Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u\in W_0^1{L}^{\Phi }\left(\Omega \right)and-div\left({\Phi }^{\text{'}}\left(\left|\nabla u\right|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\left|u\right|\right)\frac{u}{\left|u\right|}=f\left(x,u\right)+\mu h\left(x\right)in\Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω =...