Existence and multiplicity of normalized solutions for $L^2$-supercritical Schrödinger equations on noncompact metric graphs with nonlinear point defects
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arXiv
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2025
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| _version_ | 1866909947342618624 |
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| author | He, Zhentao Ji, Chao Tao, YIfan |
| author_facet | He, Zhentao Ji, Chao Tao, YIfan |
| contents | In this paper, we study the existence and multiplicity of normalized solutions for the following $L^2$-supercritical Schrödinger equation on noncompact metric graph $\G=(\V,\E)$ with nonlinear point defects \begin{equation*}
\begin{cases} u'' = λu & \text{on every }\e \in \E, \\ \|u\|_{L^2(\mathcal{G})}^2 = μ& \\ \displaystyle\sum_{\e \succ \vv} u'_\e(\vv) = -|u(\vv)|^{p-2}u(\vv) & \text{at every }\vv \in \V, \end{cases} \end{equation*} where $p>4$, $\G$ has finitely many edges, $μ>0$ is a given constant, the parameter $λ$ is a part of the unknown which arises as a Lagrange multiplier, $\e \succ \vv$ means that the edge $\e$ is incident at $\vv$, and the notation $u'_\e(\vv)$ stands for $u'_\e(0)$ or $-u'_\e(\ell_\e)$, according to whether the vertex $\vv$ is identified with $0$ or $\ell_\e$. This work complements the study initiated by Boni, Dovetta, and Serra [J. Funct. Anal. 288 (2025), 110760], which addressed only the existence of normalized solutions for the $L^2$-subcritical ($2<p<4$) Schrödinger equation on metric graphs with nonlinear point defects. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_06445 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and multiplicity of normalized solutions for $L^2$-supercritical Schrödinger equations on noncompact metric graphs with nonlinear point defects He, Zhentao Ji, Chao Tao, YIfan Analysis of PDEs In this paper, we study the existence and multiplicity of normalized solutions for the following $L^2$-supercritical Schrödinger equation on noncompact metric graph $\G=(\V,\E)$ with nonlinear point defects \begin{equation*} \begin{cases} u'' = λu & \text{on every }\e \in \E, \\ \|u\|_{L^2(\mathcal{G})}^2 = μ& \\ \displaystyle\sum_{\e \succ \vv} u'_\e(\vv) = -|u(\vv)|^{p-2}u(\vv) & \text{at every }\vv \in \V, \end{cases} \end{equation*} where $p>4$, $\G$ has finitely many edges, $μ>0$ is a given constant, the parameter $λ$ is a part of the unknown which arises as a Lagrange multiplier, $\e \succ \vv$ means that the edge $\e$ is incident at $\vv$, and the notation $u'_\e(\vv)$ stands for $u'_\e(0)$ or $-u'_\e(\ell_\e)$, according to whether the vertex $\vv$ is identified with $0$ or $\ell_\e$. This work complements the study initiated by Boni, Dovetta, and Serra [J. Funct. Anal. 288 (2025), 110760], which addressed only the existence of normalized solutions for the $L^2$-subcritical ($2<p<4$) Schrödinger equation on metric graphs with nonlinear point defects. |
| title | Existence and multiplicity of normalized solutions for $L^2$-supercritical Schrödinger equations on noncompact metric graphs with nonlinear point defects |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.06445 |