Multiple nodal solutions of the Kirchhoff-type problem with a cubic term
In this article, we are interested in the following Kirchhoff-type problem (0.1)−a+b∫RN∣∇u∣2dxΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN),\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-03-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2022-0225 |
| _version_ | 1828355382425157632 |
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| author | Wang Tao Yang Yanling Guo Hui |
| author_facet | Wang Tao Yang Yanling Guo Hui |
| author_sort | Wang Tao |
| collection | DOAJ |
| description | In this article, we are interested in the following Kirchhoff-type problem (0.1)−a+b∫RN∣∇u∣2dxΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN),\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where a,b>0,N=2a,b\gt 0,N=2 or 3, the potential function VV is radial and bounded from below by a positive number. Because the nonlocal b∣∇u∣L2(RN)2Δub| \nabla u\hspace{-0.25em}{| }_{{L}^{2}\left({{\mathbb{R}}}^{N})}^{2}\Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣u∣2u| u\hspace{-0.25em}{| }^{2}u. This causes that not all function in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer kk, equation (0.1) admits a radial nodal solution Uk,4b{U}_{k,4}^{b} having exactly kk nodes. Moreover, we show that the energy of Uk,4b{U}_{k,4}^{b} is strictly increasing in kk and for any sequence {bn}\left\{{b}_{n}\right\} with bn→0+,{b}_{n}\to {0}_{+}, up to a subsequence, Uk,4bn{U}_{k,4}^{{b}_{n}} converges to Uk,40{U}_{k,4}^{0} in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}), which is a radial nodal solution with exactly kk nodes of the classical Schrödinger equation −aΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN).\left\{\begin{array}{l}-a\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}).\end{array}\right. Our results extend the existence result from the super-cubic case to the cubic case. |
| first_indexed | 2024-04-14T02:40:09Z |
| format | Article |
| id | doaj.art-776006117d8b431d8723246b304a1693 |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-04-14T02:40:09Z |
| publishDate | 2022-03-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-776006117d8b431d8723246b304a16932022-12-22T02:17:09ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2022-03-011111030104710.1515/anona-2022-0225Multiple nodal solutions of the Kirchhoff-type problem with a cubic termWang Tao0Yang Yanling1Guo Hui2College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. ChinaCollege of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. ChinaCollege of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. ChinaIn this article, we are interested in the following Kirchhoff-type problem (0.1)−a+b∫RN∣∇u∣2dxΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN),\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where a,b>0,N=2a,b\gt 0,N=2 or 3, the potential function VV is radial and bounded from below by a positive number. Because the nonlocal b∣∇u∣L2(RN)2Δub| \nabla u\hspace{-0.25em}{| }_{{L}^{2}\left({{\mathbb{R}}}^{N})}^{2}\Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣u∣2u| u\hspace{-0.25em}{| }^{2}u. This causes that not all function in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer kk, equation (0.1) admits a radial nodal solution Uk,4b{U}_{k,4}^{b} having exactly kk nodes. Moreover, we show that the energy of Uk,4b{U}_{k,4}^{b} is strictly increasing in kk and for any sequence {bn}\left\{{b}_{n}\right\} with bn→0+,{b}_{n}\to {0}_{+}, up to a subsequence, Uk,4bn{U}_{k,4}^{{b}_{n}} converges to Uk,40{U}_{k,4}^{0} in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}), which is a radial nodal solution with exactly kk nodes of the classical Schrödinger equation −aΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN).\left\{\begin{array}{l}-a\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}).\end{array}\right. Our results extend the existence result from the super-cubic case to the cubic case.https://doi.org/10.1515/anona-2022-0225kirchhoff-type equationsnodal solutionsmiranda theoremgersgorin disk theorem35a1535j2035j50 |
| spellingShingle | Wang Tao Yang Yanling Guo Hui Multiple nodal solutions of the Kirchhoff-type problem with a cubic term Advances in Nonlinear Analysis kirchhoff-type equations nodal solutions miranda theorem gersgorin disk theorem 35a15 35j20 35j50 |
| title | Multiple nodal solutions of the Kirchhoff-type problem with a cubic term |
| title_full | Multiple nodal solutions of the Kirchhoff-type problem with a cubic term |
| title_fullStr | Multiple nodal solutions of the Kirchhoff-type problem with a cubic term |
| title_full_unstemmed | Multiple nodal solutions of the Kirchhoff-type problem with a cubic term |
| title_short | Multiple nodal solutions of the Kirchhoff-type problem with a cubic term |
| title_sort | multiple nodal solutions of the kirchhoff type problem with a cubic term |
| topic | kirchhoff-type equations nodal solutions miranda theorem gersgorin disk theorem 35a15 35j20 35j50 |
| url | https://doi.org/10.1515/anona-2022-0225 |
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