Localized semiclassical states for Hamiltonian elliptic systems in dimension two

TL;DR

This paper constructs localized semiclassical states for a Hamiltonian elliptic system in two dimensions, demonstrating concentration near local minima of the potential and analyzing decay and positivity under superlinear conditions.

Contribution

It introduces a new approach to find semiclassical states concentrating at local minima in 2D Hamiltonian systems with subcritical exponential growth.

Findings

Existence of semiclassical states concentrating at local minima.

States exhibit decay and positivity when nonlinearities are superlinear.

Method combines reduction, variational, and penalization techniques.

Abstract

In this paper, we consider the Hamiltonian elliptic system in dimension two\begin{equation}\label{1.5}\aligned \left\{ \begin{array}{lll} -\epsilon^2\Delta u+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\ -\epsilon^2\Delta v+V(x)v=f(u)\ & \text{in}\quad \mathbb{R}^2, \end{array}\right.\endaligned \end{equation} where $V\in C(\mathbb{R}^2)$ has local minimum points, and $f,g\in C^1(\mathbb{R})$ are assumed to be either superlinear or asymptotically linear at infinity and of subcritical exponential growth in the sense of Trudinger-Moser inequality. Under only a local condition on $V$, we obtain a family of semiclassical states concentrating around local minimum points of $V$. In addition, in the case that $f$ and $g$ are superlinear at infinity, the decay and positivity of semiclassical states are also given. The proof is based on a reduction method, variational methods and penalization techniques.