Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities - Gao, Fashun; Moroz, Vitaly; Yang, Minbo; Zhao, Shunneng (Calculus of variations and partial differential equations, 2022)
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.