Uniqueness results for semilinear polyharmonic boundary value problems on conformally contractible domains. II

We continue Part I of this paper on polyharmonic boundary value problems (−Δ)^mu=f(u) on , , 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−1s, λ?0, with a supercritical p>(n+2m)/(n−2m) 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+2m)/(n−2m) is subcritical we give lower bounds for the L^∞-norm of non-trivial solutions. For certain unbounded conformally contractible domains, 1<p<(n+2m)/(n−2m) subcritical and λ?0 we show that the only smooth solution in H2m−1(Ω) is u≡0. Finally, on a bounded conformally contractible domain uniqueness of non-trivial solutions for f(s)=λ(1+|s|^p−1s), p>(n+2m)/(n−2m), supercritical and small λ>0 is proved. Solutions are critical points of a functional on a suitable space X. The theorems are proved by finding one-parameter groups of transformations on X which strictly reduce the values of . Then the uniqueness principle of Part I can be applied.