Let be a Riemannian manifold of dimension . The authors deal with fourth-order operators defined by , where is the Laplacian on and is a real number. Such operators are closely related to the Paneitz operator . Here is the scalar curvature of .
The authors study on the Sobolev space , the completion of with respect to the norm . The main content of the paper is the construction of a kind of best possible Sobolev constant. To be more precise, put ; then
where the last norm is the usual norm of . Here the authors study the problem of finding optimal constants [resp. ] for these inequalities in the case of in the following sense: such that with the property that is valid} and analogously for . The result is a sharp constant , . The question of whether there exists a function such that these optimal constants are attained is partially solved. If is a conformally flat metric the constant is attained in (S2). The results are applied to the fourth-order differential equation . Here the authors obtain a sufficient condition for the existence of a smooth positive solution.
The paper is rather technical; however, this will be unavoidable in the subject.