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Critical dimensions and higher order Sobolev inequalities with remainder terms. (English) Zbl 0990.46021
P. Pucci and J. Serrin [J. Math. Pures Appl. 69, 55–83 (1990; Zbl 0717.35032)] conjectured that certain space dimensions behave “critically” in a semilinear polyharmonic eigenvalue problem and up to now only a weakened version of this conjecture could be shown. The authors of the paper under review 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. Thanks to H. Brezis and E. Lieb [J. Funct. Anal. 62, 73–86 (1985; Zbl 0577.46031)] this result is already known for the space \(H_0^1\) in dimension \(n=3\); the authors extend it to spaces \(H_0^K\) (\(K>1\)) in the “presumably” critical dimensions. The main tools are positivity results and a decomposition method with respect to dual cones.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B33 Critical exponents in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
35J60 Nonlinear elliptic equations
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