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Normalized solutions for the mixed dispersion nonlinear Schrödinger equations with four types of potentials and mass subcritical growth
- Source :
- Electronic Research Archive, Vol 31, Iss 7, Pp 3759-3775 (2023)
- Publication Year :
- 2023
- Publisher :
- AIMS Press, 2023.
-
Abstract
- This paper is devoted to considering the attainability of minimizers of the $ L^2 $-constraint variational problem $ m_{\gamma, a} = \inf \, \{J_{\gamma}(u):u\in H^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}} \vert u\vert^2 dx = a^2 \} {, } $ where $ J_{\gamma}(u) = \frac{\gamma}{2}\int_{\mathbb{R}^{N}} \vert\Delta u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} \vert\nabla u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} V(x)\vert u\vert^2 dx-\frac{1}{2\sigma+2}\int_{\mathbb{R}^{N}} \vert u\vert^{2\sigma+2} dx, $ $ \gamma > 0 $, $ a > 0 $, $ \sigma\in(0, \frac{2}{N}) $ with $ N\ge 2 $. Moreover, the function $ V:\mathbb{R}^{N}\rightarrow [0, +\infty) $ is continuous and bounded. By using the variational methods, we can prove that, when $ V $ satisfies four different assumptions, $ m_{\gamma, a} $ are all achieved.
Details
- Language :
- English
- ISSN :
- 26881594
- Volume :
- 31
- Issue :
- 7
- Database :
- Directory of Open Access Journals
- Journal :
- Electronic Research Archive
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.168bcdd14a463a9ed65da1080d81c5
- Document Type :
- article
- Full Text :
- https://doi.org/10.3934/era.2023191?viewType=HTML