  In this paper, we prove several regularity results for the heterogeneous,
two-phase free boundary problems $\mathcal
{J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla u)+\lambda_{+}
(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+gu\big)\text{d}x\rightarrow
\text{min}$ under non-standard growth conditions. Included in such problems are
heterogeneous jets and cavities of Prandtl-Batchelor type with $\gamma=0$,
chemical reaction problems with $0<\gamma<1$, and obstacle type problems with
$\gamma=1$. Our results hold not only in the degenerate case of $p> 2$ for
$p-$Laplace equations, but also in the singular case of $1<p<2$, which are
extensions of \cite{LdT}.
