BERLIN — Faced with a humanitarian crisis that shows no signs of letting up, Chancellor Angela Merkel gave her fellow Germans and other Europeans a pep talk on Monday, praising those assisting the thousands seeking asylum here but warning that the Continent’s open-border policy was at risk unless all European countries showed an equal willingness to help.

The chancellor, whose summer was interrupted by an urgent effort to secure a third financial aid package for Greece, has found herself under fire for what political opponents and even some members of her coalition government have called foot-dragging in dealing with the migration crisis.

About 800,000 people are expected to seek asylum in Germany this year, but overcrowded reception centers, a lack of personnel and outbreaks of far-right hatred against some of the new arrivals have left Germany, and some of its European partners, struggling in recent months to cope with the influx.