Suppose $f \in L^{2}$. I know that
$$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0 & \text{on } \partial\Omega \end{array}\right.,$$
where $\Omega \subset \mathbb{R}^{N}$ is open and bounded, has a weak solution $u \in H^{1}_{0}(\Omega)$. I can obtain this solution using the Riesz Representation Theorem.
But, how obtain a weak soution for this equation with Neumann boundary conditions? And in the case where $\Omega = \mathbb{R}^{N}$?
In both cases I have been tried use the Lax Milgran Theorem, but I can't show the coercitive condition.
Another question: What the regularity of these weak solutions? I know that if $f$ is $C^{0,\alpha}$ and the problem has Dirichlet conditions, then the solution is $C^{2}$, am I right?
Thanks in advance.