Let $(\mathcal{M},g)$$({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$-bundle $E$ over $\mathcal{M}$${M}$, is it then true that
$$Lu=0$$
for $u\in\Gamma^{\infty}_{c}(E)$ implies that $u=0$, or in other words, there are no compactly-supported smooth homogeneous solutions? I think such a result could be proven via some ``unique continuation property of elliptic system''. However, while looking in the literature, I didn't find a suitable version for this situation.
Examples of such operator $L$ I have in mind is for example the connection Laplacian $$\Delta_{C}:=g^{ij}\nabla_{i}\nabla_{j}:\Gamma^{\infty}(T^{\ast}\mathcal{M}^{\otimes k})\to\Gamma^{\infty}(T^{\ast}\mathcal{M}^{\otimes k})$$$$\Delta_{C}:=g^{ij}\nabla_{i}\nabla_{j}:\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})\to\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$$ with Levi-Civita connection $\nabla$ or closely related, the Hodge-de Rham Laplacian $$\Delta_{H}:=\mathrm{d}\delta+\delta\mathrm{d}:\Omega^{k}(\mathcal{M})\to\Omega^{k}(\mathcal{M})$$$$\Delta_{H}:=\mathrm{d}\delta+\delta\mathrm{d}:\Omega^{k}({M})\to\Omega^{k}({M})$$ with exterior derivative $\mathrm{d}$ and codifferential $\delta$.
