A remark on large even integers of the form $p+P_3$

Let $N$ denotes a sufficiently large even integer, $p$ denotes a prime and $P_r$ denotes an integer with at most $r$ prime factors. In this paper, we study the solutions of the equation $N-p=P_3$ and consider two special cases where $p$ is small, and $p,P_3$ are within short intervals.