \ y\in \mathbb{R}^N,\\ u_1,u_2>0, \end{cases} \end{equation*} where $N\geq 5, p,q\in(1,\infty)$ with $\frac1{p+1}+\frac1{q+1}=\frac{N-2}N$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. At first, under suitable conditions on $K_1,K_2$ and the certain range of the exponents $p,q$, we construct an unbounded sequence of non-radial positive vector solutions, whose energy can be made arbitrarily large. Moreover, we prove a type of non-degeneracy result by use of various Pohozaev identities, which is of great interest independently. The indefinite linear operator and strongly coupled nonlinearities make the Hamiltonian-type systems in stark contrast both to the systems of Gradient type and to the single critical elliptic equations in the study of the prescribed curvature problems. It is worth noting that, in higher-dimensional cases $(N\geq5)$, there have been no results on the existence of infinitely many bubbling solutions to critical elliptic systems, either of Hamiltonian or Gradient type.
