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On a class of (non)local superposition operators of arbitrary order

Serena Dipierro, Sven Jarohs, Enrico Valdinoci

TL;DR

The paper presents a unified variational framework for superpositions of operators of arbitrary positive order, defined via measures on the order parameter and angular kernels, thereby bridging local and nonlocal dynamics. It introduces the operator family $L_{m,s}$ and the global operator $L$ through signed measures, constructs corresponding energy spaces ${\mathcal{X}}^s$ and ${\mathcal{X}}(\Omega)$, and establishes fundamental embeddings and Fourier representations. Two nonlinear existence results are obtained: (i) a mountain-pass solution for subcritical nonlinearities with Carathéodory growth, and (ii) solutions for jumping nonlinearities in a Dancer-Fučík-type setting under small coupling, including a critical-growth case with a nonlinearity at the Sobolev exponent. The framework subsumes polyharmonic, fractional, and anisotropic operators, enabling robust analysis of mixed-order problems and providing tools for existence theory in a broad class of nonlinear nonlocal models with potential applications to physics and biology.

Abstract

In this paper we introduce a very general setting dealing with the superposition of operators of any positive order and provide a systematic study of them. We also provide examples and counterexamples, as well as characterizing properties of the measures and the functional spaces under consideration. Moreover, we present some applications regarding the existence theory for a class of nonlinear problems involving superposition operators of arbitrary (possibly fractional) order.

On a class of (non)local superposition operators of arbitrary order

TL;DR

The paper presents a unified variational framework for superpositions of operators of arbitrary positive order, defined via measures on the order parameter and angular kernels, thereby bridging local and nonlocal dynamics. It introduces the operator family and the global operator through signed measures, constructs corresponding energy spaces and , and establishes fundamental embeddings and Fourier representations. Two nonlinear existence results are obtained: (i) a mountain-pass solution for subcritical nonlinearities with Carathéodory growth, and (ii) solutions for jumping nonlinearities in a Dancer-Fučík-type setting under small coupling, including a critical-growth case with a nonlinearity at the Sobolev exponent. The framework subsumes polyharmonic, fractional, and anisotropic operators, enabling robust analysis of mixed-order problems and providing tools for existence theory in a broad class of nonlinear nonlocal models with potential applications to physics and biology.

Abstract

In this paper we introduce a very general setting dealing with the superposition of operators of any positive order and provide a systematic study of them. We also provide examples and counterexamples, as well as characterizing properties of the measures and the functional spaces under consideration. Moreover, we present some applications regarding the existence theory for a class of nonlinear problems involving superposition operators of arbitrary (possibly fractional) order.

Paper Structure

This paper contains 23 sections, 45 theorems, 348 equations.

Table of Contents

  1. Introduction
  2. Some motivations and goals
  3. A class of homogeneous integrodifferential operators of arbitrary order
  4. Superposition operators of arbitrary order
  5. Existence theory for superposition operators of arbitrary order
  6. Organization of the paper
  7. On the measures sigma-s
  8. On the operator Lms
  9. Some notation and preliminary results on Lms
  10. Limits as s to 0+ and as s to m-
  11. Variational definitions and properties
  12. Some useful embeddings
  13. Some remarks and examples of superposition operators
  14. Definitions and observations on the energy space
  15. Some examples
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^N$ be an open bounded set. Let $\mu$ satisfy mu-assumption1, mu-assumption2 and mu-assumption1b and $L$ be as in defi:superposition op. Let $f:\Omega\times \mathbb{R}\to\mathbb{R}$ be a Charathéodory function satisfying nonlinearprob-assumptions. Suppose that nonempty x

Theorems & Definitions (96)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 86 more