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 are far from my area of expertise, but maybe it will nevertheless be useful if I mention what I consider an aspect of existence and uniqueness theorems that I consider particularly important and useful.  That aspect is not the conclusion of the theorem ("there exists a unique function such that $\dots$") but rather the hypotheses. These describe the sort of differential equation under consideration and the initial or boundary conditions that the solution is supposed to satisfy. The theorem is essentially saying that those initial or boundary conditions are the right ones for that particular sort of equation. In other words, they tell you what conditions, in addition to the differential equation, can safely be imposed (i.e., a solution still exists when you impose the conditions) and suffice to pin down the solution completely (uniqueness). Intuitively, they tell us how much flexibility is available in the solutions of the differential equation.
