Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN
In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,in ℝN,u∈H2ℝN, where a>0, b≥0, λ is a positive parameter, α∈N−2,N, 5≤N≤8, V:ℝN⟶ℝ is a potential function, and Iα is a Riesz potent...
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Wiley
2022-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2022/5820136 |
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| author | Li Zhou Chuanxi Zhu |
| author_facet | Li Zhou Chuanxi Zhu |
| author_sort | Li Zhou |
| collection | DOAJ |
| description | In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,in ℝN,u∈H2ℝN, where a>0, b≥0, λ is a positive parameter, α∈N−2,N, 5≤N≤8, V:ℝN⟶ℝ is a potential function, and Iα is a Riesz potential of order α. Here, 2∗∗=2N/n−4 with N≥5 is the Sobolev critical exponent, and Δ2u=ΔΔu is the biharmonic operator. Under certain assumptions on Vx and fu, we prove that the equation has ground state solutions by variational methods. |
| format | Article |
| id | doaj-art-f228bb0f2798447fb2f0106886c52dce |
| institution | Kabale University |
| issn | 1687-9139 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-f228bb0f2798447fb2f0106886c52dce2025-02-03T01:20:36ZengWileyAdvances in Mathematical Physics1687-91392022-01-01202210.1155/2022/5820136Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝNLi Zhou0Chuanxi Zhu1Zhejiang University of Science & TechnologyDepartment of MathematicsIn this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,in ℝN,u∈H2ℝN, where a>0, b≥0, λ is a positive parameter, α∈N−2,N, 5≤N≤8, V:ℝN⟶ℝ is a potential function, and Iα is a Riesz potential of order α. Here, 2∗∗=2N/n−4 with N≥5 is the Sobolev critical exponent, and Δ2u=ΔΔu is the biharmonic operator. Under certain assumptions on Vx and fu, we prove that the equation has ground state solutions by variational methods.http://dx.doi.org/10.1155/2022/5820136 |
| spellingShingle | Li Zhou Chuanxi Zhu Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN Advances in Mathematical Physics |
| title | Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN |
| title_full | Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN |
| title_fullStr | Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN |
| title_full_unstemmed | Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN |
| title_short | Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN |
| title_sort | ground state solution for a fourth order elliptic equation of kirchhoff type with critical growth in rn |
| url | http://dx.doi.org/10.1155/2022/5820136 |
| work_keys_str_mv | AT lizhou groundstatesolutionforafourthorderellipticequationofkirchhofftypewithcriticalgrowthinrn AT chuanxizhu groundstatesolutionforafourthorderellipticequationofkirchhofftypewithcriticalgrowthinrn |