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Ground states for the Hartree energy functional in the critical case

Tommaso Pistillo

TL;DR

This work analyzes the Hartree energy functional in $\mathbb{R}^3$ with convolution potentials in $L^\infty+L^{3/2,\infty}$, allowing for attractive, long-range interactions including Coulomb-type and inverse-power potentials. By combining Lorentz-space estimates with a refined concentration-compactness framework, the authors prove the existence of ground states for a large mass range, establish positivity and regularity, and derive the Euler–Lagrange equation and enhanced regularity results. They then study the associated time-dependent Hartree equation, proving global well-posedness under a smallness condition and orbital stability of the ground-state set, thereby linking variational structures to dynamical behavior. These results broaden the class of admissible potentials and provide a robust variational-dynamical theory for nonlocal nonlinear Schrödinger-type equations.

Abstract

We consider the problem of finding a minimizer $u$ in $ H^1(\mathbb{R}^3)$ for the Hartree energy functional with convolution potential $w$ in $L^\infty(\mathbb{R}^3)+L^{3/2,\infty}(\mathbb{R}^3)$ with $L^\infty$ part vanishing at infinity. This class includes sums of potentials of the kind $-\frac{1}{|x|^α}$, $0<α\le2$, together with the case $w$ in $L^{3/2}(\mathbb{R}^3)$. We prove the existence of such groundstates for a wide range of $L^2$ 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.

Ground states for the Hartree energy functional in the critical case

TL;DR

This work analyzes the Hartree energy functional in with convolution potentials in , allowing for attractive, long-range interactions including Coulomb-type and inverse-power potentials. By combining Lorentz-space estimates with a refined concentration-compactness framework, the authors prove the existence of ground states for a large mass range, establish positivity and regularity, and derive the Euler–Lagrange equation and enhanced regularity results. They then study the associated time-dependent Hartree equation, proving global well-posedness under a smallness condition and orbital stability of the ground-state set, thereby linking variational structures to dynamical behavior. These results broaden the class of admissible potentials and provide a robust variational-dynamical theory for nonlocal nonlinear Schrödinger-type equations.

Abstract

We consider the problem of finding a minimizer in for the Hartree energy functional with convolution potential in with part vanishing at infinity. This class includes sums of potentials of the kind , , together with the case in . We prove the existence of such groundstates for a wide range of 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.

Paper Structure

This paper contains 12 sections, 18 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Preliminaries
  5. Functional Inequalities in Lorentz Spaces
  6. Concentration Compactness Results
  7. Proof of Theorem \ref{['main theorem']}
  8. Existence of the minimizer
  9. Properties of the minimizer
  10. Time dependent Hartee equation
  11. Proof of Theorem \ref{['evo thm']}
  12. Orbital stability

Key Result

Theorem 1.1

Let $0\not\equiv w=w_1+w_2\in L^\infty(\mathbb{R}^3)+L^{3/2,\infty}(\mathbb{R}^3)$ be an even function such that there exists $u\in H^1(\mathbb{R}^3)$ for which $\int(w\ast|u|^2)|u|^2<0$ and such that $w_1(x)\xrightarrow{|x|\rightarrow\infty}0$. Define and Then, set and If $\lambda_*<\lambda<\lambda^*$, then problem definition minimizer has a solution $u_*\in\mathcal{S}_\lambda$. Moreover, eve

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3: About $\lambda_*$
  • Remark 1.4: About $\lambda^*$
  • Remark 1.5: About the regularity of the minimizer
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1: Inclusion properties
  • Lemma 2.2: Hölder Inequality in Lorentz spaces
  • Lemma 2.3: Young Inequality in Lorentz spaces
  • ...and 32 more