On a singular and nonhomogeneous N-Laplacian equation involving critical growth

In this paper we apply minimax methods to obtain existence and multiplicity of weak solutions for singular and nonhomogeneous elliptic equation of the form − Δ N u = f ( x , u ) | x | a + h ( x ) in Ω , where u ∈ W 0 1 , N ( Ω ) , Δ N u = div ( | ∇ u | N − 2 ∇ u ) is the N-Laplacian, a ∈ [ 0 , N ) , Ω is a smooth bounded domain in R N ( N ⩾ 2 ) containing the origin and h ∈ ( W 0 1 , N ( Ω ) ) ⁎ = W − 1 , N ′ is a small perturbation, h ≢ 0 . Motivated by a singular Trudinger–Moser inequality, we study the case when f ( x , s ) has the maximal growth on s which allows to treat this problem variationally in the Sobolev space W 0 1 , N ( Ω ) .