Authorities in Munich are overwhelmed by the influx of refugees streaming into the Bavarian capital, with more than 10,000 arriving on Saturday.
Germany has so far taken the lion's share of migrants, admitting 450,000 people this year, with Chancellor Angela Merkel's decision to relax asylum rules for Syrians drawing praise from the refugees, but also sharp criticism from domestic allies and counterparts abroad.
In Munich, where more people were expected to arrive overnight Saturday-Sunday, regional officials have sounded the alarm and urged other states in Germany - seen as the promised land by many of those seeking safe haven in Europe - to do their bit.
"We no longer know what to do with refugees," mayor Dieter Reiter said, amid fears many of the new arrivals would have to spend the night outdoors.
Reiter added that he was "very concerned with the developments" noting that if other areas took in several hundred refugees it would "help to avoid chaos".
"Munich and Bavaria can't overcome this great challenge alone," a spokeswoman for the Bavarian authorities said, adding the city was struggling to find beds for all the additional people.
As the newcomers arrived, some onlookers at Munich station held welcome signs to greet them. But there were far fewer than several days ago when cheering volunteers handed out groceries and children's toys.
As the continent scrambles to respond to the biggest movement of people since World War II, sharp divisions have emerged between the European Union's 28 member states, at both a government level and on the streets.
Munich made its appeal for help as tens of thousands of Europeans held rallies in both support of and opposition to refugees.
Tens of thousands marched through London waving placards saying "Refugee lives matter" and "No human being is illegal".
Britain's newly elected Labour Party leader and veteran socialist Jeremy Corbyn drew huge cheers when he addressed the crowd from the back of a truck.
"Open your hearts and open your minds," the opposition chief said, "towards supporting people who are desperate, who need somewhere safe to live, want to contribute to our society, and are human beings just like all of us."
In Copenhagen, 30,000 people turned out to express solidarity with asylum seekers, while similar rallies drew thousands in Madrid and Hamburg.
But at the same time, thousands took to the streets in eastern Europe to voice their opposition to the influx, their numbers dwarfing those attending a handful of pro-migrant rallies.
"Islam will be the death of Europe" chanted protesters at a rally in Warsaw which was attended by nearly 5,000 people and began with prayers identifying many marchers as Catholics.
Hundreds also demonstrated in Prague and in the Slovak capital Bratislava, some holding banners reading: "You're not welcome here so go home".
The International Organisation for Migration said Friday that more than 430,000 people have crossed the Mediterranean to Europe this year, with 2,748 dying en route or going missing.
The emergency has exposed deep rifts with the EU, with "frontline" states Italy, Greece and Hungary buckling under the strain and European Commission proposals for sharing 160,000 of the new arrivals in a quota scheme facing resistance from eastern members.