Consider a nonlinear PDE $N(x,D\,\cdot\, ,\cdot)$ and a 'problem' involving it stated in a naive function space $F$. This problem may involve the statement of initial/boundary conditions, or the presence of a forcing term (like $f$ in the nonhomogeneous equation $N(x,u,Du)=f$ ).
I am interested in showing that sufficiently strong solution exists on a dense subset of $F$.
For example, the existence of strong solutions for problems(forcing term this case) in dense subset of $F$ has been investigated equations like $$ \Delta u+|u|^{p-1}u=0. $$
Many methods have been developed in order to prove 'almost sure' global existence results for these equations, meaning that sufficiently strong solutions exists with probability 1 on given subsets of $F$, with respect to some not too impulsive probability measure given on $F$.
However, I couldn't find a generic argument for showing this kind of density property.
Is there any? Where can I find it?
(Why) is it studied less than almost sure existence?
The context is too general to obtain sensible answers.
According to Hadamard, a problem is well-posed when it admits a unique solution for each choice of the data, and such solution is a uniformly continuous function of the data. Now, a uniformly continuous function defined on a dense subset of a complete metric space can be uniquely extended to a continuous function defined on the whole metric space.
This means that any Hadamard-well-posed problem will never exhibit the phenomenon you are looking for.