Solutions of the equation $\Delta u=f(x,y)e^{cu}$ in some special cases

The equation $\Delta u=f(x,y)e^{cu}$ is considered in the case when the coefficient $f(x,y)$ is the absolute value or the square of the absolute value of some holomorphic function. In this case a full description of the general solution of the equation is presented. Several problems usually posed for elliptic equations are also discussed.