I would like to derive a solution to the following equation in $R^{3}$:
$$-\Delta u(x) + e^{u(x)} - e^{-u(x)} = \delta(x)$$where $\Delta$ is the Laplace Operator and $\delta(x)$ is the Dirac's delta function.
I know how to solve $-\Delta u(x) = \delta(x)$ in the sense of weak convergence but I'm stuck on the exponential term of the equation above.