Blow-up analysis and a priori bounds for NLS equations on metric graphs

TL;DR

This paper analyzes the blow-up behavior and establishes a priori bounds for solutions of nonlinear Schrödinger equations on metric graphs, revealing exponential decay and relationships between solution properties.

Contribution

It introduces the first global exponential decay results for solutions on graphs and links Morse index bounds to solution behavior and blow-up points.

Findings

Existence of a finite set of blow-up points with exponential decay away from them.

Bound on the number of blow-up points is estimated by Morse index.

Derived various a priori bounds on solutions in L^ and L^2 norms.

Abstract

We consider, on a connected metric graph $\mathcal{G}$, a family of nonlinear Schr\"odinger equations $$ -u'' + W_n(x) u + \lambda_n u = \rho_n(x)|u|^{p-2}u, \quad n \in \mathbb{N}. \qquad (*) $$ We assume that $p > 2$, $(W_n)$, $(\rho_n) \subseteq L^{\infty}(\mathcal{G})$ with $\rho_n \geq 0$, $|W_n|_{L^\infty(\mathcal{G})}$ and $|\rho_n|_{L^\infty(\mathcal{G})}$ are bounded and $\lambda_n \to +\infty$. 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 $\lambda_n \to + \infty$ 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.