The argument you have presented is an adaptation of the Lax-Milgram theorem 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, coercive 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 recent booktextbook by Chipot or the free monograph by Showalter where the approach is explained in detail.

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The argument you have presented is an adaptation of the Lax-Milgram theorem 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 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, coercive 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 recent book by Chipot where the approach is explained in detail.