Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications
Date
2022-12-21
Authors
Molica Bisci, Giovanni
Servadei, Raffaella
Zhang, Binlin
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this article we study an equation driven by the nonlocal integrodifferential operator -LK 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 -LK 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.
Description
Keywords
Fractional Laplacian, Integrodifferential operator, Nonlocal problems, Eigenvalue and eigenfunction, Asymmetric nonlinearities, Variational methods, Critical point theory, Saddle point theorem
Citation
Molica Bisci, G., Servadei, R., & Zhang, B. (2022). Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications. <i>Electronic Journal of Differential Equations, 2022</i>(85), pp. 1-21.
Rights
Attribution 4.0 International