A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator

Abstract.

We prove a Sobolev inequality with remainder term for the imbedding $\mathcal{D}^{m,2}(\mathbb{R}^N)$ $\hookrightarrow$ $L^{2N/(N-2m)}(\mathbb{R}^N)$, $m\in\mathbb{N}$ arbitrary, generalizing a corresponding result of Bianchi and Egnell for the case m = 1. We also show that the manifold of least energy solutions $u\in\mathcal{D}^{m,2}(\mathbb{R}^N)$ of the equation $(-\Delta)^m u = \vert u\vert^{4m/(N-2m)}u$ is a nondegenerate critical manifold for the corresponding variational integral. Finally we generalize the results of J. M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology to the biharmonic operator.