im struggling how to get a weak formulation for the following problem. After this i want to apply the Lax-Milgram Theorem to show that there exists a unique solution to this problem.
$$(*)\begin{cases} -\Delta u +c \partial_{x}^{-1}u=f, & \text{in } \Omega=(0,1)^2 \\ u=0, & \text{on } \partial\Omega \end{cases}$$ where $$\partial_{x}^{-1}u(x,y)=\int_{0}^{x} u(z,y) dz$$ and $$f \in L^2(\Omega) \text{ and } c\in \mathbb{R} \text{ is a constant.}$$
I know, the common way is to multiply by a testfunction from a suitable vectorspace V and use integration by parts. My main problem is how to handle the $ \partial_{x}^{-1} $ operator on the step int by parts and get the bilinear form. Thx in advance!