Nontrivial solutions for a class of Hamiltonian strongly degenerate elliptic system

Abstract

In this paper, we study the existence of nontrivial to the following Hamiltonian strongly degenerate elliptic system in whole space $\{\begin{array}{l}-{\mathrm{\Delta }}_{\lambda }u+V(x)u=K_1(x)f(v)\text{in }{\mathbb{R}}^N,\\ -{\mathrm{\Delta }}_{\lambda }v+V(x)v=K_2(x)g(u)\text{in }{\mathbb{R}}^N,\end{array}$where $V,K_1,K_2:{\mathbb{R}}^N\to \mathbb{R},N\ge 3$ are nonnegative functions with $K_i\in L^{\mathrm{\infty }},i=1,2$ and V might unbounded or vanish at infinity. Here ${\mathrm{\Delta }}_{\lambda }$ is the strongly degenerate operator and the nonlinearity terms $f,g:\mathbb{R}\to \mathbb{R}$ are continuous functions and satisfy some assumptions that will be specified later. Due to the unbounded or vanishing of potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results.

Keywords:

• Hamiltonian elliptic system
• variational methods
• strongly degenerate
• cerami sequences
• unbounded or vanishing potentials

1991 Mathematics Subject Classifications:

• 35J50
• 35J60
• 35J70

Acknowledgments

A part of this paper was completed while the author visited the Vietnam Institute for Advanced Study in Mathematics (VIASM) in 2021. The author would like to thank VIASM for its hospitality and support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by Hanoi Pedagogical University 2 (HPU2) under [grant number HPU2.CS-2021.01].