Advances in Nonlinear Analysis (Aug 2025)
Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
Abstract
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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