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−ΔNu=f(x,u)|x|a+h(x)in Ω, where u∈W01,N(Ω), ΔNu=div(|∇u|N−2∇u)ΔNu=div(|∇u|N−2∇u) is the N  -Laplacian, a∈[0,N)a∈[0,N), Ω   is a smooth bounded domain in RNRN (N⩾2N⩾2) containing the origin and h∈(W01,N(Ω))⁎=W−1,N′ is a small perturbation, h≢0h≢0. Motivated by a singular Trudinger–Moser inequality, we study the case when f(x,s)f(x,s) has the maximal growth on s   which allows to treat this problem variationally in the Sobolev space W01,N(Ω).