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Nonexistence and Existence of Solutions with Prescribed Norms for Nonlocal Elliptic Equations with Combined Nonlinearities

Author

Listed:
  • Baoqiang Yan

    (School of Mathematical Sciences, Shandong Normal University, Jinan 250000, China)

  • Donal O’Regan

    (School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 TK33 Galway, Ireland)

  • Ravi P. Agarwal

    (Department of Mathematics, Texas A & M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA)

Abstract

In this paper, we study the nonlocal equation ∫ R N | u ( x ) | 2 d x γ Δ u = λ u + μ | u | q − 2 u + | u | p − 2 u , x in R N having a prescribed mass ∫ R N | u ( x ) | 2 d x = c 2 , where N ≥ 3 , μ , γ ∈ ( 0 , + ∞ ) , q ∈ ( 2 , 2 ∗ ) , c is a positive constant, p , q ∈ ( 2 , 2 ∗ ) with p ≠ q and 2 ∗ = 2 N N − 2 . This research is meaningful from a physical point of view. Using variational methods, we present some results on the nonexistence and existence of solutions under different cases p and q which improve upon the previous ones via topological theory.

Suggested Citation

  • Baoqiang Yan & Donal O’Regan & Ravi P. Agarwal, 2022. "Nonexistence and Existence of Solutions with Prescribed Norms for Nonlocal Elliptic Equations with Combined Nonlinearities," Mathematics, MDPI, vol. 11(1), pages 1-32, December.
  • Handle: RePEc:gam:jmathe:v:11:y:2022:i:1:p:75-:d:1014357
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