4.5 Article

CONCENTRATED SOLUTIONS FOR A CRITICAL ELLIPTIC EQUATION

期刊

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
卷 42, 期 8, 页码 4061-4094

出版社

AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcds.2022046

关键词

Critical Sobolev exponent; Pohozaev identity; non-existence; existence; uniqueness

资金

  1. NSFC [12071364, 11871387, 11771167]
  2. Technology Foundation of Guizhou Province [[2001] ZK008]
  3. Fundamental Research Funds for the Central Universities [WUT: 2020IA003]
  4. China Scholarship Council

向作者/读者索取更多资源

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.
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.

作者

我是这篇论文的作者
点击您的名字以认领此论文并将其添加到您的个人资料中。

评论

主要评分

4.5
评分不足

次要评分

新颖性
-
重要性
-
科学严谨性
-
评价这篇论文

推荐

暂无数据
暂无数据