Let's suppose $f$ is solution of the above equation, set $ F(x_1, x_2) = \frac{1}{(f(x_1)+f(x_2))^2} $ and $b' > b$. Since $ D\log(b'/b) = D\log(b'/a) - D\log(b/a) $ we have $ \int^{b'}_{b} \int^{b'}_{b} F(x_1, x_2) d x_1 d x_2 = \int^{b'}_{a} \int^{b'}_{a} F(x_1, x_2) d x_1 d x_2 - \int^{b}_{a} \int^b_a F(x_1, x_2) d x_1 d x_2 $ therefore $ \int^{b}_{a} \int^{b'}_{b} F(x_1, x_2) d x_1 d x_2 + \int^{b'}_{b} \int^{b}_{a} F(x_1, x_2) d x_1 d x_2 = 0 $ but $F(x_1, x_2)$ is non-negative so $F(x_1, x_2) = 0$ almost everywhere.
No solution can exist.