What do we need Sobolev-spaces $W^{k, p}$ with $p \neq 2$ for? In the course on PDE's I took this semester we talked a lot about the theory of Sobolev-spaces $W^{k, p}(\Omega)$ for $\Omega \subset \mathbf{R}^n$ an open set, $k \in \mathbf{N}$ and $1 \leq p \leq \infty$. But then we only used the Sobolev-spaces with $p = 2$ to deal with PDE's, since we can use the Riesz-Representation theorem on a suitable subspace of $W^{k, 2}(\Omega)$ (which is a Hilbert-space). 
For $p \neq 2$, we can't use the Riesz-Representation theorem, so we don't get weak solutions for $W^{k, p}(\Omega)$.
My question now is: Why do we introduce general Sobolev-spaces, instead of just limiting ourselves to $W^{k, 2}(\Omega)$? Are there any applications to the theory of PDE's (or other parts of mathematics) of Sobolev-spaces with $p \neq 2$? 
Thanks!
 A: For example, you can find a lot of papers about p-Laplacian elliptic boundary value problem with Dirichlet boundary condition (Neumann boundary condition or Robin boundary condition, etc...) whose solutions are  $W^{1,p}_0(
\Omega)$ (or $W^{1,p}(\Omega)).$
See https://en.wikipedia.org/wiki/P-Laplacian and the references therein.
See also Intuition and applications for the p-Laplacian  or  https://mathoverflow.net/questions/66418/on-the-physics-background-of-p-laplacian-equation
A: One reason might be that the parameter $p$ affects when we can easily extract continuous or in general $C^{m}$ functions from a Sobolev space. See the here for more information. Maybe to apply the linked results  you might want to use another value of $p$. 
To relate this specifically to PDE, maybe the following happens. You have your weak solution $u$, but you do not know if it is nice. But if you can find the right $p$ so that $u\in W^{k,p}$, then maybe you can apply the linked result to show that $u$ is nice.
