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Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs

Simone Dovetta, Louis Jeanjean, Enrico Serra

Abstract

We consider the existence of normalized solutions to nonlinear Schrödinger equations on noncompact metric graphs in the $L^2$ supercritical regime. For sufficiently small prescribed mass ($L^2$ norm), we prove existence of positive solutions on two classes of graphs: periodic graphs, and noncompact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an $L^2$ supercritical nonlinearity extended on the whole graph and unraveling the role of topology in the existence of solutions.

Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs

Abstract

We consider the existence of normalized solutions to nonlinear Schrödinger equations on noncompact metric graphs in the supercritical regime. For sufficiently small prescribed mass ( norm), we prove existence of positive solutions on two classes of graphs: periodic graphs, and noncompact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an supercritical nonlinearity extended on the whole graph and unraveling the role of topology in the existence of solutions.
Paper Structure (11 sections, 17 theorems, 132 equations, 5 figures)

This paper contains 11 sections, 17 theorems, 132 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A general existence result for bounded Palais-Smale sequences
  4. NLS equations on the real line
  5. Gagliardo-Nirenberg inequalities on graphs
  6. Properties of the Lagrange multiplier
  7. A general energy estimate for positive solutions
  8. Mountain pass structure and a priori estimates
  9. Graphs with finitely many edges: proof of Theorem \ref{['thm:exHalf']}
  10. Periodic graphs: proof of Theorem \ref{['thm:exper']}
  11. Proof of Theorems \ref{['thm:neg']}--\ref{['thm:prop_p']}

Key Result

Theorem 1.1

Let $\mathcal{G}\in\mathbf{G}$ be a noncompact metric graph with finitely many edges and at least one pendant or one signpost. Then, for every $p>6$ there exists $\mu_{p,\mathcal{G}}>0$, depending on $p$ and $\mathcal{G}$, such that, for every $\mu\in(0,\mu_{p,\mathcal{G}}]$, problem nlsG admits a s

Figures (5)

  • Figure 1: A noncompact graph in $\mathbf{G}$ with finitely many vertices and edges.
  • Figure 2: Examples of periodic graphs.
  • Figure 3: A star graph with $6$ half-lines glued together at their common origin.
  • Figure 4: The tadpole graph, i.e. a loop attached to a half-line.
  • Figure 5: The $\mathcal{T}$-graph, i.e. a pendant attached to a full line.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 25 more