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

Abstract

We study polyharmonic boundary value problems (− Δ) m u= f( u), m∈ 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.