I found my old phone N810. I bought new battery.I started charging, but there is no indication. When I press the button power, phone show picture and then turns off. Is hardware problem?

Hi. Most likely the battery is not making tight contact.

Have you tested old battery? If battery has not used over year then try this:
1. connect charger only some seconds and then disconnect
2. connect charger again and wait about 5 minutes
3. Try Power on, it shows Nokia picture and then looks like it turns off.
4. Wait about 5 minutes and power on again, now it should start, if not jump to 1.

I don't know why this works, but I have managed get work many looks like broken Nokias get to work. If somebody know why this work can explain to all of us.

Teroyk's mentioned solution has merit.
The problem reminds me of how ...with not enough battery charge, the n8x0 can end up bootlooping.

As far as the aforementioned problem goes, a number of battery issues like it ...up to and including a battery that has so low a charge it cannot charge, especially when attempting to charge via the battery in the n8x0 device and charging with the charge cable.
The battery having so little a charge remaining to begin with.
Add to that that the device while attempting to charge thus, does cause a drain on the battery with the system startup lighting and logo before going dark to charge.

A good remedy for batteries and some of the aforementioned issues is to invest in an external n8x0 battery charger, which are or were...quite cheap.
Or a universal battery charger.
This is also an excellent method of determining if a battery is the problem or not.

There also is the high likelihood that the issue is most certainly the battery.
There hasn't...to my knowledge...been any new genuine n810 or n800 batteries made in many years.
It has been a long time since I could find even original used batteries .
There are however many cheap attempts at n8x0 batteries.
Some well meaning attempts ...some not.
So be aware when looking for "new" batteries.

There is a solution to this ...inevitability .
Using a n old dead battery...using the casing ...rather... casing, even a cheap knock off battery casing.
And carefully removing the guts of it, one can hook up wiring to the insides of the battery posts, that lead out of a drilled and rubber gasketed hole in the removable back access (since leaving the battery cover off the device means it will not boot )
The wiring leading to a " step down" that is "tuneable " ..and from there to an electric outlet.
Thereby making ones device "future - proof"