In this paper, with the aid of weighted value sharing we study the uniqueness problems of
meromorphic functions when certain nonlinear differential polynomials generated by them share
a nonconstant polynomial with weight two. The result of the paper not only improves the results
due to the present first author [Bull. Math. Anal. Appl., 2(2010), 106-118] and of Zhang and
Xu [Comput. Math. Appl., 61(2011), 722-730], at the same time finds a possible answer of an
open question posed by Zhang and Xu.