## kernel of Laplace + Potential

Let $M$ be a compact Riemannian manifold, and $V,f \in C^{\infty}(M)$. Suppose there exists a solution $u$ of $$(\Delta + V)u=f.$$ Are there any conditions that will ensure the existence of a function in the kernel of $\Delta+V$?

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 Could you be more specific about what kinds of condition you are looking for? As stated, the question seems very general – Yemon Choi Nov 11 2011 at 22:03 I'm sorry. I can't be more specific, I wouldn't know how to. – unknown (google) Nov 11 2011 at 22:17 OK: do you mean conditions on $M$? Conditions on $V$? If I take $M$ to be an $n$-torus and $V=0$, is that too specific? – Yemon Choi Nov 11 2011 at 22:22 Ok, let's say conditions on V. If $V=0$ constant functions will do the job (let's assume the manifold doesn't have boundary). Let's say $v \neq 0$. Thank you for your time. – unknown (google) Nov 11 2011 at 22:38