Some nonlinear problems for the superposition of fractional operators with Neumann boundary conditions

TL;DR

This paper develops an existence theory for nonlinear nonlocal problems involving superpositions of fractional operators with Neumann boundary conditions, introducing new functional analysis tools and eigenvalue techniques.

Contribution

It presents a general framework for superpositions of fractional operators, including novel analytical methods and eigenvalue analysis for existence results.

Findings

Existence results for superpositions of fractional operators.

Application of Mountain Pass and Linking methods.

Framework encompasses sums of fractional Laplacians and other operators.

Abstract

We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators of mixed order. The setting that we introduce is very general and comprises, for instance, the sum of two fractional Laplacians, or of a fractional Laplacian and a Laplacian, as particular cases (the situation in which there are infinitely many operators, and even a continuous distribution of operators, can be considered as well). New bits of functional analysis are introduced to deal with this problem. An eigenvalue analysis divides the existence theory into two streams, one related to a Mountain Pass method, the other to a Linking technique.