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

Abstract

The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_Ω(G(|\nabla u|)+qF(u^+)+hu+λ_{+}χ_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $W^{1,G}(\Omega )$ with $u-\phi \in W_0^{1,G}(\Omega )$, for a given function $\phi$, 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 ${\lambda }_{+}\ge 0$, 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 {}{${\lambda }_{+}\ge 0$} 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 ${\lambda }_{+}>0$.

Regularities of minimizers are established for $F$ satisfying Lieberman's conditions in the first version. We will renew the results in the second version where regularities of minimizers are addressed for $F$ satisfying subcritical conditions