Nonlocal eigenvalue problems and superposition operators

We study the spectral theory of mixed local and nonlocal operators with lower-order terms in the right-hand side of the equation.This kind of problems is motivated by the analysis of superposition operators of mixed order and with the 'wrong sign' of the lower-order terms with respect to the classical elliptic theory.Our results include:● convergence to classical cases when the right-hand side of the eigenvalue equations 'localizes', recovering the simplicity and sign-definiteness of eigenfunctions in the limit;● a detailed analysis of disconnected domains, showing that, unlike the classical case, any eigenfunction associated with the first eigenvalue must change sign, and that the first eigenvalue of a union of disconnected domains is strictly smaller than that of its individual components;● examples in which the first eigenvalue is either simple or non-simple in disconnected domains;● a regularity theory that underpins these results.