A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in RN
A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in RN
复制标题
RN 中具有多个 Hardy-Sobolev 临界指数的非线性椭圆偏微分方程
DOI：
10.1016/j.jde.2021.05.027
复制
发表时间：
2021
影响因子：
2.4
通讯作者：
W. Zou
中科院分区：
文献类型：
--
作者：
X. Zhong;W. Zou
在本文中，我们将研究在$\mathbb{R}^N$中涉及多个哈代 - 索伯列夫临界指数的以下偏微分方程：
$$
\begin{cases}
\Delta u + \sum_{i = 1}^{l} \lambda_i\frac{u^{2^*(s_i) - 1}}{\vert x\vert^{s_i}} + u^{2^ * - 1} = 0\quad 在\ \mathbb{R}^N中,\\
u \in D_{0}^{1,2}(\mathbb{R}^N).
\end{cases}
$$
其中$0 < s_i < 2$，$2^*(s_i)=\frac{2(N - s_i)}{N - 2}$，$N\geq 3$，$\lambda_i > 0$对于$1\leq i\leq k$；$\lambda_i < 0$对于$k + 1\leq i\leq l$。我们开发了一种有趣的方法来研究这类涉及混合符号参数的方程。我们证明了正基态解的存在性。还研究了最小能量解的正则性。
In this paper, we will study the following PDE in $\R^N$ involving multiple Hardy-Sobolev critical exponents:.$$\begin{cases}.\Delta u+\sum_{i=1}^{l}\lambda_i \frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\;\hbox{in}\;\R^N,\\.u\in D_{0}^{1,2}(\R^N),.\end{cases}$$.where $00$ for $1\leq i\leq k$; $\lambda_i<0$ for $k+1\leq i\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence of the positive ground state solution. The regularity of the least-energy solution is also investigated.