Normalized positive solutions for Schrödinger equations with potentials in unbounded domains
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arXiv
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| Format: | Preprint |
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2022
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| author | Lancelotti, Sergio Molle, Riccardo |
| author_facet | Lancelotti, Sergio Molle, Riccardo |
| contents | The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$ -Δu+λu+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(Ω),\quad\int_Ωu^2dx=ρ^2,\quadλ\in\mathbb{R}, $$ where $Ω=\mathbb{R}^N$ or $\mathbb{R}^N\setminusΩ$ is a compact set, $ρ>0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in \left(2,2+\frac 4 N\right)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption $(D_ρ)$, or if $\|V\|_{L^q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption $(D_ρ)$ is fulfilled. There are no assumptions on the size of $\mathbb{R}^N\setminusΩ$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $Ω=\mathbb{R}^N$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_12090 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normalized positive solutions for Schrödinger equations with potentials in unbounded domains Lancelotti, Sergio Molle, Riccardo Analysis of PDEs The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$ -Δu+λu+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(Ω),\quad\int_Ωu^2dx=ρ^2,\quadλ\in\mathbb{R}, $$ where $Ω=\mathbb{R}^N$ or $\mathbb{R}^N\setminusΩ$ is a compact set, $ρ>0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in \left(2,2+\frac 4 N\right)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption $(D_ρ)$, or if $\|V\|_{L^q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption $(D_ρ)$ is fulfilled. There are no assumptions on the size of $\mathbb{R}^N\setminusΩ$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $Ω=\mathbb{R}^N$. |
| title | Normalized positive solutions for Schrödinger equations with potentials in unbounded domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2208.12090 |