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- W2769904115 abstract "Let $Omegasubset mathbb{R}^N$ ($Ngeq 3$) be an open domain which is not necessarily bounded. The sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{alpha,beta,lambda,mu}(Omega) Big(int_Omega big(lambda frac{|u|^{2^*(s)}}{|x|^s}+mu frac{|v|^{2^*(s)}}{|x|^s}+2^*(s)kappa frac{|u|^alpha |v|^beta}{|x|^s}big)dxBig)^{frac{2}{2^*(s)}}$$ $$leq int_Omega big(|nabla u|^2+|nabla v|^2big)dx$$ for $(u,v)in {mathscr{D}}:=D_{0}^{1,2}(Omega)times D_{0}^{1,2}(Omega)$ 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. For this goal, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$begin{cases} -Delta u-lambda frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=kappaalpha frac{1}{|x|^{s_2}}|u|^{alpha-2}u|v|^betaquad &hbox{in};Omega, -Delta v-mu frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=kappabeta frac{1}{|x|^{s_2}}|u|^{alpha}|v|^{beta-2}vquad &hbox{in};Omega, (u,v)in mathscr{D}:=D_{0}^{1,2}(Omega)times D_{0}^{1,2}(Omega), end{cases}$$ where $s_1,s_2in (0,2), alpha>1,beta>1, lambda>0,mu>0,kappaneq 0, alpha+betaleq 2^*(s_2)$. 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, $Omega=mathbb{R}_+^N$ or $Omega=mathbb{R}^N$). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, {it etc.}" @default.
- W2769904115 created "2017-12-04" @default.
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- W2769904115 date "2017-11-27" @default.
- W2769904115 modified "2023-09-27" @default.
- W2769904115 title "On a double-variable inequality and elliptic systems involving critical Hardy-Sobolev exponents" @default.
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