Esrogim can be found in the shuk (market) that have on them a sort of "gartel" shape around the middle. That is to say a sort of indentation that make the esrog appear as if it's wearing a "gartel" (belt). Is there any known source for using such an esrog specifically, or any discussion about this in the earlier poskim? Why is it that some people desire to have such an esrog so much?
An esrog with a gartel is not mentioned in any of the classic poskim. According to Wikipedia:
. . some look for an etrog with a gartel—an hourglass-like strip running around the middle, more commonly found on the Moroccan citron. According to researchers, this gartel indicates when the bearing tree was infected by a certain virus or viroid, which decreases the albedo on the specific spot. These viroids have been around since at least the time of Bar Kokhba (circa 130 CE), based on the fact that archaeologists have unearthed a mosaic dated to that time which depicts an etrog with a gartel. Only the etrog is found to be susceptible to these viroids, proving again that the etrog is genetically pure and has not changed significantly over the centuries.
Also, according to here some consider it more beautiful.
What about the gartel, the indentation encircling some esrogim? Note that although other types of fruits may show this phenomenon, the gartel is unique to the esrog among citrus fruits. Some Chassidim prefer an esrog with a gartel for various reasons, and Rabbi Sternbuch writes that he prefers such an esrog because he views it as the surest sign that it’s not a murkav, even though it is not usually cited as one of the signs of a pure esrog. In fact, the gartel can be shown to be an ancient phenomenon, as there are coins from the period of the Bar Kochva revolt that consistently show an esrog with a gartel, as do the mosaics from the 6th century Beit Alpha (Gallil) Shul and Caesarea.