Construction of solutions for the polyharmonic equation via local Pohozaev identities

Abstract

We consider the following problem involving critical exponent and polyharmonic operator: $$\begin{aligned} (-\Delta )^mu+V(|y'|,y'')u=u^{m^*-1}, \;u>0, \; u \in {\mathcal {D}}^{m,2}({\mathbb {R}}^{N}), \end{aligned}$$ where $$m^*=\frac{2N}{N-2m},\; N\ge 4m+1$$ , $$ m \in {\mathbb {N}}_+$$ , $$(y',y'') \in {\mathbb {R}}^{2} \times {\mathbb {R}}^{N-2}$$ and $$V(|y'|,y'')$$ is a bounded non-negative function in $${\mathbb {R}}^{+} \times {\mathbb {R}}^{N-2}$$ . By using a finite reduction argument and local Pohozaev type identities, we will show that if $$N \ge 4m+1$$ and $$r^{2m}V(r,y'')$$ has a stable critical point $$(r_0,y_0'')$$ , then the above problem has infinitely many solutions, whose energy can be arbitrarily large.