Find a bilinear form and a linear form

Question

My problem: Let $\Omega$ a smooth, bounded open set of $\mathbb{R}^N$. We want to show the existence of $u \in H_0^1(\Omega)$ solution of $$ \left\{\begin{array}{l} V(x) \cdot \nabla_x u-\Delta_x u=f, \quad \text { a.e. in } \Omega, \\ u(x)=0, \text { a.e. in } \partial \Omega, \end{array}\right. $$ where $f \in L^2(\Omega)$ and $V: \Omega \rightarrow \mathbb{R}^N$ is a smooth function such that $\operatorname{div} V=\sum_{i=1}^N \partial_i V=0$.
Find $a(\cdot, \cdot)$ a bilinear form, and $L(\cdot)$ a linear form such that the equation above is equivalent to $$ a(u, v)=L(v), $$ for any $v \in H_0^1(\Omega)$.
My attempt: I intend to use Lax-Milgram theorem but i don't know how to choose the suitable condition for test function. Please help me to solve this problem. This is the first time i have approached to this field. If I make any mistake, please let me know.

Answer 1

Suppose that $u\in C^2(\Omega) \cap C(\overline \Omega)$ satisfies the PDE in the classical sense. Let $\phi \in C^\infty_0(\Omega)$. Multiplying the PDE by $\phi$ then integrating over $\Omega$ gives $$\int_\Omega \phi V \cdot \nabla u \, dx - \int_\Omega \phi \Delta u\, dx = \int_\Omega f \phi \, dx. $$ Integrating by parts gives $$\int_\Omega \phi \Delta u \, dx = -\int_\Omega \nabla u \cdot\nabla \phi \, dx. $$ This implies that$$ \int_\Omega \phi V \cdot \nabla u \, dx +\int_\Omega \nabla u \cdot \nabla \phi \, dx = \int_\Omega f \phi \, dx. $$

Hence, if we now have that $u \in H^1_0(\Omega)$, a suitable weak formulation would be $a(u,v) = L(v)$ for all $v\in C^\infty_0(\Omega)$ (by density this then holds for all $v\in H^1_0(\Omega)$) where $$a(u,v)= \int_\Omega v V \cdot \nabla u \, dx +\int_\Omega \nabla u \cdot \nabla v \, dx $$ and $$ L(v) =\int_\Omega fv \, dx. $$