Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
This study deals with the existence of nodal solutions for the following gauged nonlinear Schrödinger equation with zero mass: −Δu+hu2(∣x∣)∣x∣2+∫∣x∣+∞hu(s)su2(s)dsu=∣u∣p−2u,x∈R2,-\Delta u+\left(\frac{{h}_{u}^{2}\left(| x| )}{{| x| }^{2}}+\underset{| x| }{\overset{+\infty }{\int }}\frac{{h}_{u}\left(...
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| Language: | English |
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De Gruyter
2024-11-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2024-0055 |
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| author | Deng Yinbin Liu Chenchen Yang Xian |
| author_facet | Deng Yinbin Liu Chenchen Yang Xian |
| author_sort | Deng Yinbin |
| collection | DOAJ |
| description | This study deals with the existence of nodal solutions for the following gauged nonlinear Schrödinger equation with zero mass: −Δu+hu2(∣x∣)∣x∣2+∫∣x∣+∞hu(s)su2(s)dsu=∣u∣p−2u,x∈R2,-\Delta u+\left(\frac{{h}_{u}^{2}\left(| x| )}{{| x| }^{2}}+\underset{| x| }{\overset{+\infty }{\int }}\frac{{h}_{u}\left(s)}{s}{u}^{2}\left(s){\rm{d}}s\right)u={| u| }^{p-2}u,\hspace{1.0em}x\in {{\mathbb{R}}}^{2}, where p>6p\gt 6 and hu(s)=12∫0sru2(r)dr{h}_{u}\left(s)=\frac{1}{2}{\int }_{0}^{s}r{u}^{2}\left(r){\rm{d}}r. By variational methods, we prove that for any integer k≥0k\ge 0, the above equation has a nodal solution wk{w}_{k} which changes sign exactly kk times. Moreover, we also prove that wk{w}_{k} belongs to L2(R2){L}^{2}\left({{\mathbb{R}}}^{2}) provided p>10p\gt 10. |
| format | Article |
| id | doaj-art-9231a67fc5cc43fd8217fafd3ee75862 |
| institution | DOAJ |
| issn | 2191-950X |
| language | English |
| publishDate | 2024-11-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-9231a67fc5cc43fd8217fafd3ee758622025-08-20T02:50:48ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2024-11-011313027610.1515/anona-2024-0055Nodal solutions for a zero-mass Chern-Simons-Schrödinger equationDeng Yinbin0Liu Chenchen1Yang Xian2School of Mathematics and Statistics & Key Laboratory of Nonlinear Analysis and Applications, Central China Normal University, Wuhan 430079, ChinaSchool of Mathematics and Statistics, Central China Normal University, Wuhan 430079, ChinaSchool of Mathematics and Information Science, Guangxi University, Nanning 530004, ChinaThis study deals with the existence of nodal solutions for the following gauged nonlinear Schrödinger equation with zero mass: −Δu+hu2(∣x∣)∣x∣2+∫∣x∣+∞hu(s)su2(s)dsu=∣u∣p−2u,x∈R2,-\Delta u+\left(\frac{{h}_{u}^{2}\left(| x| )}{{| x| }^{2}}+\underset{| x| }{\overset{+\infty }{\int }}\frac{{h}_{u}\left(s)}{s}{u}^{2}\left(s){\rm{d}}s\right)u={| u| }^{p-2}u,\hspace{1.0em}x\in {{\mathbb{R}}}^{2}, where p>6p\gt 6 and hu(s)=12∫0sru2(r)dr{h}_{u}\left(s)=\frac{1}{2}{\int }_{0}^{s}r{u}^{2}\left(r){\rm{d}}r. By variational methods, we prove that for any integer k≥0k\ge 0, the above equation has a nodal solution wk{w}_{k} which changes sign exactly kk times. Moreover, we also prove that wk{w}_{k} belongs to L2(R2){L}^{2}\left({{\mathbb{R}}}^{2}) provided p>10p\gt 10.https://doi.org/10.1515/anona-2024-0055gauged schrödinger equationnodal solutionzero massvariational methods35j2035q5535r09 |
| spellingShingle | Deng Yinbin Liu Chenchen Yang Xian Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation Advances in Nonlinear Analysis gauged schrödinger equation nodal solution zero mass variational methods 35j20 35q55 35r09 |
| title | Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation |
| title_full | Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation |
| title_fullStr | Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation |
| title_full_unstemmed | Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation |
| title_short | Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation |
| title_sort | nodal solutions for a zero mass chern simons schrodinger equation |
| topic | gauged schrödinger equation nodal solution zero mass variational methods 35j20 35q55 35r09 |
| url | https://doi.org/10.1515/anona-2024-0055 |
| work_keys_str_mv | AT dengyinbin nodalsolutionsforazeromasschernsimonsschrodingerequation AT liuchenchen nodalsolutionsforazeromasschernsimonsschrodingerequation AT yangxian nodalsolutionsforazeromasschernsimonsschrodingerequation |