The argument you have presented is an adaptation of [the Lax-Milgram theorem][1] which is essentially equivalent to the Riesz representation theorem (and generally speaking both of these results hold only in the Hilbert space framework). The Lax-Milgram theorem fails for the Laplace equation in $L^p$-spaces with $p\neq2$. Instead, some analogous results based on the ideas of coercivity, duality and monotonicity can be obtained in any reflexive Banach space.
 
The Dirichlet problem for the $p$-Laplace operator  
$$-\nabla(|\nabla u|^{p-2}\nabla u)=f,\quad x\in\Omega,\qquad (*)$$
$$u=0,\qquad x\in\partial\Omega,$$
might be a "correct" $L^p$-analogue of the problem described in the question.

The right hand side of $(*)$ gives rise to the mapping $A: W_{0}^{1,p}\to(W_{0}^{1,p})^{*}$
defined by the identity
$$\langle Au,v\rangle=\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla v\ dx\quad \mbox{for all }  v\in W_{0}^{1,p}.$$
A straightforward check shows that $A$ satisfies the conditions of the following theorem
(which might be viewed as an $L^p$-analogue of the Lax-Milgram theorem).
> **Theorem.** Let $A$ be a strictly [monotone][2], [coercive][3] operator from a reflexive Banach space $E$ to its dual $E^{* }$.   If $A$ is continuous on finite-dimensional subspaces of $E$ then for every $f\in E^{*}$ there exists a unique solution to the problem
$$Au=f.$$  



Have a look at the [textbook][4] by Chipot or  [the free monograph][5] by Showalter where the approach is explained in detail.

  


  [1]: https://en.wikipedia.org/wiki/Babu%C5%A1ka%E2%80%93Lax%E2%80%93Milgram_theorem
  [2]: http://en.wikipedia.org/wiki/Monotonic_function
  [3]: http://en.wikipedia.org/wiki/Coercive_function
  [4]: http://books.google.co.uk/books?id=WFwQ6a6fTIoC&printsec=frontcover&dq=chipot+elliptic+equations&source=bl&ots=j9iNmIjzyE&sig=I_MuBc4JWnYooBtRc9SNvHPtCGY&hl=en&ei=k6QWTM3JHcOT4ga3tsTODA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBkQ6AEwAA#v=onepage&q&f=false
  [5]: http://www.ams.org/publications/online-books/surv49-index