# On supercritical nonlinear Schr\"{o}dinger equations with ellipse-shaped potentials

## TL;DR

This paper investigates the existence, concentration, and blow-up behavior of solutions to a supercritical nonlinear Schrödinger equation with ellipse-shaped potentials in two dimensions, especially as the nonlinearity approaches the critical exponent.

## Contribution

It establishes the existence of excited solutions for nonlinearities near the critical value and describes their blow-up behavior as the nonlinearity approaches criticality.

## Key findings

- Existence of excited solutions for q close to 2.
- Precise description of blow-up formation of solutions.
- Analysis of limiting behavior as q approaches 2.

## Abstract

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad \mathbb{R}^2,\\ \int_{\mathbb{R}^2}|u|^2\,dx =1,\\ \end{array} \right. \end{equation*} where $\mu_q$ is the Lagrange multiplier. For ellipse-shaped potentials $V(x)$, we show that for $q>2$ close to $2$, the equation admits an excited solution $u_q$, and furthermore, we study the limiting behavior of $u_q$ when $q\to 2_+$. Particularly, we describe precisely the blow-up formation of the excited state $u_q$.