i've a pi-3b+ it boots raspbian but does not boot ubuntu-core. When i write ubuntu-core image for pi-3 stable from here the red led lights only. When i write ubuntu-core image for pi-3 edge from here the red led lights 4 times long and 3 times short. The power supply i use is 5.2V 1.35A
The Raspberry Pi 3 Model B+ was released on 2018-03-14 and need updated firmware. It is very likely that the images from ubuntu you are using does not already have these firmware updates.
ubuntu-core-16-pi3.img.xz is dated from 2018-01-10 22:19 and
ubuntu-core-16-armhf+raspi3.img.xz is dated from 2018-01-03 08:40.