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

In this paper 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 infinity. Moreover, we get 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 proved along the present paper. This monotonicity property is of independent interest and represents the nonlocal counterpart of a famous result got by de Figueiredo and Gossez in the setting of uniformly elliptic operators.