Critical dimensions and higher order Sobolev inequalities with remainder terms

Abstract

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 constant 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 \( H_0^1 \) in dimension n=3; we extend it to the spaces \( H_0^K \) (K>1) in the ''presumably'' critical dimensions. Crucial tools are positivity results and a decomposition method with respect to dual cones.