Existence of solutions for a class of Kirchhoff-type equations with indefinite potential
Abstract In this paper, we consider the existence of solutions of the following Kirchhoff-type problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R 3 , u ∈ H 1 ( R 3 ) , $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-08-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-021-01550-5 |
| _version_ | 1818583804971843584 |
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| author | Jian Zhou Yunshun Wu |
| author_facet | Jian Zhou Yunshun Wu |
| author_sort | Jian Zhou |
| collection | DOAJ |
| description | Abstract In this paper, we consider the existence of solutions of the following Kirchhoff-type problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R 3 , u ∈ H 1 ( R 3 ) , $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$ where a , b > 0 $a,b>0$ are constants, and the potential V ( x ) $V(x)$ is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory. |
| first_indexed | 2024-12-16T08:11:07Z |
| format | Article |
| id | doaj.art-472af40d33ce4c1d8fc74f4e2e90cdc8 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-16T08:11:07Z |
| publishDate | 2021-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-472af40d33ce4c1d8fc74f4e2e90cdc82022-12-21T22:38:20ZengSpringerOpenBoundary Value Problems1687-27702021-08-012021111310.1186/s13661-021-01550-5Existence of solutions for a class of Kirchhoff-type equations with indefinite potentialJian Zhou0Yunshun Wu1School of Mathematical Sciences, Guizhou Nromal UniversitySchool of Mathematical Sciences, Guizhou Nromal UniversityAbstract In this paper, we consider the existence of solutions of the following Kirchhoff-type problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R 3 , u ∈ H 1 ( R 3 ) , $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$ where a , b > 0 $a,b>0$ are constants, and the potential V ( x ) $V(x)$ is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.https://doi.org/10.1186/s13661-021-01550-5Kirchhoff-type equationVariational methodsPalais–Smale conditionLocal linkingMorse theory |
| spellingShingle | Jian Zhou Yunshun Wu Existence of solutions for a class of Kirchhoff-type equations with indefinite potential Boundary Value Problems Kirchhoff-type equation Variational methods Palais–Smale condition Local linking Morse theory |
| title | Existence of solutions for a class of Kirchhoff-type equations with indefinite potential |
| title_full | Existence of solutions for a class of Kirchhoff-type equations with indefinite potential |
| title_fullStr | Existence of solutions for a class of Kirchhoff-type equations with indefinite potential |
| title_full_unstemmed | Existence of solutions for a class of Kirchhoff-type equations with indefinite potential |
| title_short | Existence of solutions for a class of Kirchhoff-type equations with indefinite potential |
| title_sort | existence of solutions for a class of kirchhoff type equations with indefinite potential |
| topic | Kirchhoff-type equation Variational methods Palais–Smale condition Local linking Morse theory |
| url | https://doi.org/10.1186/s13661-021-01550-5 |
| work_keys_str_mv | AT jianzhou existenceofsolutionsforaclassofkirchhofftypeequationswithindefinitepotential AT yunshunwu existenceofsolutionsforaclassofkirchhofftypeequationswithindefinitepotential |