Normalized solutions for Choquard equations with general nonlinearities

Shuai Yuan; Sitong Chen; Xianhua Tang; Shuai Yuan; Sitong Chen; Xianhua Tang

doi:10.3934/era.2020017

Electronic Research Archive

2020, Volume 28, Issue 1: 291-309. doi: 10.3934/era.2020017

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Normalized solutions for Choquard equations with general nonlinearities

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received: 01 November 2019 Revised: 01 January 2020
Primary: 58F15, 58F17; Secondary: 53C35

In this paper, we prove the existence of positive solutions with prescribed $ L^{2} $-norm to the following Choquard equation:
$ \begin{equation*} -\Delta u-\lambda u = (I_{\alpha}*F(u))f(u), \ \ \ \ x\in \mathbb{R}^3, \end{equation*} $
where $ \lambda\in \mathbb{R}, \alpha\in (0,3) $ and $ I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R} $ is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any $ c>0 $, the above equation possesses at least a couple of weak solution $ (\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^- $ such that $ \|\bar{u}_c\|_{2}^{2} = c $.
- Choquard equations,
- normalized solution,
- variational method,
- minimax method,
- weak solutions
Citation: Shuai Yuan, Sitong Chen, Xianhua Tang. Normalized solutions for Choquard equations with general nonlinearities[J]. Electronic Research Archive, 2020, 28(1): 291-309. doi: 10.3934/era.2020017

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Abstract

In this paper, we prove the existence of positive solutions with prescribed $ L^{2} $-norm to the following Choquard equation:

$ \begin{equation*} -\Delta u-\lambda u = (I_{\alpha}*F(u))f(u), \ \ \ \ x\in \mathbb{R}^3, \end{equation*} $

where $ \lambda\in \mathbb{R}, \alpha\in (0,3) $ and $ I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R} $ is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any $ c>0 $, the above equation possesses at least a couple of weak solution $ (\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^- $ such that $ \|\bar{u}_c\|_{2}^{2} = c $.

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