Construction of solutions for the critical polyharmonic equation with competing potentials

In this paper, we consider the following critical polyharmonic equation {align*}%{abs} ( -Δ)^m u+V(|y'|,y'')u=Q(|y'|,y'')um^*-1, u>0, y=(y',y'')∈ R^3× RN-3, {align*} where $N>4m+1$, m∈ N⁺, m^*=2N/N-2m, $V(|y'|,y'')$ and $Q(|y'|,y'')$ are bounded nonnegative functions in R⁺× RN-3. By using the reduction argument and local Poho{z}aev identities, we prove that if $Q(r,y'')$ has a stable critical point (r₀,y₀'') with r₀>0, Q(r₀,y₀'')>0, D^α Q(r₀,y₀'')=0 for any |α|≤ 2m-1 and B₁V(r₀,y₀'')-B₂∑|α|=2mD^α Q(r₀,y₀'')∫RNy^α U0,1m^*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.