Blow-up analysis and a priori bounds for NLS equations on metric graphs
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910189667483648 |
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| author | Carrillo, Pablo De Coster, Colette Galant, Damien Jeanjean, Louis Troestler, Christophe |
| author_facet | Carrillo, Pablo De Coster, Colette Galant, Damien Jeanjean, Louis Troestler, Christophe |
| contents | We consider, on a connected metric graph $\mathcal{G}$, a family of nonlinear Schrödinger equations $$
-u'' + W_n(x) u + λ_n u = ρ_n(x)|u|^{p-2}u,
\quad n \in \mathbb{N}.
\qquad (*) $$ We assume that $p > 2$, $(W_n)$, $(ρ_n) \subseteq L^{\infty}(\mathcal{G})$ with $ρ_n \geq 0$, $|W_n|_{L^\infty(\mathcal{G})}$ and $|ρ_n|_{L^\infty(\mathcal{G})}$ are bounded and $λ_n \to +\infty$. Given $n \in \mathbb{N}$, we call "solution" a function $u_n \in H^1(\mathcal{G})$ which satisfies (*) for that $n\in \mathbb{N}$ together with the Kirchhoff conditions at the vertices. Focusing on the limiting behavior of sequences $(u_n) \subseteq H^1(\mathcal{G})$ of solutions as $λ_n \to + \infty$ and assuming that the Morse index $m(u_n)$ of $u_n$ is uniformly bounded, we establish, the existence of a finite subset of blow-up points away from which, up to a subsequence, $|u_n|$ has a global exponential decay. These points are generally a strict subset of the blow-up points, and their number is estimated by the bound on the Morse index of $(u_n)$. It is the first time that this global exponential decay property is established on graphs even if one consider only signed solutions. In the last part of the paper we derive various results of a priori bounds on the solutions in $L^\infty$ and $L^2$. Our blow-up analysis, combined with ODE arguments allows, for frequently considered classes of graphs, to obtain a fairly complete picture of the relationships between the number of nodal regions, Morse index, $L^\infty$ and $L^2$ norms of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02879 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blow-up analysis and a priori bounds for NLS equations on metric graphs Carrillo, Pablo De Coster, Colette Galant, Damien Jeanjean, Louis Troestler, Christophe Analysis of PDEs 35J60, 47J30 We consider, on a connected metric graph $\mathcal{G}$, a family of nonlinear Schrödinger equations $$ -u'' + W_n(x) u + λ_n u = ρ_n(x)|u|^{p-2}u, \quad n \in \mathbb{N}. \qquad (*) $$ We assume that $p > 2$, $(W_n)$, $(ρ_n) \subseteq L^{\infty}(\mathcal{G})$ with $ρ_n \geq 0$, $|W_n|_{L^\infty(\mathcal{G})}$ and $|ρ_n|_{L^\infty(\mathcal{G})}$ are bounded and $λ_n \to +\infty$. Given $n \in \mathbb{N}$, we call "solution" a function $u_n \in H^1(\mathcal{G})$ which satisfies (*) for that $n\in \mathbb{N}$ together with the Kirchhoff conditions at the vertices. Focusing on the limiting behavior of sequences $(u_n) \subseteq H^1(\mathcal{G})$ of solutions as $λ_n \to + \infty$ and assuming that the Morse index $m(u_n)$ of $u_n$ is uniformly bounded, we establish, the existence of a finite subset of blow-up points away from which, up to a subsequence, $|u_n|$ has a global exponential decay. These points are generally a strict subset of the blow-up points, and their number is estimated by the bound on the Morse index of $(u_n)$. It is the first time that this global exponential decay property is established on graphs even if one consider only signed solutions. In the last part of the paper we derive various results of a priori bounds on the solutions in $L^\infty$ and $L^2$. Our blow-up analysis, combined with ODE arguments allows, for frequently considered classes of graphs, to obtain a fairly complete picture of the relationships between the number of nodal regions, Morse index, $L^\infty$ and $L^2$ norms of solutions. |
| title | Blow-up analysis and a priori bounds for NLS equations on metric graphs |
| topic | Analysis of PDEs 35J60, 47J30 |
| url | https://arxiv.org/abs/2605.02879 |