Let $\Omega\subset \mathbb{R}^n$ be open, bounded and connected with $C^1$ boundary. Suppose $q\in L^\infty(\Omega)$ and that $q\geq C>0$ almost everywhere. Consider the following PDE problem: $$-\nabla\cdot(\nabla u) + qu = 0\quad\text{in }\Omega$$ and with zero boundary condition, i.e. $u|_{\partial\Omega} = 0$.
Show that there exists a unique $u\in L^2(\Omega)$ solving the problem.
I've been given a hint that I can use a calculus of variation approach and formulate a minimization problem. For example I know that the PDE $$-\nabla\cdot(\nabla u) = f$$
can be written in weak formulation as $a(u,v) = \ell(v)$ with
$$a(u,v) := \int_\Omega \nabla u\cdot \nabla v\,\mathrm{d}x\quad\quad\text{and }\quad\quad \ell(v) := \int_\Omega fv\,\mathrm{d}x$$ where $v\in H_0^1(\Omega)$ is a test function.
For this problem I know that the uniqueness of the solution arises from Lax-Milgram theorem or equivalently from the minimization problem $$\min\limits_{v\in H_0^1(\Omega)}\left\{\frac12 a(v,v) - \ell(v)\right\}. $$ I believe I have to formulate a similar minimization problem and prove there exists a unique minimizer. However, if I do the usual approach my $\ell$ will depend on $u$ which is not good, right? Does anyone have an idea?