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

[摘要] We continue Part I of this paper on polyharmonic boundary value problems (-Delta)(m) u = f(u) on Omega subset of R-n, m epsilon N, with Dirichlet boundary conditions. Here Omega 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) = lambdas + \s\(p-1) s, lambda less than or equal to 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-infinity-norm of non-trivial solutions. For certain unbounded conformally contractible domains, 1 < p < (n + 2m)/(n - 2m) subcritical and lambda greater than or equal to 0 we show that the only smooth solution in H2m-1 (Omega) is u equivalent to 0. Finally, on a bounded conformally contractible domain uniqueness of non-trivial solutions for f(s) = lambda(1 + \s\(p-1) s), p > (n + 2m)/(n - 2m), supercritical and small lambda > 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. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-11-01

[关键词] polyharmonic operator;uniqueness;Pohozaev's identity;conformally contractible domains