New type of solutions for the critical polyharmonic equation

Abstract

In this paper, we consider the following critical polyharmonic equation${(-\Delta )}^{m}u+V\left(\right|y^{\prime }|,y^{″})u=u^{m^{⁎}-1},u>0,y=(y^{\prime },y^{″})\in R^3\times R^{N-3},$ where $m^{⁎}=\frac{2N}{N-2m}$, $N>4m+1$, $m\in N^{+}$, and $V\left(\right|y^{\prime }|,y^{″})$ is a bounded nonnegative function in $R^{+}\times R^{N-3}$. By using the reduction argument and local Pohoz̆aev identities, we prove that if $r^{2m}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, 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.