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

Abstract

Let $\Omega$ be a $C^1$ open bounded domain in ${\mathbb{R}}^N$ ($N\ge 3$) 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: $$\{\begin{array}{ll}& \Delta u+{\lambda }_1\frac{u^{2^{\ast }(s_1)-1}}{|x{|}^{s_1}}+{\lambda }_2\frac{u^{2^{\ast }(s_2)-1}}{|x{|}^{s_2}}+{\lambda }_3\frac{u^p}{|x{|}^{s_3}}=0in\Omega ,\\ & u(x)>0in\Omega ,u(x)=0on\partial \Omega ,\end{array}$$ where $0<s_2<s_1<2,0\le s_3<2,2^{\ast }(s_i):=\frac{2(N-s_i)}{N-2},0\ne {\lambda }_i\in \mathbb{R},{\lambda }_2>0,1<p\le 2^{\ast }(s_3)-1$. 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.

26 pages, Communications in Contemporary Mathematics,2015