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Volume 38, Issue 2
Strong Instability of Standing Waves for a Type of Hartree Equations

Chenglin Wang & Jian Zhang

DOI: 10.4208/jpde.v38.n2.2

J. Part. Diff. Eq., 38 (2025), pp. 142-154.

Published online: 2025-06

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  • Abstract

In this paper, we study the following three-dimensional Schrödinger equation with combined Hartree-type and power-type nonlinearities $$i\partial_t\psi+\Delta\psi+(|x|^{-2}*|\psi|^2)\psi+|\psi|^{p-1}\psi=0$$with $1 < p < 5.$ Using standard variational arguments, the existence of ground state solutions is obtained. And then we prove that when $p≥3,$ the standing wave solution $e^{ iωt}u_ω(x)$ is strongly unstable for the frequency $ω>0.$

  • Keywords

Hartree equation, standing wave, variational arguments, strong instability, blowup.

  • AMS Subject Headings

35Q55, 35B35, 35A15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-38-142, author = {Wang , Chenglin and Zhang , Jian}, title = {Strong Instability of Standing Waves for a Type of Hartree Equations}, journal = {Journal of Partial Differential Equations}, year = {2025}, volume = {38}, number = {2}, pages = {142--154}, abstract = {

In this paper, we study the following three-dimensional Schrödinger equation with combined Hartree-type and power-type nonlinearities $$i\partial_t\psi+\Delta\psi+(|x|^{-2}*|\psi|^2)\psi+|\psi|^{p-1}\psi=0$$with $1 < p < 5.$ Using standard variational arguments, the existence of ground state solutions is obtained. And then we prove that when $p≥3,$ the standing wave solution $e^{ iωt}u_ω(x)$ is strongly unstable for the frequency $ω>0.$

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v38.n2.2}, url = {http://global-sci.org/intro/article_detail/jpde/24213.html} }
TY - JOUR T1 - Strong Instability of Standing Waves for a Type of Hartree Equations AU - Wang , Chenglin AU - Zhang , Jian JO - Journal of Partial Differential Equations VL - 2 SP - 142 EP - 154 PY - 2025 DA - 2025/06 SN - 38 DO - http://doi.org/10.4208/jpde.v38.n2.2 UR - https://global-sci.org/intro/article_detail/jpde/24213.html KW - Hartree equation, standing wave, variational arguments, strong instability, blowup. AB -

In this paper, we study the following three-dimensional Schrödinger equation with combined Hartree-type and power-type nonlinearities $$i\partial_t\psi+\Delta\psi+(|x|^{-2}*|\psi|^2)\psi+|\psi|^{p-1}\psi=0$$with $1 < p < 5.$ Using standard variational arguments, the existence of ground state solutions is obtained. And then we prove that when $p≥3,$ the standing wave solution $e^{ iωt}u_ω(x)$ is strongly unstable for the frequency $ω>0.$

Wang , Chenglin and Zhang , Jian. (2025). Strong Instability of Standing Waves for a Type of Hartree Equations. Journal of Partial Differential Equations. 38 (2). 142-154. doi:10.4208/jpde.v38.n2.2
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