A class of Hartree-Fock systems with null mass via Nehari-Pohozaev with logarithmic interactions

TL;DR

The paper tackles a zero-mass, two-component Hartree-Fock type system in the plane with a logarithmic kernel. It develops a variational framework on spaces $W^\lambda$ and introduces a Nehari-Pohozaev manifold $\mathcal{M}_{\beta}$ to obtain least-energy weak solutions, establishing regularity and a Pohozaev identity to justify the natural constraint. It proves the existence of nonnegative ground states and reveals a sharp beta-threshold $\beta>2^{p-1}-1$ that differentiates vectorial from semitrivial ground states, with precise asymptotic behavior as $\beta\to0^+$ and $\beta\to\infty$. The results advance understanding of nonlocal, logarithmic-interaction systems in low dimensions and provide a rigorous variational pathway through the zero-mass regime. Key methods include the Nehari-Pohozaev framework, a Quantitative Deformation Lemma, and detailed regularity/Pohozaev analyses.

Abstract

We establish the existence and qualitative properties of nontrivial solutions for a class of Hartree-Fock type systems defined over the whole space $\mathbb{R}^2$. By introducing a suitable Nehari-Pohozaev manifold, we prove the existence, regularity and we describe the asymptotic behavior of solutions with respect to the interaction parameter $β> 0$. In particular, we show that the system admits either a vector ground state or a semitrivial ground state solution, depending on the magnitude of $β$.

Theorems & Definitions (43)

• Theorem 1.1: Existence and asymptotic behavior of solutions
• Theorem 1.2: Regularity and Pohozaev identity
• Remark 1.3: Ground state solution
• Theorem 1.4: Vectorial and semi-trivial solutions
• Remark 1.5
• Remark 1.6
• Proposition 2.1
• Proposition 3.1
• proof
• Lemma 3.2