On a double-variable inequality and elliptic systems involving critical Hardy-Sobolev exponents

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ ($N\ge 3$) be an open domain which is not necessarily bounded. The sharp constant and extremal functions to the following kind of double-variable inequalities $$S_{\alpha ,\beta ,\lambda ,\mu }(\Omega )\left({\int }_{\Omega }\right(\lambda \frac{|u{|}^{2^{\ast }(s)}}{|x{|}^s}+\mu \frac{|v{|}^{2^{\ast }(s)}}{|x{|}^s}+2^{\ast }(s)\kappa \frac{|u{|}^{\alpha }|v{|}^{\beta }}{|x{|}^s})dx{)}^{\frac{2}{2^{\ast }(s)}}$$ $$\le {\int }_{\Omega }(|\nabla u{|}^2+|\nabla v{|}^2)dx$$ for $(u,v)\in \mathcal{D}:=D_0^{1,2}(\Omega )\times D_0^{1,2}(\Omega )$ 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: $$\{\begin{array}{llc}-\Delta u-\lambda \frac{|u{|}^{2^{\ast }(s_1)-2}u}{|x{|}^{s_1}}=\kappa \alpha \frac{1}{|x{|}^{s_2}}|u{|}^{\alpha -2}u|v{|}^{\beta }& in\Omega ,-\Delta v-\mu \frac{|v{|}^{2^{\ast }(s_1)-2}v}{|x{|}^{s_1}}=\kappa \beta \frac{1}{|x{|}^{s_2}}|u{|}^{\alpha }|v{|}^{\beta -2}v& in\Omega ,(u,v)\in \mathcal{D}:=D_0^{1,2}(\Omega )\times D_0^{1,2}(\Omega ),\end{array}$$ where $s_1,s_2\in (0,2),\alpha >1,\beta >1,\lambda >0,\mu >0,\kappa \ne 0,\alpha +\beta \le 2^{\ast }(s_2)$. Here, $2^{\ast }(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We mainly study the critical case (i.e., $\alpha +\beta =2^{\ast }(s_2)$) 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.}

78 pages. arXiv admin note: substantial text overlap with arXiv:1504.01005, arXiv:1504.02939

References in corpus (1)

• A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents