I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the introduction of some Sobolev Space. My quite vague question is the following one: Is it possible to give a meaning to the previous problem and to prove the existence of at least a solution of the previous problem in the space of distributions without introducing Sobolev Spaces?
Here's a rough approach which I do not believe needs to use Sobolev space theory, though it's perhaps not what you are looking for: Assuming some boundary regularity (I believe that $\partial \Omega \in C^2$ is sufficient), one can merely recast the problem as solving for a Green's function. It is then possible to prove the existence of the Green's function for Laplace's equation using somewhat elementary means, if I recall.