Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
In this article, we investigate the following nonlinear Kirchhoff equation with Sobolev critical growth: −a+b∫R3∣∇u∣2dxΔu+λu=μf(u)+∣u∣4u,inR3,u>0,∫R3∣u∣2dx=m2,inR3,(Pm)\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta...
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De Gruyter
2025-07-01
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| Online Access: | https://doi.org/10.1515/math-2025-0182 |
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| author | Zhang Shiyong Zhang Qiongfen |
| author_facet | Zhang Shiyong Zhang Qiongfen |
| author_sort | Zhang Shiyong |
| collection | DOAJ |
| description | In this article, we investigate the following nonlinear Kirchhoff equation with Sobolev critical growth: −a+b∫R3∣∇u∣2dxΔu+λu=μf(u)+∣u∣4u,inR3,u>0,∫R3∣u∣2dx=m2,inR3,(Pm)\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu f\left(u)+{| u| }^{4}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\gt 0,\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right.\hspace{2.0em}\hspace{2.0em}\left({P}_{m}) where μ\mu is a positive parameter, a,b>0a,b\gt 0, and the frequency λ\lambda appears as a positive Lagrange multiplier. The nonlinearity ff is more general and satisfies Sobolev subcritical conditions. With the assistance of the Pohožaev constraints and the Sobolev subcritical approximation method, we have achieved a couple of the normalized ground-state solutions to (Pm)\left({P}_{m}) and the asymptotic behavior of the ground-state is also studied. |
| format | Article |
| id | doaj-art-2b2df07869f245a0b6e0d72bcdac2fa8 |
| institution | Kabale University |
| issn | 2391-5455 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | De Gruyter |
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| series | Open Mathematics |
| spelling | doaj-art-2b2df07869f245a0b6e0d72bcdac2fa82025-08-20T03:58:48ZengDe GruyterOpen Mathematics2391-54552025-07-0123130533010.1515/math-2025-0182Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growthZhang Shiyong0Zhang Qiongfen1School of Mathematics and Statistics, Guilin University of Technology, Guilin, Guangxi 541004, P. R. ChinaSchool of Mathematics and Statistics, Guilin University of Technology, Guilin, Guangxi 541004, P. R. ChinaIn this article, we investigate the following nonlinear Kirchhoff equation with Sobolev critical growth: −a+b∫R3∣∇u∣2dxΔu+λu=μf(u)+∣u∣4u,inR3,u>0,∫R3∣u∣2dx=m2,inR3,(Pm)\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu f\left(u)+{| u| }^{4}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\gt 0,\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right.\hspace{2.0em}\hspace{2.0em}\left({P}_{m}) where μ\mu is a positive parameter, a,b>0a,b\gt 0, and the frequency λ\lambda appears as a positive Lagrange multiplier. The nonlinearity ff is more general and satisfies Sobolev subcritical conditions. With the assistance of the Pohožaev constraints and the Sobolev subcritical approximation method, we have achieved a couple of the normalized ground-state solutions to (Pm)\left({P}_{m}) and the asymptotic behavior of the ground-state is also studied.https://doi.org/10.1515/math-2025-0182normalized ground-state solutionsnonlinear kirchhoff equationcritical growthpohozaev manifold34c3735a1537j45 |
| spellingShingle | Zhang Shiyong Zhang Qiongfen Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth Open Mathematics normalized ground-state solutions nonlinear kirchhoff equation critical growth pohozaev manifold 34c37 35a15 37j45 |
| title | Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth |
| title_full | Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth |
| title_fullStr | Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth |
| title_full_unstemmed | Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth |
| title_short | Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth |
| title_sort | normalized ground states for the sobolev critical kirchhoff equation with at least mass critical growth |
| topic | normalized ground-state solutions nonlinear kirchhoff equation critical growth pohozaev manifold 34c37 35a15 37j45 |
| url | https://doi.org/10.1515/math-2025-0182 |
| work_keys_str_mv | AT zhangshiyong normalizedgroundstatesforthesobolevcriticalkirchhoffequationwithatleastmasscriticalgrowth AT zhangqiongfen normalizedgroundstatesforthesobolevcriticalkirchhoffequationwithatleastmasscriticalgrowth |