Multiple solutions to asymptotically linear problems driven by superposition operators

In this paper, we investigate the existence and multiplicity of weak solutions to problems involving a superposition operator of the type $${\int }_{[0,1]}(-\mathrm{\Delta }{)}^{s}ud\mu (s),$$ for a signed measure $\mu$ on the interval of fractional exponents $[0,1]$, when the nonlinearity is subcritical and asymptotically linear at infinity; thus, we deal with a perturbation of the eigenvalue problem for the superposition operator. We use variational tools, extending to this setting well-known results for the classical and the fractional Laplace operators.