Spike layered solutions for elliptic systems on Riemannian Manifolds
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916927059787776 |
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| author | Kannoth, Anusree R Manna, Bhakti Bhusan |
| author_facet | Kannoth, Anusree R Manna, Bhakti Bhusan |
| contents | In this article, we study the following Hamiltonian system: \begin{equation*}
\begin{cases}
\begin{aligned}
&-\varepsilon^{2}Δ_{g} u +u = |v|^{q-1}v,
&-\varepsilon^{2}Δ_{g} v +v = |u|^{p-1}u && \text{ in } \mathcal{M},
& \quad u,v >0 && \text{ in } \mathcal{M},
\end{aligned}
\end{cases} \end{equation*} where $\mathcal{M}$ is a smooth, compact and connected Riemannian manifold of dimension $N\geq 3$ without boundary. The exponents $p,q>1$ are assumed to lie below the critical hyperbola, ensuring subcritical growth conditions. We investigate a sequence of least energy critical points of the associated dual functional and analyze their concentration behavior as $\varepsilon \to 0$. Our main result shows that the sequence of solutions exhibits point concentration, with the concentration occurring at a point where the scalar curvature of $\mathcal{M}$ attains its maximum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20300 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spike layered solutions for elliptic systems on Riemannian Manifolds Kannoth, Anusree R Manna, Bhakti Bhusan Analysis of PDEs Primary 35J10, 35J35, 35J65 In this article, we study the following Hamiltonian system: \begin{equation*} \begin{cases} \begin{aligned} &-\varepsilon^{2}Δ_{g} u +u = |v|^{q-1}v, &-\varepsilon^{2}Δ_{g} v +v = |u|^{p-1}u && \text{ in } \mathcal{M}, & \quad u,v >0 && \text{ in } \mathcal{M}, \end{aligned} \end{cases} \end{equation*} where $\mathcal{M}$ is a smooth, compact and connected Riemannian manifold of dimension $N\geq 3$ without boundary. The exponents $p,q>1$ are assumed to lie below the critical hyperbola, ensuring subcritical growth conditions. We investigate a sequence of least energy critical points of the associated dual functional and analyze their concentration behavior as $\varepsilon \to 0$. Our main result shows that the sequence of solutions exhibits point concentration, with the concentration occurring at a point where the scalar curvature of $\mathcal{M}$ attains its maximum. |
| title | Spike layered solutions for elliptic systems on Riemannian Manifolds |
| topic | Analysis of PDEs Primary 35J10, 35J35, 35J65 |
| url | https://arxiv.org/abs/2506.20300 |