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This paper deals with the following critical elliptic systems of Hamiltoniantype, which are variants of the critical Lane-Emden systems and analogous tothe prescribed curvature problem: \begin{equation*} \begin{cases} -\Deltau_1=K_1(y)u_2^{p},\ y\in \mathbb{R}^N,\\ -\Delta u_2=K_2(y)u_1^{q}, \ y\in\mathbb{R}^N,\\ u_1,u_2>0, \end{cases} \end{equation*} where $N\geq 5,p,q\in(1,\infty)$ with $\frac1{p+1}+\frac1{q+1}=\frac{N-2}N$, $K_1(y)$ and$K_2(y)$ are positive radial potentials. At first, under suitable conditions on$K_1,K_2$ and the certain range of the exponents $p,q$, we construct anunbounded sequence of non-radial positive vector solutions, whose energy can bemade arbitrarily large. Moreover, we prove a type of non-degeneracy result byuse of various Pohozaev identities, which is of great interest independently.The indefinite linear operator and strongly coupled nonlinearities make theHamiltonian-type systems in stark contrast both to the systems of Gradient typeand to the single critical elliptic equations in the study of the prescribedcurvature problems. It is worth noting that, in higher-dimensional cases$(N\geq5)$, there have been no results on the existence of infinitely manybubbling solutions to critical elliptic systems, either of Hamiltonian orGradient type.

Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type