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

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.