A general theory for the $(s, p)$-superposition of nonlinear fractional operators

We consider the continuous superposition of operators of the form \[ [0, 1]× (1, N) (-Δ)_p^s \,u\,dμ(s,p), \] where μ denotes a signed measure over the set [0, 1]× (1, N), joined to a nonlinearity satisfying a proper subcritical growth. The novelty of the paper relies in the fact that, differently from the existing literature, the superposition occurs in both s and p. Here we introduce a new framework which is so broad to include, for example, the scenarios of the finite sum of different (in both s and p) Laplacians, or of a fractional p-Laplacian plus a p-Laplacian, or even combinations involving some fractional Laplacians with the "wrong" sign. The development of this new setting comes with two applications, which are related to the Weierstrass Theorem and a Mountain Pass technique. The results obtained contribute to the existing literature with several specific cases of interest which are entirely new.