New type of solutions for the critical polyharmonic equation

2024/05/25 by Wenjing Chen, Chen, Wenjing, Zexi Wang +1
Mathematics · #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2405.16095

openalex publication_date 2024/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the following critical polyharmonic equation % ( -Δ)^m u+V(|y'|,y'')u=u^{m^*-1}, ugt;0, y=(y',y'')∈ ℝ³× ℝ^N-3, where m^*=(2N)/(N-2m), N>4m+1, m∈ ℕ⁺, and V(|y'|,y'') is a bounded nonnegative function in ℝ⁺× ℝ^N-3. By using the reduction argument and local Pohouzaev identities, we prove that if r^2mV(r,y'') has a stable critical point (r₀,y₀'') with r₀>0 and V(r₀,y₀'')>0, then the above problem has a new type of solutions, which concentrate at points lying on the top and the bottom circles of a cylinder.

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