Revenue maximization
Firm 1's objective is to maximize profits :
$$\max_{q_1\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_1 - cq_1$$

Solving the above problem, we get the best response function of firm 1 as
$$q_1 = \dfrac{a - c - bq_2}{2b}$$

Firm 2's objective is to maximize revenue subject to the constraint that its profits are non-negative :
$$\max_{q_2\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_2$$ $$\text{s.t. }\left(a - b(q_1+q_2)\right)q_2 - cq_2 \geq 0$$

Solving the above problem, we get the best response function of firm 2 as
\begin{eqnarray*} q_2 =\begin{cases}  \dfrac{a - bq_1}{2b} &amp; \text{if } a - bq_1 \geq 2c \\ \dfrac{a - c - bq_1}{b} &amp; \text{if } a -  bq_1 &lt; 2c \end{cases}\end{eqnarray*}

Solving the best response functions for $q_1$ and $q_2$ yields
\begin{eqnarray*} (q_1^*, q_2^*) = \begin{cases} \left(\dfrac{a-2c}{3b}, \dfrac{a+c}{3b}\right) &amp; \text{if } a \geq 2c\\
 \left( 0, \dfrac{a-c}{b}\right) &amp; \text{if } a &lt; 2c \end{cases} \end{eqnarray*}