I'm facing the problem of proving existence/non-existence of a (unique and or weak) non-trivial solution to the following PDE on the Riemannian manifold ($S^2\times S^2$, $g$) with $g$ smooth (but not quite the round) metric tensor:
$$Pu=0,\quad P=\Delta+A:C^\infty(S^2\times S^2)\rightarrow C^\infty(S^2\times S^2)$$
with $u$ real-valued, $\Delta=\ast\mathrm{d}\ast\mathrm{d}$ the Laplacian and $A$ containing first- and zeroth-order terms with smooth non-constant coefficients.
Ellipticity is evident from the Laplacian, hence $P$ is Fredholm. Unfortunately, the index (which only depends on the principal symbol, i.e. $\sigma(\Delta)$) vanishes, so I can't use it to show existence of solutions. I also thought about proving things locally and then patching using partitions of unity but this doesn't seem to be working.
Although there is an abundance of literature about existence/regularity theory on $\mathbb{R}^n$ for second-order linear elliptic operators, the only reference I found that contains similar considerations are the notes by Melrose (and to the best of my very limited understanding, he does not talk about existence at all, only about regularity).
So I'd appreciate both, good references about the topic and concrete hints how to show that there does(n't) exist some $u$. Cheers!