On homomorphisms into Weyl modules corresponding to partitions with two parts

Let K be an infinite field of characteristic p> 0 and let lambda, mu be partitions, where mu has two parts. We find sufficient arithmetic conditions on p, lambda, mu for the existence of a nonzero homomorphism Delta(lambda) -> Delta(mu) of Weyl modules for the general linear group GL(n)(K). Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.