  We construct some $n$-dimensional eigenvarieties for finite slope
overconvergent eigenforms over some unitary Shimura varieties with signature
$(1,n-1)\times(0,n)\times\cdots\times(0,n)$ by adapting
Andreatta-Iovita-Pilloni's method. We also show that there are some Galois
pseudo-characters over our eigenvarieties by studying analytic continuation of
finite slope eigenforms over these Shimura varieties.
