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*} \begin{cases} -Δu + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -Δv + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N, \end{cases} \end{equation*} where $(y^{\prime },y^{\prime \prime })\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$, $V(|y^{\prime }|,y^{\prime \prime })\not\equiv 0$ is a bounded, nonnegative function on ${\mathbb{R}}_{+}\times {\mathbb{R}}^{N-2}$ and $p,q>1$ lie on the critical hyperbola: \[ \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}. \] By applying the finite-dimensional reduction method and local Pohozaev identities combined with the Green representation formula and technical analysis, we show that, under the assumptions that $N\ge 5$, $(p,q)$ lies in a certain admissible range, and $r^{2}V(r,y^{\prime \prime })$ has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large.