A perturbed nonlinear elliptic PDE with two Hardy-Sobolev critical exponents

Let $\mathrm{\Omega }$ be a $C^1$ open bounded domain in ${\mathbb{R}}^N$ ($N\ge 3$) with $0\in \partial \Omega$\mathrm{.}Supposethat$\partial\Omega$is$C^2$at$0$ and the mean curvature of $\mathrm{\partial }\mathrm{\Omega }$ at $0$ is negative. Consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: $$\{\begin{array}{ll}& \mathrm{\Delta }u+{\lambda }_1\frac{u^{2^{\ast }(s_1)-1}}{\mathrm{\mid }x{\mathrm{\mid }}^{s_1}}+{\lambda }_2\frac{u^{2^{\ast }(s_2)-1}}{\mathrm{\mid }x{\mathrm{\mid }}^{s_2}}+{\lambda }_3\frac{u^p}{\mathrm{\mid }x{\mathrm{\mid }}^{s_3}}=0\text{in}\mathrm{\Omega },\\ & u(x)>0\text{in}\mathrm{\Omega },u(x)=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $0<s_2<s_1<2, 0\leq s_3<2, 2^(s_i):=\frac{2(N-s_i)}{N-2}, 0\neq \lambda_i\in \R, \lambda_2>0, 1< p\leq 2^(s_3)-1$. The existence of ground state solution is studied under different assumptions via the concentration compactness principle and the Nehari manifold method. We also apply a perturbation method to study the existence of positive solution.