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

摘要

We study polyharmonic boundary value problems (-Delta)(m) u = f (u), m epsilon N, with Dirichlet boundary conditions on bounded and unbounded conformally contractible domains in R-n. Such domains can be contracted to a point (bounded case) or to infinity (unbounded case) by one-parameter groups of conformal maps. The class of star-shaped domain is a subclass. The problem has variational structure. This allows us to derive a sufficient condition for uniqueness by studying the interaction of one-Parameter transformation groups with the underlying functional L. If the transformation group strictly reduces the values of L then uniqueness of the critical point of L follows. The proof is inspired by E. Noether's theorem on symmetries and conservation laws. Applications of the uniqueness principle are given in Part II of this paper. (C) 2003 Elsevier Inc. All rights reserved. [References: 9]