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On Elliptic Systems involving critical Hardy-Sobolev exponents (1504.01005v2)

Published 4 Apr 2015 in math.AP

Abstract: Let $\Omega\subset \RN$ ($N\geq 3$) be an open domain which is not necessarily bounded. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -\Delta u-\lambda \frac{|u|{2*(s_1)-2}u}{|x|{s_1}}=\kappa\alpha \frac{1}{|x|{s_2}}|u|{\alpha-2}u|v|\beta\quad &\hbox{in}\;\Omega,\ -\Delta v-\mu \frac{|v|{2*(s_1)-2}v}{|x|{s_1}}=\kappa\beta \frac{1}{|x|{s_2}}|u|{\alpha}|v|{\beta-2}v\quad &\hbox{in}\;\Omega,\ (u,v)\in \mathscr{D}:=D_{0}{1,2}(\Omega)\times D_{0}{1,2}(\Omega), \end{cases}$$ where $s_1,s_2\in (0,2), \alpha>1,\beta>1, \lambda>0,\mu>0,\kappa\neq 0, \alpha+\beta\leq 2*(s_2)$. Here, $2*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We mainly study the critical case (i.e., $\alpha+\beta=2*(s_2)$) when $\Omega$ is a cone (in particular, $\Omega=\R_+N$ or $\Omega=\RN$). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, {\it etc.} In particular, the sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{\alpha,\beta,\lambda,\mu}(\Omega) \Big(\int_\Omega \big(\lambda \frac{|u|{2*(s)}}{|x|s}+\mu \frac{|v|{2(s)}}{|x|s}+2^(s)\kappa \frac{|u|\alpha |v|\beta}{|x|s}\big)dx\Big){\frac{2}{2*(s)}}$$ $$\leq \int_\Omega \big(|\nabla u|2+|\nabla v|2\big)dx$$ for $(u,v)\in {\mathscr{D}} $ will be explored. Further results about the sharp constant $S_{\alpha,\beta,\lambda,\mu}(\Omega)$ with its extremal functions when $\Omega$ is a general open domain will be involved.

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