Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities
In this paper, we study the multiplicity of solutions for the following fourth-order Kirchhoff type problem involving concave-convex nonlinearities and indefinite weight function $ \begin{equation*} \Delta^2u-\left(a+b\int_{ \mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+V(x)u = \lambda f(x)|u|^{q-2}...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-03-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2022044?viewType=HTML |
| _version_ | 1811341480162230272 |
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| author | Zijian Wu Haibo Chen |
| author_facet | Zijian Wu Haibo Chen |
| author_sort | Zijian Wu |
| collection | DOAJ |
| description | In this paper, we study the multiplicity of solutions for the following fourth-order Kirchhoff type problem involving concave-convex nonlinearities and indefinite weight function
$ \begin{equation*} \Delta^2u-\left(a+b\int_{ \mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+V(x)u = \lambda f(x)|u|^{q-2}u+|u|^{p-2}u, \end{equation*} $
where $ u\in H^2(\mathbb{R}^N)(4 < N < 8) $, $ \lambda > 0, 1 < q < 2, 4 < p < 2_\ast(2_\ast = 2N/(N-4)) $, $ f(x) $ satisfy suitable conditions, and $ f(x) $ may change sign in $ \mathbb{R}^N $. Using Nehari manifold and fibering maps, the existense of multiple solutions is established. Moreover, the existence of sign-changing solution is obtained for $ f(x)\equiv0 $. Our results generalize some recent results in the literature. |
| first_indexed | 2024-04-13T18:56:46Z |
| format | Article |
| id | doaj.art-d4e4102953a84ceba250d3ba80925af3 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-13T18:56:46Z |
| publishDate | 2022-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-d4e4102953a84ceba250d3ba80925af32022-12-22T02:34:14ZengAIMS PressElectronic Research Archive2688-15942022-03-0130383084910.3934/era.2022044Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearitiesZijian Wu0Haibo Chen1School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, ChinaSchool of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, ChinaIn this paper, we study the multiplicity of solutions for the following fourth-order Kirchhoff type problem involving concave-convex nonlinearities and indefinite weight function $ \begin{equation*} \Delta^2u-\left(a+b\int_{ \mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+V(x)u = \lambda f(x)|u|^{q-2}u+|u|^{p-2}u, \end{equation*} $ where $ u\in H^2(\mathbb{R}^N)(4 < N < 8) $, $ \lambda > 0, 1 < q < 2, 4 < p < 2_\ast(2_\ast = 2N/(N-4)) $, $ f(x) $ satisfy suitable conditions, and $ f(x) $ may change sign in $ \mathbb{R}^N $. Using Nehari manifold and fibering maps, the existense of multiple solutions is established. Moreover, the existence of sign-changing solution is obtained for $ f(x)\equiv0 $. Our results generalize some recent results in the literature.https://www.aimspress.com/article/doi/10.3934/era.2022044?viewType=HTMLfourth-order kirchhoff type problemsmultiple solutionsindefinite weight functionsnehari manifoldfibering maps |
| spellingShingle | Zijian Wu Haibo Chen Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities Electronic Research Archive fourth-order kirchhoff type problems multiple solutions indefinite weight functions nehari manifold fibering maps |
| title | Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities |
| title_full | Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities |
| title_fullStr | Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities |
| title_full_unstemmed | Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities |
| title_short | Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities |
| title_sort | multiple solutions for the fourth order kirchhoff type problems in mathbb r n involving concave convex nonlinearities |
| topic | fourth-order kirchhoff type problems multiple solutions indefinite weight functions nehari manifold fibering maps |
| url | https://www.aimspress.com/article/doi/10.3934/era.2022044?viewType=HTML |
| work_keys_str_mv | AT zijianwu multiplesolutionsforthefourthorderkirchhofftypeproblemsinmathbbrninvolvingconcaveconvexnonlinearities AT haibochen multiplesolutionsforthefourthorderkirchhofftypeproblemsinmathbbrninvolvingconcaveconvexnonlinearities |