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.

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.

If a solution $u$ exists in the whole plane, it seems that $f$ must be a bounded entire function, hence a constant, from which it follows $u\equiv0$. Perhaps this can be proved by Pohozaev's calculus.

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