Supercritical Trudinger-Moser inequalities with logarithmic weights in dimension two

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.