Let $M$ be a smooth Riemannian manifold and $E$ be a finite-rank vector bundle over $M$ equipped with a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$, i.e. $\langle\cdot,\cdot\rangle$ is a positive-definite inner product on each fibre. This induces an inner product on the level of (compactly-supported) sections via $$(\cdot,\cdot):=\int_{M}\langle\cdot,\cdot\rangle\,\mathrm{vol}_{g}.$$
Now, if $M$ is compact and $L:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$ a formally self-adjoint elliptic operator, then the famous Fredholm alternatives states that the elliptic equation $$Lu=f$$ for some source $f\in\Gamma^{\infty}(E)$ has a smooth solution $u\in\Gamma^{\infty}(E)$ if and only if $(f,g)=0$ for all $g\in\ker(L)$.
If $M$ is non-compact, is there a similar statement, i.e. if $f\in\Gamma^{\infty}_{c}(E)$ does there exists a compactly-supported smooth solution $u\in\Gamma^{\infty}_{c}(E)$ if and only if $f$ is orthogonal to every element in $\ker(L\vert_{\Gamma^{\infty}_{c}})$?
