关键词: Laplacien, critical Sobolev-Hardy exponent, critical Sobolev exponent, infinitely many solutions, Pohozaev identity

Abstract: In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents $ \left\{\begin{array}{ll} -\Delta u=\mu \vert u \vert ^{2^{*}-2} u +\frac{ \vert u \vert ^{2^{*}(s)-2}u}{ \vert x \vert ^{s}}+ a(x) \vert u \vert ^{q-2} u & \;{\rm in} \; \Omega, \\ u=0 & \;{\rm on} \; \partial \Omega, \end{array}\right.$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with $0\in \partial \Omega$ and all the principle curvatures of $ \partial \Omega$ at 0 are negative, $a \in \mathcal{C}^{1}(\bar{\Omega}, \mathbb{R^{\ast}}^{+}), $ $ \mu> 0, $ $0<s<2, $ $1<q<2$ and $N > 2\frac{q+1}{q -1}.$ By $2^{*}:=\frac{2 N}{N-2}$ and $2^{*}(s):=\frac{2 (N-s)}{N-2}$ we denote the critical Sobolev exponent and Hardy-Sobolev exponent, respectively.

Key words: Laplacien, critical Sobolev-Hardy exponent, critical Sobolev exponent, infinitely many solutions, Pohozaev identity