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.

In this work, we are interested in studying the existence of nontrivial weak solutions to the following class of Schrödinger equations

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

where $w(x)=(\ln (1/|x|){)}^{\beta }$ for some $\beta \in [0,1)$, the nonlinearity $f(x,s)$ behaves like $\exp ((1+h(|x|))|s{|}^{2/(1-\beta )})$ and $h$ is a continuous radial function such that $h(r)$ tends to infinity as $r$ tends to $1$. The proof involves variational methods and a new version of Trudinger-Moser inequality.

In this work, we investigated the existence of nontrivial weak solutions for the equation

$$-\mathrm{d}\mathrm{i}\mathrm{v}(w(x)\mathrm{\nabla }u)=f(x,u),x\in {\mathbb{R}}^2,$$

where $w(x)$ is a positive radial weight, the nonlinearity $f(x,s)$ possesses growth at infinity of the type $\exp (({\alpha }_0+h(|x|))|s{|}^{2/(1-\beta )})$, with ${\alpha }_0>0$, $0<\beta <1$ and $h$ is a continuous radial function that may be unbounded at infinity. To show the existence of weak solutions, we used variational methods and a new type of the Trudinger-Moser inequality defined on the whole two-dimensional space.