We continue Part I of this paper on polyharmonic boundary value problems (− Δ) m u= f( u) on Ω⊂ R n , m∈ N , with Dirichlet boundary conditions. Here Ω is a bounded or unbounded conformally contractible domain as defined in Part I. The uniqueness principle proved in Part I is applied to show the following theorems: if f( s)= λs+| s| p−1 s, λ⩽0, with a supercritical p>( n+2 m)/( n−2 m) we extend the well-known non-existence result of Pucci and Serrin (Indiana Univ. Math. J. 35 (1986) 681–703) for bounded star-shaped domains to the wider class of bounded conformally contractible domains. We give two examples of domains in this class which are not star-shaped. In the case where 1< p<( n+2 m)/( n−2 m) is subcritical we give lower bounds for the L ∞-norm of non-trivial solutions. For certain unbounded conformally contractible domains, 1< p<( n+2 m)/( n−2 m) subcritical and λ⩾0 we show that the only smooth solution in H 2m−1(Ω) is u≡0. Finally, on a bounded conformally contractible domain uniqueness of non-trivial solutions for f( s)= λ(1+| s| p−1 s), p>( n+2 m)/( n−2 m), supercritical and small λ>0 is proved. Solutions are critical points of a functional L on a suitable space X. The theorems are proved by finding one-parameter groups of transformations on X which strictly reduce the values of L . Then the uniqueness principle of Part I can be applied.