Solutions for Singular Quasilinear Equations of N-Laplacian Type with Critical Exponential Growth

In this paper, we study the following singular quasilinear equation, - Δ N u + V ( x ) | u | N - 2 u = f ( x , u ) | x | β , in R N , N ≥ 2 , where Δ N u : = div ( | ∇ u | N - 2 ∇ u ) , 0 < β < N , V ∈ C ( R N , R ) and f has critical exponential growth at infinity. We establish a compactness lemma for functional ∫ R N F ( x , u ) | x | β d x and develop some delicate analyses to deal with non-standard difficulties caused by singular term f ( x , u ) | x | β and the compactness issue. We introduce some natural assumptions on the nonlinearity f and obtain the existence of Mountain-pass type solution and ground state solution for above equation. Moreover, two distinct solutions, one has positive energy while the other one has negative energy, are established for the above equation with a perturbation.