Ground states for planar Hamiltonian elliptic systems with critical exponential growth

Abstract

This paper focuses on the study of ground states and nontrivial solutions for the following Hamiltonian elliptic system:{−Δu+V(x)u=f1(x,v),x∈R2,−Δv+V(x)v=f2(x,u),x∈R2, where V∈C(R2,(0,∞)) and f1,f2:R2×R→R have critical exponential growth. The strongly indefinite features together with the critical exponent bring some new difficulties in our analysis. In this paper, we develop a direct approach and use an approaching argument to seek Cerami sequences for the energy functional and estimate the minimax levels of such sequences. In particular, under some general assumptions imposed on the nonlinearity fi, we obtain the existence of ground states and nontrivial solutions for the above system as well as the following system in bounded domain,{−Δu=f1(v),x∈Ω,−Δv=f2(u),x∈Ω,u=0,v=0,x∈∂Ω. Our results improve and extend the related results of de Figueiredo-do Ó-Ruf (2004) [20]; (2011) [21], of Lam-Lu (2014) [30], and of de Figueiredo-do Ó-Zhang (2020) [22].