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

• Published in: Nonlinear analysis: real world applications Vol. 82; p. 104251
• Main Authors: Dipierro, Serena, Proietti Lippi, Edoardo, Sportelli, Caterina, Valdinoci, Enrico
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.04.2025
• Subjects: Fractional p-Laplacian; Nonlocal diffusion; Superposition of operators; Fractional p-Laplacian; Nonlocal diffusion; Superposition of operators
• ISSN: 1468-1218
• DOI: 10.1016/j.nonrwa.2024.104251

We consider the continuous superposition of operators of the form ∬[0,1]×(1,N)(−Δ)psudμ(s,p),where μ denotes a signed measure over the set [0,1]×(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.