I have tried an installation of Ubuntu 12.04 LTS, but this system caused battery management trouble. Is there a version of Linux that will run properly on my Asus notebook model N56VV?

try slackware, latest version of ubuntu (13.10 currently), linux mint, pinguyos..

you can save time by booting distros off usb stick with unetbootin.

you didn't say what is wrong with battery managment, seems to me like a minor problem if you can work on battery.

What sort of problems with the battery management? You can use cpufreq-set to change your preference to extend battery life. Running linux off of a usb powered device can help extend battery life by powering down the internal drive. Which should lower power usage and the need for heat management. The last time I checked, ondemand is the default for ubuntu. But there's conservative and powersave preferences, below ondemand.

$ cpufreq-info
$ cpufreq-set -g conservative

There's also changes to frequency scaling in more recent kernels which might be worth a look. It's probably less about which distro and more about which configuration suits your needs.

No acpi maybe?

What happens is that I am indeed able to work on the battery, only once: it will not recharge...
Tried installing Slackware, but cannot boot it form the USB drive. I'll keep on trying different systems. Thanks for the answers