sounds like either your DNS or default gateway isn't configured properly when you switch to static IP.
turn the DHCP back on, make a note of the values it assigns to your S3, enter these in for the static IP value and switch the DHCP off.
Having your S3 set to obtain an automatic IP shouldn't drain the battery as all it should do is ask for IP get one and use it, the next time it talks to your router will be when the lease expires (set by your router) at which point it will say, can i have the same IP? yes thanks bye or, ok i'll take this instead.
Either way, configure your phone with a correct static IP and disable the DHCP, alternatively assign the S3 a static IP in the router, sometimes called address reservation.
NB if you disable the DHCP and there is noo other DHCP server on your network then you've basically broken you wifi/network, unless you plan to manually configure each device you connect to it.