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

TL;DR

This paper introduces a broad framework for the superposition of nonlinear fractional operators across both order and type, enabling new analytical tools and applications in nonlinear analysis.

Contribution

It develops a novel theoretical framework for superposing nonlinear fractional operators in both order and type, extending previous work and including new operator combinations.

Findings

Includes finite sums of different (s,p) Laplacians

Addresses superpositions with fractional and classical Laplacians

Provides new applications of Weierstrass and Mountain Pass techniques

Abstract

We consider the continuous superposition of operators of the form \[ \iint_{[0, 1]\times (1, N)} (-\Delta)_p^s \,u\,d\mu(s,p), \] 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.