Ukraine’s armed forces are undeterred in their efforts to “break” Moscow’s will to pursue a nearly five-month war, President Volodymyr Zelenskiy says, while Russian leader Vladimir Putin hailed his military’s victory in the gruelling battle of Luhansk.

Russia captured the city of Lysychansk in eastern Ukraine on Sunday, completing its conquest of the Luhansk province, one of two regions it has demanded Ukraine cede to separatists in the Donbas region.

“There have been no significant changes on the battlefield in the past 24 hours,” Zelenskiy said in his nightly video message. 

“The armed forces of Ukraine respond, push back and destroy the offensive potential of the occupiers day after day. 

“We need to break them. It is a difficult task. It requires time and superhuman efforts. But we have no alternative.”

Earlier in the day, Putin congratulated Russian forces on “victories in the Luhansk direction”. 

Participants should “absolutely rest and recover their military preparedness” while units in other areas keep fighting, the Russian president said in a brief televised meeting with his defence minister.

The battle for Luhansk is the closest Moscow has come to achieving one of its stated objectives since its forces were defeated trying to capture Kyiv in March. 

It marks Russia’s biggest victory since it captured the southern port of Mariupol in late May.

Both sides suffered thousands of dead and wounded, while claiming they inflicted far greater losses on the other side along a loop of the Siverskyi Donets river that winds through Luhansk and Donetsk.

Relentless Russian bombardment has made a wasteland of Lysychansk, neighbouring Sievierodonetsk and surrounding towns, many of which had heavy industrial plants that defenders used as fortified bunkers. 

Russia had repeatedly tried and failed to surround the Ukrainians, eventually opting to blast them out with artillery.

Military experts said the battle could be a turning point in the war, making a big impact on both sides’ ability to fight on, even though the strategic value of the ruined cities themselves is limited.

“It’s a tactical victory for Russia but at an enormous cost,” said Neil Melvin of the RUSI think tank in London. 

“This has taken 60 days to make very slow progress,” he said. “The Russians may declare some kind of victory, but the key war battle is still yet to come.”

Ukraine could have withdrawn from Luhansk weeks ago but chose to keep fighting to exhaust the invasion force. 

It hopes the ferocious battle will leave the Russians too depleted to hold gains elsewhere.

Serhiy Gaidai, the Ukrainian governor of Luhansk, acknowledged his entire province was now effectively in Russian hands, but told Reuters: “We need to win the war, not the battle for Lysychansk … It hurts a lot, but it’s not losing the war.”

The mayor of Sloviansk said heavy shelling on Sunday had killed at least six people, including a 10-year-old girl.

Russia’s Tass news agency quoted military officials in the self-styled Donetsk Peoples’ Republic as saying three civilians died and 27 were injured in what they said was shelling by Ukrainian forces.

Reuters could not verify the battlefield accounts.

Meanwhile, Zelenskiy said “colossal” funding would be needed to rebuild Ukraine’s shattered infrastructure.

Speaking at a conference in the Swiss city of Lugano, Ukraine’s Prime Minister Denys Shmygal said the cost could reach $US750 billion – and rich Russians should help pay the bill.

The decisive battle for Ukraine was likely to take place not in the east, where Russia is mounting its main assault, but in the south, where Ukraine has begun a counter-offensive to recapture territory, RUSI’s Melvin said.

“This is where we see the Ukrainians are making progress around Kherson,” he said. 

“There are counter-attacks beginning there and it’s most likely that we’ll see the momentum swing to Ukraine as it tries to then mount a large-scale counter-offensive to push the Russians back.”

Ukraine’s hopes for a sustained counter-attack rest in part on receiving additional weapons from the West, including rockets that can neutralise Russia’s huge firepower advantage by striking deep behind the front line.