Construction of infinitely many solutions for two-component Bose-Einstein condensates with nonlocal critical interaction

Abstract

In this paper, we deal with the coupled Hartree system with axisymmetric potentials,{−Δu+P(|x′|,x″)u=α1(|x|−4⁎u2)u+β(|x|−4⁎v2)uinR6,−Δv+Q(|x′|,x″)v=α2(|x|−4⁎v2)v+β(|x|−4⁎u2)vinR6, where (x′,x″)∈R2×R4, β>max⁡{α1,α2}≥min⁡{α1,α2}>0, P(|x′|,x″) and Q(|x′|,x″) are bounded nonnegative functions in R+×R4. The system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. When the functions r2P(r,x″) and r2Q(r,x″) have a common topologically nontrivial critical point, using a finite dimensional reduction argument and developing new local Pohožaev identities, we construct infinitely many solutions of synchronized type, whose energy can be made arbitrary large. The main difficulty is caused by the non-local terms, since little is known about the nondegeneracy of the positive solutions of the limit system and the error estimates of the nonlocal parts in applying the reduction arguments and establishing the local Pohožaev identities.