Giovanni Molica Bisci ^[a] and Patrizia Pucci ^[b]

Multiple sequences of entire solutions for critical polyharmonic equations

Pages: 117-144
Received: 7 January 2019
Accepted in revised form: 26 February 2019
Mathematics Subject Classification (2010): Primary: 35J91, 35J60; Secondary: 35A01, 45A15.
Keywords: Variational methods, principle of symmetric criticality, nodal solutions.
Authors address:
[a]: Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029 Urbino, Italy
[b]: Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

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Abstract: In this paper we study the critical polyharmonic equation in \( \ \mathbb R^d \). By exploiting some algebraic-theoretical arguments developed in [2,13,20], we prove the existence of a finite number \( \ \zeta_{d} \ \) of sequences of infinitely many finite energy nodal solutions which are unbounded in the classical higher order Sobolev space \( \ \mathcal{D}^{m,2}(\mathbb R^d) \), associated to the polyharmonic operator \( \ (-\Delta)^m \), with \( \ m\in{\mathbb N} \). Taking into account the recent results contained in [20], an explicit expression of \( \ \zeta_{d} \ \) 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 \( \ \zeta_{d} \ \) 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.