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} -\Delta u + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -\Delta v + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N, \end{cases} \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$, $V(|y'|, y'') \not\equiv 0$ 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 $r^2 V(r, y'')$ has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large.