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We prove the existence of entropy solutions to unilateral problems associated to equations of the type $Au-div\left(\varphi \left(u\right)\right)=\mu \in L¹\left(\Omega \right)+W^{-1,p^{\text{'}}\left(·\right)}\left(\Omega \right)$, where A is a Leray-Lions operator acting from $W{₀}^{1,p\left(·\right)}\left(\Omega \right)$ into its dual $W^{-1,p\left(·\right)}\left(\Omega \right)$ and $\varphi \in C⁰(ℝ,{ℝ}^N)$.

We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), ${b(x,u)|}_{t=0}=b(x,u₀)$ in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in $L¹\left(Q\right)+L^{p^{\text{'}}}(0,T;W^{-1,p^{\text{'}}}\left(\Omega \right))$ and b(x,u₀) ∈ L¹(Ω).