Solution Method for Infinite-Horizon Maximization Problem
Your first question (regarding constraints on the parameters) can be answered through first and second derivative analysis. In order to satisfy strictly increasing, we need $u'&gt;0$ and to satisfy strictly concave, we need $u''&lt;0$. What does this actually mean?

$$u'(\frac{}{})=\phi p\theta c_t^{p-1}e^{\theta c_t^p}&gt;0$$
Since we know that $c_t^{p-1}e^{\theta c_t^p}&gt;0$ automatically, this reduces to:
$$\phi p\theta&gt;0$$
Next:
$$u''(\frac{}{})=\big[\phi p\theta (p-1) c_t^{p-2}+\phi p\theta c_t^{p-1}\big]e^{\theta c_t^p}&lt;0$$
Because we know that $e^{\theta c_t^p}&gt;0$ and $\phi p\theta c_t^{p-1}&gt;0$, this must mean: 
$$\phi p\theta (p-1)c_t^{p-2}&lt;0$$

By virtue of $(1)$ and the fact that $c_t^{p-2}&gt;0$, we know that:
$$p-1&lt;0$$
or
$$p&lt;1$$
Now,  what we can do is think about it critically. If $p=0$ then utility does not change when consumption changes. This is clearly untrue. If $p&lt;0$ then utility decreases when consumption increases which also seems incorrect. Therefore, our condition for $p$ is:
$$0&lt;p&lt;1$$
Now, what does this mean for $\phi \;\text{and}\; \theta$? It means: 
$$either\quad\phi , \theta&gt;0\quad or\quad \phi ,\theta&lt;0$$
Now, this looks like a typical exponential utility function, so we know that in these cases, both the coefficient for the exponent and the coefficient in the power will be negative (I will not bother proving this as I am sure it is somewhere in literature). Therefore we have the following three conditions for the parameters:
$$0&lt;p&lt;1$$
$$\theta &lt;0$$
$$\phi &lt;0$$

Now, the next step is to show that we do in fact have constant consumption. Taking our Euler Equation approach, we substitute in the constraint and take two terms in the sum,
$$...+\beta^t\phi e^{\theta (Rw_{t-1}-w_t)^p}+\beta^{t+1}\phi e^{\theta(Rw_t-w_{t+1})^p}+...$$
Differentiating w.r.t $w_t$ and doing some simplifying, we get:
$$(Rw_{t-1}-w_t)^{p-1}e^{\theta(Rw_{t-1}-w_t)^p}=\beta R(Rw_t-w_{t+1})^{p-1}e^{\theta(Rw_t-w_{t+1})^p}$$
Substituting consumption back in, we get:
$$c_t^{p-1}e^{\theta c_t^p}=\beta Rc_{t+1}^{p-1}e^{\theta c_{t+1}^p}$$
Now, since it is specified that $\beta R=1$ in the problem, we can see that the only way this equality holds is if $c_t=c_{t+1}=c^\star$

What does this mean for wealth? Will wealth also be constant? Well, let's conduct a thought experiment and suppose not. Suppose wealth increases each period ($w_{t+1}&gt;w_t\;\forall t$). This implies that as $t\rightarrow \infty$, we will see wealth grow to be infinite as well, which is wasteful and would not satisfy the transversality condition. Now suppose wealth is decreasing ($w_{t+1}&lt;w_t\;\forall t$). We can clearly see that eventually we will not have enough wealth to sustain constant consumption. Therefore we can see that $w_t=w_{t+1}=w^\star$. Now, what are the relationship between these two? Well, going back to our condition for wealth, we can see
$$w^\star=Rw^\star-c^\star$$
Rearranging, we get:
$$c^\star=(R-1)w^\star$$ 

Because we have steady state wealth, we know that $w^\star=w_0$

Now, knowing that we have steady state consumption, we know our utility function will be the same value for every period (discounted by $\beta$). So our value function simply becomes:

$$V(w)=\frac{\phi e^{((R-1)w_0)^p}}{1-\beta}$$ 

Therefore, we do not need to use the method of undetermined coefficients to find the value function.