I want to solve the following first order PDE
$$
(\star)\quad\begin{cases}
\nabla u\cdot  \nabla\xi=f \quad\text{in}\,\Omega, \\
u\mid_{\partial \Omega}=0 
\end{cases}
$$
where $\xi\in C^2(\overline{\Omega})$ and $|\nabla \xi|$ does not vanish everywhere. 

If we ignore the boundary condition, this problem is not difficult. By setting $w=e^{-\lambda \xi} u$, the original equation is equivalent to 
$$
\lambda  w+\frac{1}{|\nabla \xi|^2}\nabla w\cdot \nabla \xi=\frac{e^{-\lambda \xi} }{|\nabla \xi|^2}f 
$$

For sufficient large $\lambda$, we can consider the the following elliptic equation 
$$
-\epsilon\Delta w+\lambda w+\nabla w\cdot \Xi=F \quad \text{in}\ \Omega, 
$$
where $\epsilon>0$ is a parameter and $\Xi=\frac{1}{|\nabla \xi|^2}\nabla \xi$.  By extending the coefficients $\Xi$ and $F$  to a large domain $U$ and equip it with a necessary boundary condition. i.e. 
$$
-\epsilon\Delta w+\lambda w+\nabla w\cdot \Xi=F \quad \text{in}\ U. \quad 
w\mid_{\partial U}=0 
$$
Such elliptic equation has a unique solution $w^\epsilon$ for each $\epsilon>0$. In addition, by
multiplying both hands with $-\Delta w^\epsilon$ and integration by parts, we can show that 
$$
\|w^{\epsilon}\|_{1,U}\lesssim\|F\|_{1,U}
$$
By passing limit $\epsilon \to 0$, one can find at least one solution. 
However, this technique does not work for the case with boundary condition.

**i) For which $F$ can we ensure that the equation $(\star)$ has a solution $u\in H_0^1(\Omega)$.  
ii) Is there some papers/book for PDE for  the equation $(\star)$**