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

Baoqiang Yan (), Donal O’Regan and Ravi P. Agarwal
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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

Mathematics, 2022, vol. 11, issue 1, 1-32

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.

Keywords: constrained minimization; Pohozaev’s identity; nonlocal equation; existence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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