Molica Bisci, GiovanniServadei, RaffaellaZhang, Binlin2023-05-152023-05-152022-12-21Molica 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. 1-21.1072-6691https://hdl.handle.net/10877/16809In 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.Text21 pages1 file (.pdf)enAttribution 4.0 InternationalFractional LaplacianIntegrodifferential operatorNonlocal problemsEigenvalue and eigenfunctionAsymmetric nonlinearitiesVariational methodsCritical point theorySaddle point theoremArticle