Applications of Mathematics

Let Ω ⊂ R, n > 2, be a bounded connected domain of the class C for some θ ∈ (0, 1]. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem u ∈ W L(Ω), − div ( Φ(|∇u|) ∇u |∇u| ) + V (x)Φ(|u|) u |u| = f(x, u) + μh(x) in Ω, ∂u ∂n = 0 on ∂Ω, where Φ is a Young function such that the space W L(Ω) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V (x) is a continuous potential, h ∈ (L(Ω)) is a nontrivial continuous function, μ > 0 is a small parameter and n denotes the outward unit normal to ∂Ω.