On the existence of solutions of a Hamiltonian strongly degenerate elliptic system with potentials in ${\mathbb{R}}^N$

Abstract

In this paper, we prove the existence of nontrivial solutions to the following semilinear strongly degenerate elliptic system:

where $N\ge 3$, $V,K_1,K_2:{\mathbb{R}}^N\to \mathbb{R}$ are functions to be specified later and ${\Delta }_{\lambda }$ is the strongly degenerate operator. Using variational methods, we obtain nontrivial solutions of the above system. Our main contribution is that we are able to deal with two separable cases: (1) the function $V$ is bounded and $K_1,K_2$ might be bounded or vanishing at infinity; (2) all the functions $V$ and $K_1,K_2$ are unbounded at infinity.