An existence theory for nonlinear superposition operators of mixed fractional order

TL;DR

This paper proves the existence of multiple solutions for a broad class of nonlinear fractional problems involving superpositions of different fractional Laplacians, extending previous results to more general and complex operator combinations.

Contribution

It introduces a general existence framework for nonlinear superpositions of fractional Laplacians of mixed orders, including infinitely many operators with signed measures.

Findings

Established multiple solutions for critical fractional problems

Extended results to sums of multiple fractional p-Laplacians

Handled superpositions with signed measures and dominance conditions

Abstract

We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The results obtained are new even in the case of the sum of two different fractional $p$-Laplacians, or the sum of a fractional $p$-Laplacian and a classical $p$-Laplacian, but our framework is general enough to address also the sum of finitely, or even infinitely many, operators. In fact, we can also consider the superposition of a continuum of operators, modulated by a general signed measure on the fractional exponents. When this measure is not positive, the contributions of the individual operators to the whole superposition operator is allowed to change sign. In this situation, our structural assumption is that the positive measure on the higher fractional exponents dominates the rest of the signed measure.