Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
Let (ℳ,g)\left({\mathcal{ {\mathcal M} }},g) and (K,κ)\left({\mathcal{K}},\kappa ) be two Riemannian manifolds of dimensions NN and mm, respectively. Let ω∈C2(ℳ)\omega \in {C}^{2}\left({\mathcal{ {\mathcal M} }}) satisfy ω>0\omega \gt 0. The warped product ℳ×ωK{\mathcal{ {\mathcal M} }}{\times }_...
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De Gruyter
2025-08-01
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| Online Access: | https://doi.org/10.1515/anona-2025-0096 |
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| author | Chen Wenjing Wang Zexi |
| author_facet | Chen Wenjing Wang Zexi |
| author_sort | Chen Wenjing |
| collection | DOAJ |
| description | Let (ℳ,g)\left({\mathcal{ {\mathcal M} }},g) and (K,κ)\left({\mathcal{K}},\kappa ) be two Riemannian manifolds of dimensions NN and mm, respectively. Let ω∈C2(ℳ)\omega \in {C}^{2}\left({\mathcal{ {\mathcal M} }}) satisfy ω>0\omega \gt 0. The warped product ℳ×ωK{\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}} is the (N+m)\left(N+m)-dimensional product manifold ℳ×K{\mathcal{ {\mathcal M} }}\times {\mathcal{K}} equipped with the metric g+ω2κg+{\omega }^{2}\kappa . We consider the following elliptic system: (1)−Δg+ω2κu+h(x)u=vp−αε,in (ℳ×ωK,g+ω2κ),−Δg+ω2κv+h(x)v=uq−βε,in (ℳ×ωK,g+ω2κ),u,v>0,in (ℳ×ωK,g+ω2κ),\left\{\begin{array}{ll}-\hspace{-0.03em}{\Delta }_{g+{\omega }^{2}\kappa }u+h\left(x)u={v}^{p-\alpha \varepsilon },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\\ -{\Delta }_{g+{\omega }^{2}\kappa }v+h\left(x)v={u}^{q-\beta \varepsilon },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\\ u,v\gt 0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\end{array}\right. where Δg+ω2κ=divg+ω2κ∇{\Delta }_{g+{\omega }^{2}\kappa }={{\rm{div}}}_{g+{\omega }^{2}\kappa }\nabla denotes the Laplace-Beltrami operator on ℳ×ωK{\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}}, h(x)h\left(x) is a C1{C}^{1}-function on ℳ×ωK{\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}}, α,β>0\alpha ,\beta \gt 0 are the real numbers, ε>0\varepsilon \gt 0 is a small parameter, and (p,q)∈(1,+∞)×(1,+∞)\left(p,q)\in \left(1,+\infty )\times \left(1,+\infty ) satisfies 1p+1+1q+1=N−2N\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. For any integer k≥2k\ge 2, using the Lyapunov-Schmidt reduction, we prove that problem (1) admits a kk-peak solution concentrated along an mm-dimensional minimal submanifold of (ℳ×ωK)k{\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}})}^{k} as ε→0\varepsilon \to 0. |
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| institution | DOAJ |
| issn | 2191-950X |
| language | English |
| publishDate | 2025-08-01 |
| publisher | De Gruyter |
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| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-edb1e6051ff542ee91d367ca41ebdcc52025-08-20T03:07:00ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-08-0114126929610.1515/anona-2025-0096Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifoldsChen Wenjing0Wang Zexi1School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. ChinaSchool of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. ChinaLet (ℳ,g)\left({\mathcal{ {\mathcal M} }},g) and (K,κ)\left({\mathcal{K}},\kappa ) be two Riemannian manifolds of dimensions NN and mm, respectively. Let ω∈C2(ℳ)\omega \in {C}^{2}\left({\mathcal{ {\mathcal M} }}) satisfy ω>0\omega \gt 0. The warped product ℳ×ωK{\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}} is the (N+m)\left(N+m)-dimensional product manifold ℳ×K{\mathcal{ {\mathcal M} }}\times {\mathcal{K}} equipped with the metric g+ω2κg+{\omega }^{2}\kappa . We consider the following elliptic system: (1)−Δg+ω2κu+h(x)u=vp−αε,in (ℳ×ωK,g+ω2κ),−Δg+ω2κv+h(x)v=uq−βε,in (ℳ×ωK,g+ω2κ),u,v>0,in (ℳ×ωK,g+ω2κ),\left\{\begin{array}{ll}-\hspace{-0.03em}{\Delta }_{g+{\omega }^{2}\kappa }u+h\left(x)u={v}^{p-\alpha \varepsilon },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\\ -{\Delta }_{g+{\omega }^{2}\kappa }v+h\left(x)v={u}^{q-\beta \varepsilon },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\\ u,v\gt 0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\end{array}\right. where Δg+ω2κ=divg+ω2κ∇{\Delta }_{g+{\omega }^{2}\kappa }={{\rm{div}}}_{g+{\omega }^{2}\kappa }\nabla denotes the Laplace-Beltrami operator on ℳ×ωK{\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}}, h(x)h\left(x) is a C1{C}^{1}-function on ℳ×ωK{\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}}, α,β>0\alpha ,\beta \gt 0 are the real numbers, ε>0\varepsilon \gt 0 is a small parameter, and (p,q)∈(1,+∞)×(1,+∞)\left(p,q)\in \left(1,+\infty )\times \left(1,+\infty ) satisfies 1p+1+1q+1=N−2N\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. For any integer k≥2k\ge 2, using the Lyapunov-Schmidt reduction, we prove that problem (1) admits a kk-peak solution concentrated along an mm-dimensional minimal submanifold of (ℳ×ωK)k{\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}})}^{k} as ε→0\varepsilon \to 0.https://doi.org/10.1515/anona-2025-0096blowing-up solutionsconcentrated along minimal submanifoldsupercritical hamiltonian systemriemannian manifolds58j0535j4735b33 |
| spellingShingle | Chen Wenjing Wang Zexi Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds Advances in Nonlinear Analysis blowing-up solutions concentrated along minimal submanifold supercritical hamiltonian system riemannian manifolds 58j05 35j47 35b33 |
| title | Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds |
| title_full | Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds |
| title_fullStr | Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds |
| title_full_unstemmed | Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds |
| title_short | Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds |
| title_sort | blowing up solutions concentrated along minimal submanifolds for some supercritical hamiltonian systems on riemannian manifolds |
| topic | blowing-up solutions concentrated along minimal submanifold supercritical hamiltonian system riemannian manifolds 58j05 35j47 35b33 |
| url | https://doi.org/10.1515/anona-2025-0096 |
| work_keys_str_mv | AT chenwenjing blowingupsolutionsconcentratedalongminimalsubmanifoldsforsomesupercriticalhamiltoniansystemsonriemannianmanifolds AT wangzexi blowingupsolutionsconcentratedalongminimalsubmanifoldsforsomesupercriticalhamiltoniansystemsonriemannianmanifolds |