it might be more accurate to say the Enterprise possesses more shield capacity, owing to it's large size and the need to enclose it.
Simply speaking = possible but unlikely given other on-screen evidence.
A ships warp core is it's energy source, it's engine. All things being equal, a physically larger engine denotes more ability to generation power.
In real life perhaps... nothing in trek we've seen indicates that a large core is able to output higher energy levels.
But that's not the warp core, that would be the antimatter storage or bottles, which are separate.
Voyager... early seasons, Janeway mentions during a tour that her ships Warp Core is able to go on for 3 years without refilling.
This can be superimposed onto other ships most-likely in terms of how long can a ship's core operate at peak efficiency before it has to be replenished.
The Enterprise-D was not mentioned on-screen that it can go on for 5 years in deep space, rather it's a non-canon statement (apparently per the TM - which isn't canon)... I'm only entertaining the notion for comparative purposes.
In which case, a smaller ship like the Intrepid could produce equal power levels like the Galaxy, but 'burn itself out' after 3 years, whereas the Galaxy can last 2 years extra.
These two statements counterdict each other. If phaser emissions don't lose power over distance, then there wouldn't be a maximum effective range of only a single light second.
They do not contradict each other.
I said the maximum effective range of phasers is 1 light second.
Just because we've seen ships within spitting distance to one another demonstrates idiocy of the VFX department in an attempt to make things look 'cool'.
The phaser array isn't just what you see running along the outside of the hull. There would be a internal mechanism extending down into the ship. Regardless of the length of the strip, Voyager has less internal volume to have the same size mechanism as the Enterprise.
If the Defiant was able to pulverize the Dominion bug ships with pulse phasers at equally close range like the Oddysey was, then it demonstrates that small ships like the Defiant can be comparable or surpass larger ones in terms of destructive fire power.
Remember that the Defiant is a ship which was completely stripped down of virtually all luxuries which likely contributed to it being 'overpowered'.
In the same regard, the Intrepid class of the 24th century only has 2 holodecks and we've seen it house a crew of only 150.
Given the very small amount of 'amenities' for the crew on the Intrepid, it's core could probably easily power weapons and shields to the same levels of those on the Galaxy class.
As I said, equal power capacity in a smaller package, but the core empties faster compared to larger ships.
Same is most likely evident for the Defiant.
The Galaxy class can last longer in deep space due to it's size, but smaller ships producing such power levels would likely last shorter amount of time.
This wouldn't eliminate larger ships from the equation though.
Why would it?
SF sees it perfectly amicable to construct large ships and supplement their fleet with smaller ones of equal fire power/defenses which would usually be situated close to home or serve for deep space exploration (though in shorter intervals) like in the case of the Intrepid class (it's more than capable after all).