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

Let $Ω$ be a $C^1$ open bounded domain in $\R^N$ ($N\geq 3$) with $0\in \partial Ω$. Suppose that $\partialΩ$ is $C^2$ at $0$ and the mean curvature of $\partialΩ$ at $0$ is negative. Consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: $\begin{cases} &Δu+λ_1 \frac{u^{2^*(s_1)-1}}{|x|^{s_1}}+λ_2\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}+λ_3\frac{u^p}{|x|^{s_3}}=0\;\quad \hbox{in}\;Ω,\\ &u(x)>0\;\hbox{in}\;Ω,\;\, u(x)=0\;\hbox{on}\;\partialΩ, \end{cases}$ where $0<s_2<s_1<2, 0\leq s_3<2, 2^*(s_i):=\frac{2(N-s_i)}{N-2}, 0\neq λ_i\in \R, λ_2>0, 1< p\leq 2^*(s_3)-1$. 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.

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