Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918222445412352 |
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| author | Yu, Junwei |
| author_facet | Yu, Junwei |
| contents | We study the existence and multiplicity of positive normalized solutions with prescribed $L^{2}$-norm for the Sobolev critical Schrödinger equation $-ΔU + V(x) U = λU + |U|^{2^*-2} U$ in $\mathbb{R}^N$, $\int_{\mathbb{R}^N} U^2\,dx = ρ^2$, where $N \ge 3$, $V\ge 0$ is a trapping potential, $λ\in \mathbb{R}$ and $2^*=\frac{2N}{N-2}$. Our first result is that the existence of local minimum solutions for $ρ\in (0, ρ^*)$, for some suitable $ρ^* > 0$, under appropriate assumptions on the potential. These solutions correspond to ground states. Our second result concerns the existence of mountain pass solutions, under the same assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential Yu, Junwei Analysis of PDEs 35J20, 35B33, 35Q55, 35Q89, 35J61 We study the existence and multiplicity of positive normalized solutions with prescribed $L^{2}$-norm for the Sobolev critical Schrödinger equation $-ΔU + V(x) U = λU + |U|^{2^*-2} U$ in $\mathbb{R}^N$, $\int_{\mathbb{R}^N} U^2\,dx = ρ^2$, where $N \ge 3$, $V\ge 0$ is a trapping potential, $λ\in \mathbb{R}$ and $2^*=\frac{2N}{N-2}$. Our first result is that the existence of local minimum solutions for $ρ\in (0, ρ^*)$, for some suitable $ρ^* > 0$, under appropriate assumptions on the potential. These solutions correspond to ground states. Our second result concerns the existence of mountain pass solutions, under the same assumptions. |
| title | Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential |
| topic | Analysis of PDEs 35J20, 35B33, 35Q55, 35Q89, 35J61 |
| url | https://arxiv.org/abs/2511.19271 |