Let $\Omega \subset \mathbb{R}^2$ be a bounded domain, $p, p' \in (1,\infty)$ be conjugate exponents and $f \in L^p(\Omega)$. Consider the problem of finding $u \in W_{0}^{1,p}(\Omega)$ such that $$ \int_{\Omega} \nabla u \cdot \nabla v \thinspace dx = \int_{\Omega} fv \thinspace dx \quad \forall v \in W_{0}^{1,p'}(\Omega) $$ The problem has a unique solution when $p=2$, but does there exist a solution when $p \neq 2$?
My idea was to show that the bounded linear mapping $A: W_{0}^{1,p} \to \left(W_{0}^{1,p'}(\Omega)\right)', \thinspace Au(v) = \int_{\Omega} \nabla u \cdot \nabla v \thinspace dx,$ is surjective, but for that I seem to need some kind of ellipticity result $\left\lVert Au \right\rVert \geq c \left\lVert u \right\rVert$ and can't get past that. I'm starting to suspect that a solution does not generally exist when $p \neq 2$. Is my suspicion correct?