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 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 . 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 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