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

Deng, Shengbing; Musso, Monica

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 59-95.

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

$$\begin{array}{cc}\underset{\partial \Omega }{\int }{e}^{k\pi (1+\mu )u^2}& \text{(0.1)}\end{array}$$

in the set $\{u\in H^1\left(\Omega \right):{\int }_{\Omega }({\left|\nabla u\right|}^2+u^2)dx=1\}$, 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.

Mots clés : Trudinger–Moser trace functional, Reduction methods

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