trigger on ObjectA (after insert){
 TriggerHelper.checkExistenceInObjectB(trigger.new);
}

public class TriggerHelper{
  public static void checkExistenceInObjectB(List<ObjectA> l){
   //some code to query objects B based on your objects A criteria
   //some code to treat or create objects B
   //insert or update your list of objects B:
   upsert listOfObjectsB;
  }
}