According to J. Liouville *Sur l'équation aux différences partielles* $\partial^2\log\lambda/\partial u\partial v\pm\lambda/2a^2$ **J. Maths. Pures & Appl. 18** (1853), pp 71-71, the general solution of this equation in the plane is given by the formula
$$u=-\log\frac{8|f'(z)|^2}{(1-|f(z)|^2)^2}\,,$$where $f$ is holomorphic. You should start from that.

You can also use the paper by H. Fujita **Bull. AMS 75** (1969), pp 132-135, which treats the equation in arbitrary space dimension.