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