On the Dirichlet problem for the n,αn,α-Laplacian with the nonlinearity in the critical growth range

Let Ω⊂RnΩ⊂Rn, n≥2n≥2, be a bounded domain. Applying the Mountain Pass Theorem we prove the existence of a non-trivial weak solution to the Dirichlet problem −div(Φ′(|∇u|)∇u|∇u|)=f(x,u) in Ω, where uu is in the Orlicz–Sobolev space W01LΦ(Ω) with a Young function of the type Φ(t)≈tnlogα(t)Φ(t)≈tnlogα(t), α0β>0.