More than one Bellman Equation
(The second equation for the value function of the unemployed should be
$$
v(w,U)= \max \{v(w,E); \,u[c,1]+\beta\int v(w', U) dF(w')\}. \quad (*)
$$
)

...how do you know when your problem solution require more than
one Bellman equation?

Whenever the state space of the problem contains discrete coordinates,
there would be  &quot;multiple&quot; value functions, indexed by the discrete coordinates.
Here the state space is $[0, \infty) \times \{E, U\}$. The second coordinate is discrete. So there are &quot;two&quot; value functions $v(w, E)$ and $v(w, U)$.

How do you realize that?

Any variable that determines the optimization problem faced by the agent is part of the state.
In this particular example, an unemployed agent has the option to switch to being employed. An employed agent has no choice but to stay in the job---&quot;...to work forever&quot;.
The choice set, therefore the decision problem, faced by the agent are different depending on his employment status.
This tells you that employment status is part of the state.
(Note that, if the employed worker has the option to leave the job, the decision problem faced by the agent are still different depending on his employment status. In that case you would still have two value functions and they would be &quot;intertwined&quot;, so to speak.)

Argue...the optimal strategy has the reservation wage property.

This is more or less immediate from the equation $(*)$.
The reservation wage $w^*$ is given by
$$
v(w^*,E) = u[c,1]+\beta\int v(w', U) dF(w').
$$
At $w = w^*$, agent is indifferent, $v(w^*, U) = v(w^*, E)$.
You would expect that for $w &lt; w^*$.
$$
v(w,U) = u[c,1]+\beta\int v(w', U) dF(w') &gt; v(w,E).
$$
This is a typical option-exercise problem.
The difference $v(w, U) - v(w,E)$ when $w &lt; w^*$ is the option value. If the offered wage $w$ is too low, the agent would rather keep the option and not exercise it.
(This is an American-type option, which can be exercised anytime. Same &quot;exercise threshold&quot; phenomenon occurs in the continuous-time setting, which is sometimes more convenient. There the threshold is given by the smooth-pasting condition.)
General Comments
The general formulation of the Bellman equation is
$$
V(s) = \max_{c \in \mathcal{C}(s)} \int_{\mathcal{S}} V(s') dF(s'; s, c),
$$
where $\mathcal{C}(s)$ is the choice set faced by the agent at state $s$ and
$s' \mapsto dF(s'; s, c)$ is the Markov transition kernel if agent makes choice $c$ at state $s$.
(For notational simplicity, assume no period utility/discounting/etc. The discussion would not change.)
Therefore, by definition, any variable that determines the optimization problem faced by the agent is part of the state.