In this paper we study the critical polyharmonic equation in the whole Euclidean space. By exploiting some algebraic-theoretical arguments developed in [2,13,20], we prove the existence of a finite number ζd of sequences of infinitely many finite energy nodal solutions which are unbounded in the classical higher order Sobolev space, associated to the polyharmonic operator. Taking into account the recent results contained in [20], an explicit expression of ζd is given 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 ζd 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 Theorem 1.1 of [2] for polyharmonic problems settled in higher dimensional Euclidean spaces. Finally, an explicit numerical comparison with Theorem 4.8 of [20] is presented.

### Multiple sequences of entire solutions for critical polyharmonic equations

#### Abstract

In this paper we study the critical polyharmonic equation in the whole Euclidean space. By exploiting some algebraic-theoretical arguments developed in [2,13,20], we prove the existence of a finite number ζd of sequences of infinitely many finite energy nodal solutions which are unbounded in the classical higher order Sobolev space, associated to the polyharmonic operator. Taking into account the recent results contained in [20], an explicit expression of ζd is given 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 ζd 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 Theorem 1.1 of [2] for polyharmonic problems settled in higher dimensional Euclidean spaces. Finally, an explicit numerical comparison with Theorem 4.8 of [20] is presented.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11576/2673990
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