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 Ω ⊂ Rn,m ∈ N, with Dirichlet boundary conditions. HereΩ 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 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.  2003 Elsevier Inc. All rights reserved.