CONCENTRATED SOLUTIONS FOR A CRITICAL ELLIPTIC EQUATION

In this paper, we are concerned with the following elliptic equation {Delta u = Q(x)u(2)(-1)*+epsilon u(3), in Omega, u > 0, in Omega, u = 0, on partial derivative Omega, where N >= 3, s is an element of [1, 2* - 1) with 2* = 2N/N-2, epsilon > 0, Omega is a smooth bounded domain in R-N. Under some conditions on Q(x), Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997, 461-483) proved that there exists a single-peak solution for small epsilon if N >= 4 and s is an element of (1, 2*- 1). And they proposed in Remark 1.7 of their paper that it is interesting to know the existence of single-peak solutions for small epsilon and s = 1. Also it was addressed in Remark 1.8 of their paper that the question of solutions concentrated at several points at the same time is still open. Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether s = 1 or s > 1.

Automated summary

In this paper, we investigate the existence of solutions for an elliptic equation. We prove the existence of single-peak solutions for certain values of the parameter s. We also study the concentration of solutions at multiple points and establish the local uniqueness of multi-peak solutions. Our results demonstrate the dependence of the concentration of solutions on the parameter s.