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 constant mag. 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.