New type of solutions for the critical polyharmonic equation

In this paper, we consider the following critical polyharmonic equation {align*}%{abs} ( -Δ)^m u+V(|y'|,y'')u=um^*-1, u>0, y=(y',y'')∈ R^3× RN-3, {align*} where m^*=2N/N-2m, $N>4m+1$, m∈ N⁺, and $V(|y'|,y'')$ is a bounded nonnegative function in R⁺× RN-3. By using the reduction argument and local Poho{z}aev identities, we prove that if r²ᵐV(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.