Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities

• Published in: Calculus of variations and partial differential equations Vol. 61; no. 6
• Main Authors: Gao, Fashun, Moroz, Vitaly, Yang, Minbo, Zhao, Shunneng
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Critical point; Identities; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; 35A15; 35J20; 35J60
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-022-02340-2

In this paper, we study a class of critical Choquard equations with axisymmetric potentials, - Δ u + V ( | x ′ | , x ′ ′ ) u = ( | x | - 4 ∗ | u | 2 ) u in R 6 , where ( x ′ , x ′ ′ ) ∈ R 2 × R 4 , V ( | x ′ | , x ′ ′ ) is a bounded nonnegative function in R + × R 4 , and ∗ stands for the standard convolution. The equation is critical in the sense of the Hardy-Littlewood-Sobolev inequality. By applying a finite dimensional reduction argument and developing novel local Pohožaev identities, we prove that if the function r 2 V ( r , x ′ ′ ) has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.