Critical dimensions and higher order Sobolev inequalities with remainder terms

Pucci and Serrin [21] conjecture that certain space dimensions behave "critically" in a semilinear polyharmonic eigenvalue problem. Up to now only a considerably weakened version of this conjecture could be shown. We prove that exactly in these dimensions an embedding inequality for higher order Sobolev spaces on bounded domains with an optimal embedding con-stant may be improved by adding a "linear" remainder term, thereby giving further evidence to the conjecture of Pucci and Serrin from a functional analytic point of view. Thanks to Brezis-Lieb [5] this result is already known for the space H10 in dimension n = 3; we extend it to the spaces HK0 (K> 1) in the \presumably " critical dimensions. Crucial tools are positivity results and a decomposition method with respect to dual cones