On the existence of ground states to Hartree-type equations in ℝ³ with a delta potential

2024/04/22 by Gustavo de Paula Ramos, Ramos, Gustavo de Paula · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Quantum Chromodynamics and Particle Interactions

paper · pdf · doi:10.48550/arxiv.2404.13806

Abstract

Consider the Hartree-type equation in ℝ³ with a delta potential formally described by i ∂_t ψ= - Δ_x ψ+ αδ₀ ψ- (I_β∗ |ψ|^p) |ψ|^{p - 2} ψ where α∈ ℝ; 0 < β< 3 and we want to solve for ψ\colon ℝ³ × ℝ → ℂ. By means of a Pohožaev identity, we show that if p = (3 + β) / 3 and α≥ 0, then the problem has no ground state at any mass μ> 0. We also prove that if (3 + β)/(3) lt; p lt; min ( (5 + β)/(3), (5 + 2 β)/(4) ), which includes the physically-relevant case p = β= 2, then the problem admits a ground state at any mass μ> 0.

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