The Europeans have an immigrant problem for the same reason we do.
They have reduced their birth rate and extended life expectancy so that no one else is available to work and pay taxes. And rather than plow back those profits into public investment, they have let them accumulate in the fortunes of their capitalist classes. So they import foreign labor to do the dirty work and so they won't have to pay anyone a decent wage, which through market forces, would immediately make more workers available.
The Europeans, like their paleface neighbors across the pond, are paying the price for centuries of European colonial imperialism in Africa, Asia and South America.