On Elliptic Systems involving critical Hardy-Sobolev exponents (Part II)

Abstract

This paper is the second part of a work devoted to the study of elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\{\begin{array}{ll}-\Delta u-\lambda \frac{|u{|}^{2^{\ast }(s_1)-2}u}{|x{|}^{s_1}}=\kappa \alpha \frac{1}{|x{|}^{s_2}}|u{|}^{\alpha -2}u|v{|}^{\beta }& in\Omega ,\\ -\Delta v-\mu \frac{|v{|}^{2^{\ast }(s_1)-2}v}{|x{|}^{s_1}}=\kappa \beta \frac{1}{|x{|}^{s_2}}|u{|}^{\alpha }|v{|}^{\beta -2}v& in\Omega ,\\ \kappa >0,(u,v)\in \mathcal{D}:=D_0^{1,2}(\Omega )\times D_0^{1,2}(\Omega ),& \end{array}$$ where $s_1\ne s_2\in (0,2),\alpha >1,\beta >1,\lambda >0,\mu >0,\kappa >0,\alpha +\beta =2^{\ast }(s_2)$. Here, $2^{\ast }(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. When $\Omega$ is a cone (especially $\Omega ={\mathbb{R}}_{+}^N$ or $\Omega ={\mathbb{R}}^N$), we study the existence of positive ground state solution.

This paper has been withdrawn due to that the work of this article has been merged into the new version of article 1504.01005

References in corpus (1)

• On Elliptic Systems involving critical Hardy-Sobolev exponents

Cited by in corpus (1)

• On Elliptic Equations and Systems involving critical Hardy-Sobolev exponents (non-limit case)