Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019

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

Multiple sequences of entire solutions for critical polyharmonic equations

Pages: 117-144
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.
[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

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.

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