Finding subgame perfect equilibrium The main part of the question is as follows 



(if you cannot read this, I can immediately write it)

My question is how to find the subgame perfect Nash equilibrium for the both cases $\bar{g_2}\ge \bar{g_1}$ and $\bar{g_2}&lt; \bar{g_1}$

My attempt is 

For firm 2 

$$max [f_2(G)-c_2(g_2)]$$
$$max [\beta ln(g_1+g_2)-g_2]$$

By FOCs

$$\beta / (g_1+g_2)- 1=0$$

$$g_2=\beta -g_1$$

For firm 1

$$max [f_2(G)-c_2(g_2)]$$
$$max [\theta ln(g_1+\beta -g_1)-g_1]$$

I cannot get FOC with respect to $g_1$ 

At this point I’m stuck. How can I proceed this solution?

And I have another question 

For the same question, I assume the following part



Again I need to find subgame perfect Nash equilibrium of this game where player 1 contributes a positive amount of public good.? 

Sorry but I could not this part completely. Any helps will be appreciated. 

Since I could not understand such type of questions for the SPE, I asked lots of time. Thanks.
Note, there is a mistake in FOC---it should be 
$$\beta/(g_1+g_2) - 1 =0,$$
so the response function of firm $2$ is
$$g_2 = \beta - g_1.$$

This means that a unit increase in a public good contributed by firm $1$ causes firm $2$ to decrease its contribution by the same unit. Then, notice that firm $1$, who correctly anticipates such a (optimal) response by firm $2$, is better off by not contributing anything (hint: $g_1$ only affects $C(g_1)$ up to $g_1 \in [0,\beta]$).

I guess you can take from here to solve the second part