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

Abstract

Let $\Omega$ be a $C^1$ open bounded domain in $\R^N$ ($N\geq 3$) with $0\in \partial \Omega$. Suppose that $\partial\Omega$ is $C^2$ at $0$ and the mean curvature of $\partial\Omega$ at $0$ is negative. Consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: $$ \begin{cases} &\Delta u+\lambda_1 \frac{u^{2^*(s_1)-1}}{|x|^{s_1}}+\lambda_2\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}+\lambda_3\frac{u^p}{|x|^{s_3}}=0\;\quad \hbox{in}\;\Omega,\\ &u(x)>0\;\hbox{in}\;\Omega,\;\, u(x)=0\;\hbox{on}\;\partial\Omega, \end{cases} $$ where $0<s_2<s_1<2, 0\leq s_3<2, 2^*(s_i):=\frac{2(N-s_i)}{N-2}, 0\neq \lambda_i\in \R, \lambda_2>0, 1< p\leq 2^*(s_3)-1$. 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.