Ground states for the Hartree energy functional in the critical case
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912773487722496 |
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| author | Pistillo, Tommaso |
| author_facet | Pistillo, Tommaso |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ground states for the Hartree energy functional in the critical case Pistillo, Tommaso Mathematical Physics Analysis of PDEs 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. |
| title | Ground states for the Hartree energy functional in the critical case |
| topic | Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2512.16513 |