Let $a(u,v)=\int_0^1 u'v'+\left( \int_0^1 u\right)\left( \int_0^1 v\right)$ be a bilinear form defined on $H^1(0,1)$. Show that for every $f \in L^2$ there exists a unique $u \in H^1$ such that \begin{align} \left( \forall v \in H^1 \right) a(u,v)=\int_0^1 fv. \end{align} My guess is that one should use the fact that $a$ is coercive in some way...
This should also correspond to a certain minimization problem.