Let $E \subset \mathbb{R}^n$ be an open set where $y_0 \in E$ is an equilibrium point of the system $y' = f(y)$, where $f \in C^1(E)$. Suppose that there is a function $V:E \rightarrow \mathbb{R}$, $V\in C^1(E)$ such that $V(y_0)=0$ and $V(y)>0$ if $y \neq y_0$. A function $V$ that satisfies those hypothesis is called a Lyapunov function. And if $V'(y) \leq 0$ $\forall y\in E$, $y_0$ is a stable point, while if $V'(y) <0$ $\forall y \in E\setminus \{y_0\}$, then $y_0$ is asymptotically stable.
The problem is that if I found a Lyapunov function $V$ such that $y_0$ is a stable point of the system $y' = f(y)$, a priori nothing guarantees the existence of another Lyapunov function $V_2$ such that $y_0$ is asymptotically stable. My question is
If $y_0$ is an equilibrium point of the system $y' = f(y)$, and I have already found a Lyapunov function $V$ proving that $y_0$ is a stable point of the system. Are there any extra conditions on $V$ that guarantees the existence or the non-existence of another Lyapunov function such that $y_0$ is asymptotically stable?