SOLUTIONS WITH LARGE NUMBER OF PEAKS FOR THE SUPERCRITICAL HENON EQUATION

This paper is concerned with the Henon equation {-Delta u = vertical bar y vertical bar(alpha) u(p+epsilon), u > 0, in B-1(0), u = 0 on partial derivative B-1(0), where B-1(0) is the unit ball in R-N (N >= 4), p = (N + 2)/(N - 2) 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/(N - 1)) when epsilon -> 0(+). Moreover, all peaks of the solutions approach the boundary of B-1(0) as epsilon goes to 0(+).