Ground states for the Hartree energy functional in the critical case

2025/12/18 by Pistillo, Tommaso
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2512.16513

openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28

Abstract

We consider the problem of finding a minimizer u in H¹(ℝ³) for the Hartree energy functional with convolution potential w in L^∞(ℝ³)+L^3/2,∞(ℝ³) with L^∞ part vanishing at infinity. This class includes sums of potentials of the kind -(1)/(|x|^α), 0<α≤2, together with the case w in L^3/2(ℝ³). We prove the existence of such groundstates for a wide range of L² masses. We also establish basic properties of the groundstates, i.e.~positivity and regularity. Lastly, we exploit the estimates we derived for the stationary problem to prove global well-posedness of the associated evolution problem and orbital stability of the set of ground states.

Citations

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