The Multiple positive solutions to a nonlocal problem with critical and supercritical nonlinear terms

  • Jie Guo orcid

    Basic Education Department, Guiyang Vocational and Technical College, Guiyang 550081, China
  • Yue Wang orcid

    School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
Article ID: 4561
DOI: https://doi.org/10.59400/adecp4561
Keywords: nonlocal problem; critical exponent; supercritical growth; Nehari method

Abstract

Nonlocal equations have emerged as a prominent research frontier in the field of nonlinear partial differential equations, while simultaneously posing pervasive challenges in theoretical modeling across disciplines including elastic vibrations and geometric analysis. To date, the academic community has developed a robust and comprehensive theoretical framework for such equations, with existing results encompassing both critical and supercritical nonlinearities. Nevertheless, when employing variational methods to investigate multiple positive solutions of nonlocal equations defined on the four-dimensional ball, the loss of embedding compactness induced by critical terms remains a core bottleneck hindering further progress. This paper explores multiple positive solutions of nonlocal equations with both critical and supercritical nonlinear terms on the four-dimensional spherical domain. To rigorously establish the existence of multiple positive solutions, we integrate the Nehari manifold framework with advanced variational techniques. We harness the Brézis–Lieb lemma to circumvent the compactness deficiency arising from critical nonlinearities, and draw upon potential function analysis to compensate for the failure of compactness conditions caused by supercritical terms, thereby rigorously proving the existence of k distinct positive solutions for the equation. This result not only generalizes some existing conclusions in the literature but also offers new insights for further research on high-dimensional nonlocal problems.

Published
2026-09-07
How to Cite
Guo, J., & Wang, Y. (2026). The Multiple positive solutions to a nonlocal problem with critical and supercritical nonlinear terms. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4561
Issue
Vol. 33 No. 3 (2026): In Progress
Section
Article

Copyright (c) 2026 Jie Guo, Yue Wang

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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