Multiple sequences of entire solutions for critical polyharmonic equations

In this paper we study the critical polyharmonic equation in R^d. By exploiting some algebraic-theoretical arguments developed in previous papers, we prove the existence of a finite number z of sequences of infinitely many finite energy nodal solutions which are unbounded in the classical higher order Sobolev space of order m, associated to the polyharmonic operator of order 2m. Taking into account recent results, we give an explicit expression of z in terms of the number of unrestricted partitions of the Euclidean dimension d, given by the celebrated Rademacher formula. Furthermore, the asymptotic behavior of the number z obtained here is a direct consequence of the classical Hardy-Ramanujan analyis based on the circle method. The main multiplicity result represents a more precise form of previous theorems for polyharmonic problems settled in higher dimensional Euclidean spaces. Finally, an explicit numerical comparison with previous results is presented.