Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent
Xiaomei Sun Wenyi Chen
In this paper, we consider the following semilinear elliptic equations with critical Hardy-Sobolev exponent:

$ -\Delta u+\lambda\frac{u}{|x-a|^2}-\gamma\frac{u}{|x|^2} =\frac{Q(x)}{|x|^s}|u|^{2^*(s)-2}u+g(x,u), u>0$ in $\Omega,$

$ \frac{\partial u}{\partial\nu}+\alpha(x)u=0 $ on $\partial\Omega. $

By variational method, the existence of positive solution is obtained.

keywords: concentration compactness principle. Critical Hardy-Sobolev exponent
Elliptic equations with cylindrical potential and multiple critical exponents
Xiaomei Sun Yimin Zhang
In this paper, we deal with the following problem: \begin{eqnarray*} -\Delta u-\lambda |y|^{-2}u=|y|^{-s}u^{2^{*}(s)-1}+u^{2^{*}-1}\ \ \ in \ \ R^N , y\neq 0\\ u\geq 0 \end{eqnarray*} where $u(x)=u(y,z): R^m\times R^{N-m}\longrightarrow R$, $N\geq 4$, $2 < m < N$, $\lambda < (\frac{m-2}{2})^2$ and $0 < s < 2$, $2^*(s)=\frac{2(N-s)}{N-2}$, $2^*=\frac{2N}{N-2}$. Using the Variational method, we proved the existence of a ground state solution for the case $0 < \lambda < (\frac{m-2}{2})^2$ and the existence of a cylindrical weak solution under the case $\lambda<0$.
keywords: Variational method cylindrical weight multiple critical exponents.

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