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

We consider the continuous superposition of operators of the form $${\iint }_{[0,1]\times (1,N)}(-\mathrm{\Delta }{)}_p^{s}ud\mu (s,p),$$ where $\mu$ denotes a signed measure over the set $[0,1]\times (1,N)$, 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.