Consider following problem from Evans
Assume $\Omega$ is connected. A function $u\in H^1(\Omega)$ is a weak solution of $$(1)\quad\left\{\begin{array}{ll} -\Delta u(x)=f & x\in\Omega\\ \frac{\partial u(x)}{\partial n}=0 & x\in\partial\Omega\end{array}\right.$$ if for all $v\in H^1(\Omega)$ there holds $$\int_\Omega\nabla u\cdot\nabla v\,dx=\int_\Omega fv\,dx \,.$$ Suppose that $f\in L^2(\Omega)$. Prove $(1)$ has a weak solution if and only if $$\int_\Omega f\,dx=0.$$
It is quite difficult. Do I need to use Lax Milgram. How would I go about problem?
