\(L^2\)-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features. (English) Zbl 1432.35212
The article focuses on the study of ground states for a particular nonlinear Schrödinger equation on \(G\), a suitable metric graph. Let us describe one of the meaningful results of the article. Let \(H^1_\mu(G)\) be the space of functions belonging to \(H^1(G)\), the Sobolev space of order one on \(G\), such that their \(L_2(G)\)-norm squares are equal to \(\mu\) (the mass constraint). Then for \(v\in H^1_\mu(G)\), the authors consider the following energy functional \(\displaystyle E(v,K)=\frac{1}{2}\int_G|v'|^2dx-\frac{1}{6}\int_K|v|^6dx\) such that \(K\) is a subgraph of \(G\) comprised by bounded edges. Consequently, according to the values of \(\mu\) and to the structure of \(G\), the authors confer the (no)existence of functions \(u\in H^1_\mu(G)\) such that \(\displaystyle E(u,K)=\inf_{v\in H_\mu^1(G)}E(v,K)\) (Theorem 1.1).
Reviewer: Mohammed El Aïdi (Bogotá)
MSC:
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 35Q40 | PDEs in connection with quantum mechanics |
| 49J40 | Variational inequalities |
Keywords:
metric graphs; NLSE; ground states; Kirchhoff Laplacian; Gagliardo-Nirenberg inequality on graphs; Sobolev space on graphs; symmetric and decreasing rearrangementsReferences:
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