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 ∫[0,1](−Δ)sudμ(s), for a signed measure μ 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.