Poincar\'e type J-equation

We introduce a two-parameter continuity path for the J-equation and use it to characterize the solvability of the J-equation for K\"ahler metrics with Poincar\'e type singularities along a divisor $D$, allowing simple normal crossings and self-intersections. On K\"ahler surfaces, we show that the classical subsolution condition in the smooth setting implies solvability in the Poincar\'e type setting for any smooth divisor $D$. As a consequence, if $X$ contains no curves of negative self-intersections and $K_X[D]$ is ample, then the K-energy is bounded from below on any Poincar\'e type K\"ahler class. In the smooth divisor case, we further analyze the asymptotic behavior of solutions near $D$, and show that existence of a Poincar\'e type solution implies existence of a solution to the J-equation on $D$.