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The aim of this paper is to study the heterogeneous optimization problem  \begin{align*} \mathcal {J}(u)=\int_{\Omega}(G(|\nablau|)+qF(u^+)+hu+\lambda_{+}\chi_{\{u>0\}} )\text{d}x\rightarrow\text{min},\end{align*} in the class of functions $ W^{1,G}(\Omega)$ with $ u-\varphi\inW^{1,G}_{0}(\Omega)$, for a given function $\varphi$, where $W^{1,G}(\Omega)$is the class of weakly differentiable functions with $\int_{\Omega}G(|\nablau|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditionsof Lieberman's type that allow for a different behavior at $0$ and at $\infty$.{}{Moreover, $F$ allows for a subcritical growth.} Given functions $q,h$ andconstant $\lambda_+\geq 0$, we address several regularity results forminimizers of $\mathcal {J}(u)$, including local $C^{1,\alpha}-$, and localLog-Lipschitz continuities for minimizers of $\mathcal {J}(u)$ with$\lambda_+=0$, and {}{$\lambda_+\geq 0$} respectively. We also establish growthrate near the free boundary for each non-negative minimizer of $\mathcal{J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. Furthermore, underadditional assumption that $F\in C^1([0,+\infty); [0,+\infty))$, localLipschitz regularity is carried out for non-negative minimizers of $\mathcal{J}(u)$ with $\lambda_{+}>0$.

A minimum problem with free boundary and subcritical growth in Orlicz spaces