In the past I saw climbers climb alone on multi-pitch routes only by themselves (but not free solo). What's a correct technique to do this?
I know how to do it with two ropes, rope ascenders and on top rope, but is there a way to do it without?
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Climbing is inherently dangerous. Soloing is even more so. Please learn from experienced people and in person, not from Internet. So this answer mainly describes physical principles, supported by some experience that I have. Here is an excellent tutorial, by a climber more experienced than me, that explains the technique in much more detail.
There is a belay device, Soloist, designed specifically for roped soloing. However, soloing is also possible with the usual grigri device. To add to the general warning on the danger of soloing, let me note that using grigri for this purpose is prohibited by the manufacturer (however, it's possible and some people do it).
First, anchor one end of the rope. If you fall, this anchor will be pulled up, so take this into account when building it.
Then, grab the rope a few metres from the anchored point, and load it into your grigri. Give it a quick pull to verify that the grigri locks (if you loaded it the wrong way, it won't).
Arrange the rope (starting from the other end) on the floor to make sure you can pull it up, and it will not get tangled when unsupervised (while you are climbing). (Note: it's better to stuff it into a backback to eliminate the chance of it getting stuck, but I don't know how to do it properly)
Start climbing the usual way, placing protection regularly. While climbing, feed rope through the grigri when you have the need and opportunity to do so. With grigri, feeding the rope is especially awkward and may require both hands; some people "mod" their grigri by drilling a hole to ease that process.
You will have two strands of rope going down; you should pay attention and clip only the right one (the one that is anchored).
You should avoid coming to the awkward situation when the rope stops you from climbing further (grigri "thinks" that you fell and locks). If a difficult section is ahead, where you cannot afford fiddling around with the rope, you should feed yourself enough rope to pass all of it.
When you arrive to the end of a pitch, build an anchor (might want to make it so it can support upward loading direction, to serve for next pitch), descend to the previous anchor, collect the protection gear and climb back up, using the top-rope technique.
Things that can go bad when soloing:
For this answer I'll discuss the devices and considerations specific to lead soloing, rather than describe a particular technique.
There are very few devices on the market for roped solo lead climbing. Rock Exotica's Soloist device used a camming mechanism similar to that of Petzl's GriGri. (As mentioned, the GriGri is not intended for solo climbing.) The successor to the Soloist is the Silent Partner, which uses a rotating drum around which a clove hitch is tied. A centrifugal clutch locks the drum when the rope moves too quickly, similar to some seatbelt mechanisms.
It is important to realize that solo belay devices have their limitations. For example, the Soloist will not catch a head-first fall. If you are falling backwards on a lower-angle slab, or catch the rope with your leg, the rope will be exiting the device at an angle that will not trigger the cam.
Many people consider the GriGri an auto-locking device but it is technically only an "assisted braking" device. While it will lock-off in many cases, this should be considered an added bonus rather than an integral feature. You should assume that a GriGri will at some point decide not to lock.
The Silent Partner is very convenient as it feeds the rope as you go. However, this also means that near the top of a pitch the weight of the payed-out rope can cause it to self-feed additional slack. This can be mitigated by attaching some thin cord or a hefty rubber band to the rope and a piece of protection. The cord should be weak enough to break in the event of a fall so as not to increase the fall factor. It's also possible to tie the rope to the protection piece, but this will effectively shorten the rope and increase the fall factor. (Also, while the Silent Partner is probably the best choice for lead soloing, it's not a very good rappel device.)
Finally, there is the question of device orientation. Some of these devices are rather large, and some need to be fixed in position (e.g. the Soloist requires a chest harness). A fall can cross-load the attaching carabiners and break them. Years ago this happened to a guy soloing El Capitan in Yosemite using a GriGri; the DMM belay 'biner holding the GriGri snapped first, after which the GriGri itself was damaged and unusable. Accordingly, some people use steel maillons rather than aluminum carabiners.
As with rappelling, you should always back up your system. The guy in the El Capitan story had backup knots; that's the only reason he didn't end up as chunky salsa on the rocks below. One traditional way of doing this is to attach loops of rope to your harness by clove-hitching them through carabiners. This will also help the rope feed better through the device.
When soloing you usually don't have the benefit of a partner who can go for help. This means that your self-rescue skills need to be well-practiced. You should be able to ascend or descend the rope if needed, as well as know how to escape the belay.
If you are climbing outside of cell phone range you need to be willing to drag yourself back to the trailhead, or at least to some place with other people. If you've broken both ankles hitting a ledge you may encounter added difficulty. I suggest ensuring your first aid kit has material for splints and a decent amount of ibuprofen.
In addition to the other answers, a blog post by Andy Kirkpatrick (a leading proponent of the art of solo roped climbing) on his description of what he does, can be seen at http://www.andy-kirkpatrick.com/articles/view/rope_soloing_101_part_1