Existence and uniqueness of PDE with solutions in $W^{k,p}$ with $p \neq 2$? I just realised that i have never seen the space $W^{k,p}$, $p\neq 2$, used in showing existence/uniqueness to some PDE. Usually books/lectures build up theory about $W^{k,p}$ (like certain compact embeddings) spaces and then immediately show well-posedness to some PDE in the space $H^k$. Typically with Poisson's equation.
I am very curious about how to show existence to PDEs where their solutions live in $W^{k,p}$, $p \neq 2$. Obviously these spaces are not Hilbert so Lax-Milgram goes out of the window. Can someone give me a cool example or cite some book that goes through an existence proof?
Thanks
 A: Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, $p\in (1,\infty)$, $q\in (1,\infty)$, $\frac{1}{p}+\frac{1}{q}=1$, $f\in L^q(\Omega)$. Consider the problem $$\tag{1}
\left\{ \begin{array}{rl}
 -\operatorname{div}(|\nabla u|^{p-2}\nabla u)=f &\mbox{ in $\Omega$} \\
  u\in W_0^{1,p}(\Omega) &\mbox{}
       \end{array} \right.
$$
The operator $ \Delta_pu=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is called $p$-Laplacian. We say that $u\in W_0^{1,p}(\Omega)$ is a solution (weak solution) of (1) if forall $\phi\in C_0^\infty(\Omega)$ $$\tag{2}\int_\Omega |\nabla u|^{p-2}\nabla u\nabla \phi=\int_\Omega f\phi$$ 
One way to solve the equation (2) is to consider the energy functional $F:W_0^{1,p}(\Omega)\rightarrow\mathbb{R}$ associated to it: $$F_p(u)=\frac{1}{p}\int_\Omega|\nabla u|^p -\int_\Omega fu$$
It is a pleasurable exercise to show that $F_p$ is a strictly convex functional and hence it must have an unique minimizer. Moreover you can show that if $u\in W_0^{1,p}$ is the minimizer, hence $$\langle F_p'(u),\phi\rangle=\int_\Omega |\nabla u|^{p-2}\nabla u\nabla \phi-\int_\Omega f\phi=0,\ \forall\ \phi\in W_0^{1,p}$$
To get a problem where the solution lies in $W^{k,p}$ with $k>1$ you need to ask more differentiability, for example, in the $p$-laplacian problem we need only that $k=1$.
Note 1: Observe that standard Dirichlet problem is included here and the same technique applies to solve this problem.
Note 2: If you ask more regularity of $f$, for example $f\in L^{r}$ with $r>\frac{N}{p}$, you can show that in fact $u\in C_{loc}^{1,\alpha}(\Omega)$ for some $\alpha\in (0,1)$.
