Why is it useful to show the existence and uniqueness of solution for a PDE? Don't get me wrong, I understand that  it is important in mathematics to qualitatively study the problems given. But I would like to know to what extent this helps, for example, to actually solve the problem. 
I am reading books that deal with variational approach for elliptic PDEs like the Laplacian. Apart from transforming the problem into one of minimization of functionals, the main goal is to show the existence and uniqueness of a solution for the given PDE.
The only thing I can think of is that, for example, the Laplacian can be solved by using separation of variables. If you manage to show that the solution in separated variables is indeed a solution, you have found the solution. 

PDE's often come up from the modelling of a system. Existence of a solution means that the assumptions are consistent. Uniqueness means that they are sufficient to determine the system behaviour.
Answering these questions has a pivotal role in assessing the legitimacy of a model.
