Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 85, pp. 1-21.

Title: Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications
       
Authors: Giovanni Molica Bisci (Univ. degli Studi di Urbino Carlo Bo, Italy)
         Raffaella Servadei (Univ. degli Studi di Urbino Carlo Bo, Italy)
         Binlin Zhang (Shandong Univ. of Science and Tech., Qingdao, China)
         
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.