Abstract:

In the paper, we used variational method to consider the following quasilinear elliptic equation

$ -\sum\limits_{i=1}^{N}\frac{\partial}{\partial x_i}\left(|\triangledown u|^{p-2}\frac{\partial u}{\partial x_i}\right)-\mu\frac{|u|^{p-2}u}{|x|^p}=|u|^{p^*-2}u+\lambda g(x)\quad u\in {\cal D}^{1, p}(\mathbb{R} ^N), $

we show that there exists two nontrivial solutions for our problem, one solution is a local minimum and the other is of mountain pass type.

Key words: Quasilinear, Concentration-compactness, Mountian pass