I've notice that if monsters are around, the regeneration of the stamina bar is slower. How come and is it possible to make it faster in combat ?
The obvious reason for stamina to regenerate slower in combat is balance - it's nice to be able to sprint more while out of combat, but limiting stamina regen in battle adds depth to combat.
You can use armor with fortify stamina regen enchantments to give yourself a passive boost to stamina regen both in and out of combat, as well as a weapon enchanted with absorb stamina to help you replenish stamina in combat. Having a weapon with absorb stamina will allow you to power attack non-stop (it doesn't matter whether you have 1 or 200 stamina, as long as it's not at 0 you can do a power attack), so tread carefully if you enjoy balanced combat.
From the UESP wiki, "Stamina" article:
This was most likely done to balance the combat in the game and not make it overpowered, encouraging you to manage or optimize your use of Stamina. The same thing is done with Magicka, which regeneration rate also decreases while in combat.
Increasing Stamina Capacity and Regeneration Rate
Use Fortify Stamina potions or enchantments to increase your Stamina capacity and Fortify Stamina Regen potions or enchantments to increase your Stamina regeneration rate. Also, see the list of Fortify Stamina and Fortify Stamina Regen items in the game.
Try to look for or enchant weapons with the Absorb Stamina enchantment, which will allow you to absorb stamina every time you hit an opponent. Some strong unique weapons that have the Absorb Stamina enchantment are Volendrung (Enchantment: Absorb 50 points of Stamina) and the Nightingale Blade (Absorb Stamina enchantment: Absorb Stamina, 25 pts for 1 sec).
Redguards are a preferable player race choice if you want to have the Adrenaline Rush greater power (Stamina regenerates 10x faster for 60 seconds, once per day), which is useful in situations where you need to use a lot of Stamina in combat.
Also, a Stamina tip (from a Reddit /r/Skyrim post):
I am thinking that this is also the same case in combat, but I haven't been able to test this yet.
If you want to use a mod, you could try Magicka and Stamina Combat Regen, it can increase combat Stamina Regen rate from 1.5x to 15x the normal rate.
If you combine the Redguard Adrenaline Rush greater (racial) power with a mod like No Racial Power Cooldown, then you will effectively make that greater power permanent and will have no need to worry about Stamina regeneration anymore. A mod that will allow you to use your racial power more frequently, but less "cheaty" is the Auto-cast Racial Powers Plugin (Redguard: Adrenaline Rush will now have a 25% chance to activate when hit by an aggressor when Stamina reaches 25% or below and Health is at 80% or below).
The applications of stamina in combat are (found here):
Let us consider specifically the Power Attack, which is (imo) the primary use of stamina in combat.
The formula for stamina cost of a power attack is (found here):
What this formula does not explicitly state is that you only need 1 stamina in order to start a power attack.
Thus I would propose you create a weapon of your choice that has Absorb Stamina enchantment with only 1 point of magnitude (this is not fool proof if you use bows). This weapon is vastly superior to the same weapon with the maximum available Absorb Stamina magnitude. Here's why:
To answer your question:
1. Your stamina regenerates slower when you are in combat (or enemies are near).
2. Don't be so concerned about having a lot of stamina in combat, rather strive to have enough to do what it lets you do (power attack). For that I recommend the Absorb Stamina (1 point) weapon.
Use stamina potions. As far as I know that's the only way to regenerate quickly. :/