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A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\mathbb{R}^N$ (1504.01133v2)

Published 5 Apr 2015 in math.AP

Abstract: In this paper, we will study the following PDE in $\mathbb{R}N$ involving multiple Hardy-Sobolev critical exponents: $$ \begin{cases} \Delta u+\sum_{i=1}{l}\lambda_i \frac{u{2(s_i)-1}}{|x|{s_i}}+u{2^-1}=0\;\hbox{in}\;\mathbb{R}N, u\in D_{0}{1,2}(\mathbb{R}N), \end{cases} $$ where $0<s_1<s_2<\cdots<s_l\<2, 2^\ast:=\frac{2N}{N-2}, \; 2^\ast(s):=\frac{2(N-s)}{N-2}$ and there exists some $k\in [1, l]$ such that $\lambda_i\>0$ 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 and non-existence of the positive ground state solution. The regularity of the least-energy solution are also investigated.

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