Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918388103643136 |
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| author | Nakayama, Hikaru |
| author_facet | Nakayama, Hikaru |
| contents | We consider the defocusing fourth-order nonlinear Schrödinger equation with potential \[ i\partial_t u + Δ^2 u + Vu + λ|u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), \] in dimensions $n \ge 5$. In the energy-critical case $p = \frac{n+4}{n-4}$, under suitable assumptions on a radial real-valued potential $V$, we prove global well-posedness for radial initial data in $H^2(\mathbb{R}^n)$. We also show that every such solution scatters in $H^2$ to a free solution of the biharmonic Schrödinger equation. The proof relies on Strichartz estimates for fourth-order Schrödinger operators with potential, equivalence of Sobolev norms associated with $Δ^2+V$ and $Δ^2$, boundedness of wave operators, perturbative stability theory, and a Morawetz-type estimate adapted to the presence of a potential. This extends earlier results for the case without potential to a class of radial potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13892 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case Nakayama, Hikaru Analysis of PDEs We consider the defocusing fourth-order nonlinear Schrödinger equation with potential \[ i\partial_t u + Δ^2 u + Vu + λ|u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), \] in dimensions $n \ge 5$. In the energy-critical case $p = \frac{n+4}{n-4}$, under suitable assumptions on a radial real-valued potential $V$, we prove global well-posedness for radial initial data in $H^2(\mathbb{R}^n)$. We also show that every such solution scatters in $H^2$ to a free solution of the biharmonic Schrödinger equation. The proof relies on Strichartz estimates for fourth-order Schrödinger operators with potential, equivalence of Sobolev norms associated with $Δ^2+V$ and $Δ^2$, boundedness of wave operators, perturbative stability theory, and a Morawetz-type estimate adapted to the presence of a potential. This extends earlier results for the case without potential to a class of radial potentials. |
| title | Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.13892 |