Europe has pushed to toughen its response to Russia's invasion of Ukraine, with Sweden following Finland in deciding to seek Nato membership and EU officials working to rescue proposed sanctions that would target Russian oil exports helping to finance the war.

On the ground, Ukrainian troops resisted attempted Russian advances and even rolled back front lines.

In a small but symbolic boost for Ukrainian morale, a patrol of soldiers recorded triumphant video of their push right up to the Russian border in the region of Kharkiv.

Ukrainian forces have already driven Russian troops back from the region's capital, reducing their ability to hit the battered city with artillery.

As fighting raged, international efforts to respond to Russia's aggression continued to pick up pace.

Swedish officials announced their intention to seek Nato membership - following a similar decision from its neighbour Finland.

Those are seismic developments for the Nordic countries that have traditionally positioned themselves as militarily "non-aligned".

Russian President Vladimir Putin has looked grimly upon the alliance's post-Cold War expansion in eastern Europe, seeing it as a threat.

But if the invasion was meant as a check on Nato, it appears to have backfired - by driving Sweden and Finland into Nato's arms and pushing members of the alliance to send massive shipments of weapons to Ukraine.

Yet the Russian leader on Monday seemed to brush off that setback, saying "there is no direct threat to Russia created by the expansion involving these countries".

"But the expansion of military infrastructure on to this territory will of course give rise to our reaction in response," he said.

Nato's secretary-general Jens Stoltenberg has said the membership process for both Finland and Sweden could be very quick - though member Turkey has cast doubts over the move.

Swedish Prime Minister Magdalena Andersson said that joining the 30-member military alliance was her country's best defence in the face of Russian behaviour.

"Unfortunately, we have no reason to believe that the trend (of Russia's actions) will reverse in the foreseeable future," she said.

As well as sending military aid to Ukraine, Europe is also working to choke off funding for the Kremlin's war, by reducing the billions of dollars it spends on imports of Russian energy.

But a proposed EU embargo on imports of Russian oil faces opposition from a small group of countries led by Hungary, which is one of a number of landlocked countries that are highly dependent on the imports, along with the Czech Republic and Slovakia.

Bulgaria also has reservations.

"We will do our best in order to deblock the situation," EU foreign policy chief Josep Borrell said.

"I cannot ensure that it is going to happen because positions are quite strong."

Western weapons deliveries to Ukraine and sanctions against Russia have helped the outgunned and outnumbered Ukrainian forces bog down the Russian advance - and even turn it back in places.

Mr Stoltenberg said on Sunday the war "is not going as Moscow had planned".

"Ukraine can win this war," he added - a remarkable assessment that would have been unthinkable for many on the eve of the invasion.

But Russia has been plagued by a series of setbacks in the war, most glaringly in its failure to overrun Kyiv, the capital, in the early stages of its February 24 invasion.

Since then, much of the fighting has shifted to the Donbas, Ukraine's eastern industrial heartland.

Determining a full picture of the unfolding battle there is difficult.

Air strikes and artillery barrages make it extremely dangerous for reporters to move around, and reporting is restricted by Ukraine and the Moscow-backed separatists it has been fighting in the Donbas for eight years.

The battlefront in the east and south of Ukraine extends over hundreds of miles, stretching men, machines and resources.

The two sides have been fighting village by village in parts.

Ukrainian forces have ground down the Russians, but are taking losses too.

In the Luhansk region of the Donbas, strikes overnight hit a hospital in Severodonetsk, killing two and wounding nine, including a child, the regional military command said on Monday.

Overnight strikes also hit other towns.

Regional military governor Serhiy Haidai said Ukrainian special forces blew up Russian-held railway bridges as part of efforts to slow the Russian offensive.

The information could not immediately be independently verified.

Meanwhile, in the south, a strike on Monday wounded three people in the Odesa region.

Air raid sirens ring out routinely in the region and its Black Sea port city, also called Odesa.

Visitors who would normally fill the city as the tourist season kicks off at this time of year are absent, and the streets of the usually lively city are quiet.

"This battle is dragging on. Russia does not stop attacking and advancing," Ukrainian military analyst Oleh Zhdanov said in recent days.

"The fate of the confrontation depends on the speed of supply of western weapons."

In a symbolic boost, a Ukrainian patrol in the Kharkiv region reached the Russian border and made a victorious video there addressed to President Volodymyr Zelensky.

The video posted on Sunday on Facebook by Ukraine's Ministry of Defence shows a dozen fighters around a wooden post - painted blue and yellow, Ukraine's colours - that they carried with them and then stood upright as a border marker, in triumph.

It was not clear exactly where the video was shot.

One soldier said the unit went "to the dividing line with the Russian Federation, the occupying country. Mr President, we have reached it. We are here".

Other fighters made victory signs and raised their fists.

Along another section of the frontier with Russia, Ukrainian border guards said they defeated a Russian attempt on Monday morning to send troops into the northern Sumy region.

The border guard service said Russian forces deployed mortars, grenade launchers and machine guns in an attempt to cover a "sabotage and reconnaissance group" crossing the border from Russia.

The border guard service said its officers returned fire and forced the Russian group to retreat back into Russia.

The area is largely rural and has not seen intense fighting in more than a month.

There was no immediate word from Russia.

Russian troops have also continued air and artillery strikes around the Azovstal steel plant in Mariupol, the last holdout of several hundred Ukrainian forces in the strategically important city, the Ukrainian General Staff said.

In an online news conference, many wives of the besieged soldiers urged the international community to help gain the release of "the entire garrison", which is suffering from a dire lack of food, water and medicine.

Turkey's presidential spokesman Ibrahim Kalin said his country had offered to evacuate wounded Ukrainian soldiers and civilians by ship from Azovstal, the official state broadcaster TRT said.

Britain's Ministry of Defence said Belarus was deploying special operations forces along its border with Ukraine and air defence, artillery and missile units to training ranges in the west of the country.

Belarus's forces have not been directly involved in the conflict, though its territory was used as a staging ground for the invasion.

But the presence of Belarusian troops near the border may keep Ukrainian troops pinned down there, preventing them from moving to support the counteroffensive in the Donbas.

Despite the fighting in the wider Kharkiv region and the threat of Russian missile attacks, many people were returning home to Kharkiv and other cities around Ukraine, Anna Malyar, deputy head of the Ministry of Defence, said.

Refugees were returning not just because of optimism that the war might ebb.

"Living somewhere just like that, not working, paying for housing, eating ... they are forced to return for financial reasons," Ms Malyar said in remarks carried by the RBK-Ukraine news agency.