Solutions with large number of peaks for the supercritical Hénon equation

Zhongyuan Liu and Shuangjie Peng

Vol. 280 (2016), No. 1, 115–139

DOI: 10.2140/pjm.2016.280.115

Abstract

This paper is concerned with the Hénon equation

$$\left\{\begin{array}{cc}-\Delta u=|y{|}^{\alpha }{u}^{p+\epsilon },u>0,\hfill & \text{ in}{B}_1\left(0\right),\hfill \\ u=0\hfill & \text{ on}\partial B_1\left(0\right),\hfill \end{array}\right.$$

where $B_1\left(0\right)$ is the unit ball in ${ℝ}^N$ ($N\ge 4$), $p=\left(N+2\right)∕\left(N-2\right)$ is the critical Sobolev exponent, $\alpha >0$ and $\epsilon >0$. We show that if $\epsilon$ is small enough, this problem has a positive peak solution which presents a new phenomenon: the number of its peaks varies with the parameter $\epsilon$ at the order ${\epsilon }^{-1∕\left(N-1\right)}$ when $\epsilon \to 0^{+}$. Moreover, all peaks of the solutions approach the boundary of $B_1\left(0\right)$ as $\epsilon$ goes to $0^{+}$.

Keywords

peak solutions, supercritical Hénon equation, reduction method

Mathematical Subject Classification 2010

Primary: 35J60

Secondary: 35J65, 58E05

Milestones

Received: 6 November 2014

Revised: 10 May 2015

Accepted: 11 May 2015

Published: 28 December 2015