Semiclassical asymptotics of infinitely many solutions for the infinite case of a nonlinear Schrödinger equation with critical frequency

Bull. Korean Math. Soc. Published online November 10, 2021

Ariel Aguas-Barreno, Jordy Cevallos-Chávez, Juan Mayorga-Zambrano, and Leonardo Medina-Espinosa
Yachay Tech University; Arizona State University; Yachay Tech University; Pontificia Universidad Católica de Chile

Abstract : We consider a nonlinear Schr\"odinger equation with critical frequency, $\displaystyle \left( \mathrm{P}_\varepsilon \right): \ \varepsilon^2 \ \Delta v(x) - V(x) \ v(x) + |v(x)|^{p-1} \ v(x) = 0$, $x\in \mathbb{R}^N$, and $v(x) \rightarrow 0$ as $|x|\rightarrow +\infty$, for the \emph{infinite case} as described by Byeon and Wang. \emph{Critical} means that $0\leq V\in \mathrm{C}(\mathbb{R}^N)$ verifies $\mathcal{Z} = \{V = 0 \} \neq \emptyset$. \emph{Infinite} means that $\mathcal{Z} = \{x_0\}$ and that, grossly speaking, the potential $V$ decays at an exponential rate as $x\rightarrow x_0$. For the semiclassical limit, $\varepsilon \rightarrow 0$, the infinite case has a characteristic limit problem, $\displaystyle \left( \mathrm{P}_{\mathrm{inf}} \right): \ \Delta u(x) - P(x) \, u(x) + |u(x)|^{p-1}\, u(x)=0$,
$x\in \Omega$, with $u(x) = 0$ as $x\in \Omega$, where $\Omega\subseteq \mathbb{R}^N$ is a smooth bounded strictly star-shaped region related to the potential $V$. We prove the existence of an infinite number of solutions
for both the original and the limit problem via a Ljusternik-Schnirelman scheme for even functionals. Fixed
a topological level $k$ we show that $v_{k,\varepsilon}$, a solution of $(\mathrm{P}_\varepsilon)$, subconverges, up to a scaling, to a corresponding solution of $(\mathrm{P}_{\mathrm{inf}})$, and that $v_{k,\varepsilon}$ exponentially decays out of $\Omega$. Finally, uniform estimates on $\partial \Omega$ for scaled solutions of $(\mathrm{P}_\varepsilon)$ are obtained.