Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with indefinite potential in the radial case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916308754366464 |
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| author | Wang, Jun Yin, Zhaoyang |
| author_facet | Wang, Jun Yin, Zhaoyang |
| contents | In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schrödinger equation with indefinite potential \begin{equation*}
\left\{\begin{array}{l} i \partial_t u+Δu-V(x)u +|u|^{\frac{4}{N-2}}u=0,\ (x, t) \in \mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H ^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. Moreover, we get a positive ground state solution which is radially symmetric by using variational methods. This paper extends the results of \cite{KCEMF2006}(Invent. Math) to the potential equation and develops the recent conclusions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with indefinite potential in the radial case Wang, Jun Yin, Zhaoyang Analysis of PDEs 35Q55, 35R11, 37K05, 37L50 In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schrödinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+Δu-V(x)u +|u|^{\frac{4}{N-2}}u=0,\ (x, t) \in \mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H ^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. Moreover, we get a positive ground state solution which is radially symmetric by using variational methods. This paper extends the results of \cite{KCEMF2006}(Invent. Math) to the potential equation and develops the recent conclusions. |
| title | Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with indefinite potential in the radial case |
| topic | Analysis of PDEs 35Q55, 35R11, 37K05, 37L50 |
| url | https://arxiv.org/abs/2407.01588 |