Ground states of a Kirchhoff equation with the potential on the lattice graphs
<p>In this paper, we study the nonlinear Kirchhoff equation</p> <p class="disp_formula">$ \begin{align*} -\Big(a+b\int_{\mathbb{Z}^{3}}|\nabla u|^{2} d \mu\Big)\Delta u+V(x)u = f(u) \end{align*} $</p> <p>on lattice graph $ \mathbb{Z}^3 $, where $ a, b &a...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-11-01
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| Series: | Communications in Analysis and Mechanics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/cam.2023038?viewType=HTML |
| _version_ | 1827386719959252992 |
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| author | Wenqian Lv |
| author_facet | Wenqian Lv |
| author_sort | Wenqian Lv |
| collection | DOAJ |
| description | <p>In this paper, we study the nonlinear Kirchhoff equation</p>
<p class="disp_formula">$ \begin{align*} -\Big(a+b\int_{\mathbb{Z}^{3}}|\nabla u|^{2} d \mu\Big)\Delta u+V(x)u = f(u) \end{align*} $</p>
<p>on lattice graph $ \mathbb{Z}^3 $, where $ a, b > 0 $ are constants and $ V:\mathbb{Z}^{3}\rightarrow \mathbb{R} $ is a positive function. Under a Nehari-type condition and 4-superlinearity condition on $ f $, we use the Nehari method to prove the existence of ground-state solutions to the above equation when $ V $ is coercive. Moreover, we extend the result to noncompact cases in which $ V $ is a periodic function or a bounded potential well.</p> |
| first_indexed | 2024-03-08T15:50:38Z |
| format | Article |
| id | doaj.art-570c3a00f4c04274b689b332c72c0d81 |
| institution | Directory Open Access Journal |
| issn | 2836-3310 |
| language | English |
| last_indexed | 2024-03-08T15:50:38Z |
| publishDate | 2023-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Communications in Analysis and Mechanics |
| spelling | doaj.art-570c3a00f4c04274b689b332c72c0d812024-01-09T06:08:25ZengAIMS PressCommunications in Analysis and Mechanics2836-33102023-11-0115479281010.3934/cam.2023038Ground states of a Kirchhoff equation with the potential on the lattice graphsWenqian Lv0School of Mathematics, East China University of Science and Technology, Shanghai, 200237, China<p>In this paper, we study the nonlinear Kirchhoff equation</p> <p class="disp_formula">$ \begin{align*} -\Big(a+b\int_{\mathbb{Z}^{3}}|\nabla u|^{2} d \mu\Big)\Delta u+V(x)u = f(u) \end{align*} $</p> <p>on lattice graph $ \mathbb{Z}^3 $, where $ a, b > 0 $ are constants and $ V:\mathbb{Z}^{3}\rightarrow \mathbb{R} $ is a positive function. Under a Nehari-type condition and 4-superlinearity condition on $ f $, we use the Nehari method to prove the existence of ground-state solutions to the above equation when $ V $ is coercive. Moreover, we extend the result to noncompact cases in which $ V $ is a periodic function or a bounded potential well.</p>https://www.aimspress.com/article/doi/10.3934/cam.2023038?viewType=HTMLkirchhoff equationlattice graphground statesnehari manifoldvariational methods |
| spellingShingle | Wenqian Lv Ground states of a Kirchhoff equation with the potential on the lattice graphs Communications in Analysis and Mechanics kirchhoff equation lattice graph ground states nehari manifold variational methods |
| title | Ground states of a Kirchhoff equation with the potential on the lattice graphs |
| title_full | Ground states of a Kirchhoff equation with the potential on the lattice graphs |
| title_fullStr | Ground states of a Kirchhoff equation with the potential on the lattice graphs |
| title_full_unstemmed | Ground states of a Kirchhoff equation with the potential on the lattice graphs |
| title_short | Ground states of a Kirchhoff equation with the potential on the lattice graphs |
| title_sort | ground states of a kirchhoff equation with the potential on the lattice graphs |
| topic | kirchhoff equation lattice graph ground states nehari manifold variational methods |
| url | https://www.aimspress.com/article/doi/10.3934/cam.2023038?viewType=HTML |
| work_keys_str_mv | AT wenqianlv groundstatesofakirchhoffequationwiththepotentialonthelatticegraphs |