Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth

In this paper, we deal with the existence of nontrivial solutions to the following class of strongly coupled Hamiltonian systems:

$$\{\begin{array}{rcl}-\mathrm{d}\mathrm{i}\mathrm{v}(w(x)\mathrm{\nabla }u)=g(x,v),& & x\in B_1(0),\\ -\mathrm{d}\mathrm{i}\mathrm{v}(w(x)\mathrm{\nabla }v)=f(x,u),& & x\in B_1(0),\\ u=v=0& & x\in \mathrm{\partial }{B}_1(0),\end{array}$$

where $w(x)=(\log 1/|x|{)}^{\gamma }$, $0\le \gamma <1$, and the nonlinearities $f$ and $g$ possess exponential growth ranges above the exponential critical hyperbola. Our approach is based on Trudinger-Moser type inequalities for weighted Sobolev spaces and variational methods.