I'm studying a Poisson BVP: Find $u \in W^{1,p}(\Omega)$ such that $$ \int_{\Omega} \nabla u \cdot \nabla v = F(v) \quad \forall v \in W^{1,p'}(\Omega), $$ where $p$ and $p'$ are Hölder conjugates and $F$ belongs to the dual of $W^{1,p'}(\Omega)$. Note the implicit homogeneous Neumann boundary condition $\partial u / \partial \nu = 0$; consider solutions defined up to an additive constant.
My question: If a solution exists, is it unique (up to an additive constant)? I know that the answer is yes when $p=2$ due to coercivity but what about when, say, $p \in (1,2)$? This boils down to whether $$ \int_{\Omega} \nabla w \cdot \nabla v = 0 \quad \forall v \in W^{1,p'}(\Omega) $$ implies $w$ is constant, but I'm having difficulty showing that.