Electron. J. Differential Equations, Vol. 2022 (2022), No. 85, pp. 1-21.

Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications

Giovanni Molica Bisci, Raffaella Servadei, Binlin Zhang

Abstract:
In this article we study an equation driven by the nonlocal integrodifferential operator $-\mathcal L_K$ in presence of an asymmetric nonlinear term f. Among the main results of the paper we prove the existence of at least a weak solution for this problem, under suitable assumptions on the asymptotic behavior of the nonlinearity f at $\pm \infty$. Moreover, we show the uniqueness of this solution, under additional requirements on f. We also give a non-existence result for the problem under consideration. All these results were obtained using variational techniques and a monotonicity property of the eigenvalues of $-\mathcal L_K$ with respect to suitable weights, that we prove along the present paper. This monotonicity property is of independent interest and represents the nonlocal counterpart of a famous result obtained by de Figueiredo and Gossez [14] in the setting of uniformly elliptic operators.

Submitted December 8, 2022. Published December 21, 2022.
Math Subject Classifications: 35A01, 35S15, 47G20, 45G05.
Key Words: Fractional Laplacian; integrodifferential operator; nonlocal problems; eigenvalue and eigenfunction; asymmetric nonlinearities; variational methods; critical point theory; saddle point theorem.
DOI: https://doi.org/10.58997/ejde.2022.85

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Giovanni Molica Bisci
Dipartimento di Scienze Pure e Applicate (DiSPeA)
Università degli Studi di Urbino Carlo Bo
Piazza della Repubblica 13, 61029 Urbino (Pesaro e Urbino), Italy
email: giovanni.molicabisci@uniurb.it
Raffaella Servadei
Dipartimento di Scienze Pure e Applicate (DiSPeA)
Università degli Studi di Urbino Carlo Bo
Piazza della Repubblica 13, 61029 Urbino (Pesaro e Urbino), Italy
email: raffaella.servadei@uniurb.it
Binlin Zhang
College of Mathematics and Systems Science
Shandong University of Science and Technology
Qingdao 266590, China
email: zhangbinlin2012@163.com

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