Communications of the
Korean Mathematical Society CKMS

In this paper, we study the existence and nonexistence of solutions for a class of Hamiltonian strongly degenerate elliptic system with subcritical growth \begin{equation*} \begin{cases} -\Delta_\lambda u -\mu v =|v|^{p-1}v &\;\text{ in } \Omega,\\ -\Delta_\lambda v -\mu u=|u|^{q-1}u &\;\text{ in } \Omega,\\ u = v = 0 &\;\text{ on } \partial\Omega, \end{cases} \end{equation*} where $p, q>1$ and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\ge 3$. Here $\Delta_\lambda$ is the strongly degenerate elliptic operator. The existence of at least a nontrivial solution is obtained by variational methods while the nonexistence of positive solutions are proven by a contradiction argument.

Keywords: Hamiltonian elliptic system, variational methods, strongly degenerate, existence and nonexistence

MSC numbers: Primary 58B34, 58J42, 81T75