Normalized ground states for nonlinear Schrödinger equations with general Sobolev critical nonlinearities
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914929170186240 |
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| author | Liu, Manting Chang, Xiaojun |
| author_facet | Liu, Manting Chang, Xiaojun |
| contents | In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger equation \begin{equation*} \left\{ \begin{aligned} &-Δu=f(u)+ λu\quad \mbox{in}\ \mathbb{R}^{N},\\ &u\in H^1(\mathbb{R}^N), ~~~\int_{\mathbb{R}^N}|u|^2dx=c, \end{aligned} \right. \end{equation*} where $N\ge3$, $c>0$, $λ\in \mathbb{R}$ and $f$ has a Sobolev critical growth at infinity but does not satisfies the Ambrosetti-Rabinowitz condition. By analysing the monotonicity of the ground state energy with respect to $c$, we develop a constrained minimization approach to establish the existence of normalized ground state solutions for all $c>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_06908 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normalized ground states for nonlinear Schrödinger equations with general Sobolev critical nonlinearities Liu, Manting Chang, Xiaojun Analysis of PDEs Primary: 35Q55, Secondary: 35J20, 35J60, 47J30 In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger equation \begin{equation*} \left\{ \begin{aligned} &-Δu=f(u)+ λu\quad \mbox{in}\ \mathbb{R}^{N},\\ &u\in H^1(\mathbb{R}^N), ~~~\int_{\mathbb{R}^N}|u|^2dx=c, \end{aligned} \right. \end{equation*} where $N\ge3$, $c>0$, $λ\in \mathbb{R}$ and $f$ has a Sobolev critical growth at infinity but does not satisfies the Ambrosetti-Rabinowitz condition. By analysing the monotonicity of the ground state energy with respect to $c$, we develop a constrained minimization approach to establish the existence of normalized ground state solutions for all $c>0$. |
| title | Normalized ground states for nonlinear Schrödinger equations with general Sobolev critical nonlinearities |
| topic | Analysis of PDEs Primary: 35Q55, Secondary: 35J20, 35J60, 47J30 |
| url | https://arxiv.org/abs/2209.06908 |