Standing waves for quasilinear Schrödinger equations involving double exponential growth

We will focus on the existence of nontrivial, nonnegative solutions to the following quasilinear Schrödinger equation

$$\{\begin{array}{rcll}-\mathrm{d}\mathrm{i}\mathrm{v}(\log {\displaystyle \frac{e}{|x|}}\mathrm{\nabla }u)-\mathrm{d}\mathrm{i}\mathrm{v}(\log {\displaystyle \frac{e}{|x|}}\mathrm{\nabla }(u^2))u& =& g(x,u),& x\in B_1,\\ u& =& 0,& x\in \mathrm{\partial }{B}_1,\end{array}$$

where $B_1$ denotes the unit ball centered at the origin in ${\mathbb{R}}^2$ and $g$ behaves like $\mathrm{e}\mathrm{x}\mathrm{p}(e^{s^4})$ as $s$ tends to infinity, the growth of the nonlinearity is motivated by a Trudinder-Moser inequality version, which admits double exponential growth. The proof involves a change of variable (a dual approach) combined with the mountain pass theorem.