Critical points of the Trudinger–Moser trace functional with high energy levels

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^2$ with smooth boundary. In this paper we are concerned with the existence of critical points for the supercritical Trudinger–Moser trace functional

in the set $\{u\in H^1(\Omega ):{\int }_{\Omega }(|\nabla u{|}^2+u^2)dx=1\}$, where $k⩾1$ is an integer and $\mu >0$ is a small parameter. For any integer $k⩾1$ and for any $\mu >0$ sufficiently small, we prove the existence of a pair of $k$-peaks constrained critical points of the above problem.