Construction of solutions for the critical polyharmonic equation with competing potentials

2024/07/23 by Wenjing Chen, Chen, Wenjing, Zexi Wang +1
Mathematics · #advanced mathematical theories #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2408.00007

Abstract

In this paper, we consider the following critical polyharmonic equation % ( -Δ)^m u+V(|y'|,y'')u=Q(|y'|,y'')u^{m^*-1}, ugt;0, y=(y',y'')∈ ℝ³× ℝ^N-3, where N>4m+1, m∈ ℕ⁺, m^*=(2N)/(N-2m), V(|y'|,y'') and Q(|y'|,y'') are bounded nonnegative functions in ℝ⁺× ℝ^N-3. By using the reduction argument and local Pohouzaev identities, we prove that if Q(r,y'') has a stable critical point (r₀,y₀'') with r₀>0, Q(r₀,y₀'')>0, D^αQ(r₀,y₀'')=0 for any |α|≤ 2m-1 and B₁V(r₀,y₀'')-B₂∑_|α|=2mD^αQ(r₀,y₀'')∫_ℝ^Ny^αU_0,1^{m^*}dy>0, then the above problem has a family of solutions concentrated at points lying on the top and the bottom circles of a cylinder, where B₁ and B₂ are positive constants that will be given later.

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