Generalized nn-Laplacian: Quasilinear nonhomogenous problem with critical growth

Applying the generalized Moser–Trudinger inequality, the Mountain Pass Theorem and the Ekeland Variational Principle we study the existence of non-trivial weak solutions to the problem −div(Φ′(|∇u|)∇u|∇u|)+V(x)Φ′(|u|)u|u|=f(x,u)+μh(x),x∈Rn,u∈W1LΦ(Rn) where ΦΦ is a Young function such that the space W1LΦ(Rn)W1LΦ(Rn) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity f(x,t)f(x,t) has the corresponding critical growth, V(x)V(x) is a continuous potential, h∈(LΦ(Rn))∗h∈(LΦ(Rn))∗ is a nontrivial continuous function and μ>0μ>0 is a small parameter.