DONETSK — Residents say the eastern Ukrainian city of Luhansk is dying. The power grid was completely down Monday, the city government said, and fuel is running dry.

Store shelves are emptying fast, and those who haven't managed to flee must drink untreated tap water. With little medicine left, doctors are sending patients home.

As Ukrainian government forces slowly tighten their ring around the city — one of two major pro-Russian rebel strongholds — traveling in and out has become a perilous undertaking.

In an impassioned statement released over the weekend, mayor Sergei Kravchenko described a situation that is becoming more unsustainable by the day.

"As a result of the blockade and ceaseless rocket attacks, the city is on the verge of a humanitarian catastrophe," Kravchenko said. "Citizens are dying on the streets, in their courtyard and in their homes. Every new day brings only death and destruction."

Luhansk, a city of more than 400,000 people at peacetime, now has seen its population dwindle as citizens flee violence and deprivation. Located about an hour's drive from Russia, which Ukraine insists is supplying rebels with weapons and manpower, Luhansk is being fiercely fought over by all sides of the conflict.

Shelling is a daily occurrence and the targets apparently quite random. On Saturday, eight buildings were damaged by rockets. These included a school, a supermarket and several multistory apartment blocks, Luhansk city government said.

Last week, a crucial electrical transformer in Luhansk was hit by a shell, leading to an 80 percent drop in power supplies, according to a report issued Monday by an Organization for Security and Cooperation in Europe monitoring mission.

Rebels accuse the army of waging a vicious bombing campaign against the civilian population. Authorities deny they have used artillery against residential neighborhoods and in turn accuse rebels of shelling civilians as a way of discrediting the army. This claim is faithfully repeated by almost all Ukrainian media, although it has been questioned by Human Rights Watch and others.

With gas reserves all but exhausted, even those willing to brave a drive out of the city for supplies struggle to refill their cars.

A little is getting through all the same, mainly from Russia. Pro-rebel online television station Luhansk-24 on Sunday carried a report about a consignment of medicinal supplies reaching the city from the southern Russian city of Saratov.

This was a visible reminder that supply lines to Russia remain intact. With clashes taking place at several spots surrounding the city, however, maintaining a steady convoy of goods convoy is complicated.

The fight for control over the frontier has been bitter.

Authorities concede that more than 60 miles (100 kilometers) of the Russian-Ukrainian border remains in rebel hands. The government says that Russia has been flagrantly smuggling large amounts of heavy weaponry and manpower to aid the separatist cause.

Sandwiched between the border and rebel fighters, government forces have succumbed to routine defeats and humiliations, even as they appear to slowly gain the upper hand in the fight to regain control of the last rebel strongholds.

On Monday, a Russian border security official said more than 400 Ukrainian soldiers crossed into Russia.

The Russian official said the soldiers deserted the Kiev government and that the Russian side had opened a safe corridor. A Ukrainian military official, who did not give a number for the soldiers involved, said the troops were forced into Russian territory by rebel fire after running out of ammunition.

Border control official Vasily Malayev later said 180 of the soldiers were being returned to Ukraine at their own request.

One of the soldiers, medic Anton Shingera, said he was uncertain how Ukraine would treat them upon their return, but "no matter what, the main thing is that I am alive." Other soldiers, who declined to give their names, said they had fled because they had run out of ammunition.

Ukraine's government accuses the separatists of entrapping the civilian population in besieged cities like Luhansk and has pushed for the creation of humanitarian corridors.

Their priority appears to persuade as many people to evacuate cities in advance of attempting full incursions.

"We urge the peaceful population to abandon territory seized by terrorists," Andriy Lysenko, a spokesman for the government's military operation in the east, said Monday.

Those choosing to stay behind and assist people in need are facing a desperate plight.

"Doctors can do nothing but cry. Every day, wounded people come in and we can't help them. We lack even basic medicines," said Fyodor Solyanik, a doctor in the Luhansk regional hospital.

Speaking briefly over a crackling landline connection, Solyanik said patients who were due for long-planned operations are being sent home.

"This is a nightmare situation," he said. "It's not just medicine we don't have. We don't even have food and water."

With prices for basic staples rising fast, the elderly deprived of their now-suspended pension payments are hit especially hard.

Local authorities are inundated with pleas for help that they are helpless to address. In the space of one day last week, a recently created crisis center received 1,800 appeals for assistance in just one day.

According to official figures, around 100 people have been killed since fighting began in Luhansk. With public services in total disarray, the reliability of such figures is questionable.

While the local government that existed before the current conflict began is attempting to provide basic services, real authority lies with the gunmen leading the separatist government that dubs itself the Luhansk People's Republic

Yelena Gaida, 46, is one of the many thousands who have managed to get out of Luhansk.

Speaking in Donetsk, another rebel-controlled city 90 miles (150 kilometers) west of Luhansk, Gaida said she is abandoning her three-room apartment and plans to head for Russia

"Almost half the city has fled already," Gaida said. "Every day they bomb, shoot and kill."

Gaida was dismissive of the separatist cause.

"Yes, we have our Luhansk People's Republic. But life there is impossible," Gaida said. "Who needs it anymore?"