Existence and uniqueness of solution for a unilateral problem in Sobolev spaces with variable exponent

We study the existence and uniqueness of the obstacle problem associated to the equation $$ -\operatorname{div}(a(x,u,\nabla u)+\phi(u))+g(x,u)=f-\operatorname{div}F $$ in the framework of Sobolev spaces with variable exponent, where $F\in (L^{r(\cdot)}(\varOmega))^N$ and $ f\in L^{q(\cdot)}(\varOmega)$ with $$ \begin{cases} r(x) \gt \frac{N}{p(x)-1},\quad r(x)\geq p’(x)& \forall x \in \varOmega,\\ q(x) \gt \max\biggl(\frac{N}{p(x)},1\biggr),\quad q(x)\geq p’(x)& \forall x \in\varOmega, \end{cases} $$ for a log-Lipschitz function $p:\overline \varOmega \to [1,+\infty)$.