On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials

Abstract

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} -Δu + V(x) u = μ_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 $u_q$.

26 pages