Global compactness in the lower half-critical case

Establish a global compactness result for the linearly coupled Kirchhoff–Choquard system in mathbb{R}^3 when the first Choquard exponent satisfies the lower half-critical condition mathit{p}= (3+mathit{alpha})/3, where the nonlocal splitting lemma used in the noncritical and upper half-critical cases is unavailable.

Background

For exponents above the lower threshold, the paper derives global compactness through a nonlocal splitting lemma, which permits the decomposition of Palais–Smale sequences and supports the concentration–compactness analysis. At the lower half-critical exponent mathit{p}=(3+mathit{alpha})/3, the authors state that this splitting tool is unavailable.

Because the global compactness argument cannot be used in that regime, the paper instead imposes the stronger potential condition (V5) and obtains a ground state directly as a minimizer on the Nehari–Pohozaev manifold. A global compactness theory for the lower half-critical case therefore remains unresolved in the stated framework.