2022/02/28 by Lipeng Duan, Duan, Lipeng, Shuying Tian +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2202.13721
openalex publication_date 2022/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we are concerned with the following elliptic equation \begincases -Δu= Q(x)u2^*-1 +ε us,~ amp;in~Ω,
ugt;0,~ amp;in~Ω,
u=0, amp;on~∂ Ω, \endcases where N≥ 3, s∈ [1,2^*-1) with 2^*=(2N)/(N-2), ε>0, Ω is a smooth bounded domain in ℝN. 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 ε if N≥ 4 and s∈ (1,2^*-1). And they proposed in Remark 1.7 of their paper that \vskip 0.1cm``it is interesting to know the existence of single-peak solutions for small ε and s=1''.\vskip 0.1cm \noindent Also it was addressed in Remark 1.8 of their paper that \vskip 0.1cm ``the question of solutions concentrated at several points at the same time is still open''.\vskip 0.1cm \noindent 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.