Construction of solutions for a critical elliptic system of Hamiltonian type

Abstract: We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*}\left{\begin{align*} -\Delta u + V(|y'|,y'') u = v^p, \;\; \text{in} \;\; \mathbb{R}^N,\ -\Delta v + V(|y'|,y'') v = u^q, \;\; \text{in} \;\; \mathbb{R}^N,\ u, v > 0 , (u,v) \in (\dot{W}^{2,\frac{p+1}{p}} (\mathbb{R}^N) \cap L^2(\mathbb{R}^N)) \times (\dot{W}^{2,\frac{q+1}{q}} (\mathbb{R}^N) \cap L^2(\mathbb{R}^N)) ,\end{align*}\right. \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$ and $V(|y'|, y'')\not\equiv 0$ is a bounded non-negative function in $\mathbb{R}_+\times \mathbb{R}^{N-2}$, $p,q>1$ satisfying $$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$$ By using a finite reduction argument and local Pohozaev identities, under the assumption that $N\geq 5$, $(p,q)$ lies in the certain range and $r^2V(r,y'')$ has a stable critical point, we prove that the above problem has infinitely many solutions whose energy can be made arbitrarily large.