Let $\Omega \subset \mathbb{R}^n$open $u$ continuously differentiable twice and for every $\phi \in C^{\infty}_c(\Omega)$ $$0= \int_{\Omega} \langle Du,D\phi\rangle$$
Then, integrating by parts I get $$0= \int_{\Omega} \Delta u \cdot \phi$$ and by the fundamental lemma of calulus of variations and continuity of $u$ I should get $u$ is harmonic. Am I reasoning correctly?
[I assume $u$ is atleast twice differentible to work with] You should use density argument.
Set of $C_c^{\infty}(\Omega)$ dense in $C^0(\Omega)$ in $L^2$ norm.
So there exists $\phi_n\in C_c^{\infty}(\Omega) $, such that $\phi_n\to \Delta u$. So $$0=\int \Delta u \phi_n\to \int |\Delta u|^2$$ This implies $u$ is harmonic. (Non negative function with integral zero)