This question already has an answer here:
Let $R$ be a local ring (commutative with identity) ans $M$ and $N$ be finitely generated $R$-modules.
If $M\otimes_R N=0$, then $M=0$ or $N=0$.
The problem clearly seems to be an application of the Nakayama lemma. If we can show that $M=\mathfrak mM$ or $N=\mathfrak mN$, where $\mathfrak m$ is the unique maximal ideal of $M$, then by Nakayama we would have $M=0$ or $N=0$, for the Jacobson radical of $M$ is nothing but $\mathfrak m$ itself.
I am unable to figure out what to do.