# how to find the fundamental solution of $-\Delta u + e^u - e^{-u} = \delta(\vec{x})$ in 2D?

$$-\Delta u + e^u - e^{-u} = \delta(\vec{x})$$, where $$\Delta$$ is the Laplace operator and $$\delta(\vec{x})$$ is the Dirac's delta function and satisfies: $$\delta(\vec{x}) = \begin{cases} 0, & \vec{x} \neq \vec{0} \\ \infty & \vec{x} = \vec{0} \end{cases}$$ and $$\int_{R^2} f(\vec{x})\delta(\vec{x}) \,d\vec{x} = f(\vec{0})$$ where $$f$$ is any bounded and continuous function in $$\mathbb{R}^2$$.