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

Abstract

In this paper, we will study the following PDE in ${\mathbb{R}}^N$ involving multiple Hardy-Sobolev critical exponents: $$\{\begin{array}{l}\Delta u+{\sum }_{i=1}^l{\lambda }_i\frac{u^{2^{\ast }(s_i)-1}}{|x{|}^{s_i}}+u^{2^{\ast }-1}=0in{\mathbb{R}}^N,u\in D_0^{1,2}({\mathbb{R}}^N),\end{array}$$ 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 $1\le i\le k$; ${\lambda }_i<0$ for $k+1\le i\le l$. 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.

27 pages

References in corpus (2)

• A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents
• On Elliptic Systems involving critical Hardy-Sobolev exponents (Part II)