Brezis-Nirenberg type problems associated with nonlinear superposition operators of mixed fractional order

2025/04/07 by Yergen Aikyn, Aikyn, Yergen, Sekhar Ghosh +5 · 7 citations
Mathematics · #35A01 #35J60 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2504.05105

openalex publication_date 2025/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper aims to study the Brezis-Nirenberg type problem driven by the nonlinear superposition of operators of the form A_{μ, p}u:=∫_[0,1](-Δ)_p^s u d μ(s), where μ denotes the signed measure over [0, 1]. We consider nonlinear nonlocal equations associated with A_{μ, p}, involving critical nonlinearity and lower-order perturbation. Using variational techniques, we establish existence results for the critical problem by employing weak lower semicontinuity arguments under general assumptions on the perturbation term. We discuss the multiplicity results when the perturbation term vanishes at the origin. Additionally, when the lower-order term is a pure power function, we examine the Brezis-Nirenberg-type problem using the mountain pass technique. Furthermore, we address the existence of solutions to subcritical problems associated with A_{μ, p}. Our findings are novel, even in the case of the sum of two distinct fractional p-Laplacians or a combination of a fractional p-Laplacian with a classical p-Laplacian. More generally, our framework is sufficiently broad to accommodate finite sums of different fractional p-Laplacians as well as cases involving fractional Laplacians with ``wrong" signs. A key contribution of this study is the development of a unified approach that systematically addresses these problems by incorporating a broad class of operators and lower-order perturbation terms within a common theoretical framework. The results remain new even in the case of linear superposition of fractional operators of different orders.

Citations

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Cited by

• Nontrivial solutions for nonlinear problems driven by a superposition of fractional p-Laplacians with Neumann boundary conditions
• Superlinear problems involving nonlinear superposition operators of mixed fractional order
• Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent

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