| Summary: | We will focus on the existence of nontrivial, nonnegative solutions to the following quasilinear Schrödinger equation
$ \begin{equation*} \left\lbrace\begin{array}{rcll} -{\rm div} \Big(\log \dfrac{e}{|x|}\nabla u\Big) -{\rm div} \Big(\log \dfrac{e}{|x|}\nabla (u^2)\Big) u \ & = &\ g(x, u), &\ x \in B_1, \\ u \ & = &\ 0, &\ x \in \partial B_1, \end{array}\right. \end{equation*} $
where $ B_1 $ denotes the unit ball centered at the origin in $ \mathbb{R}^2 $ and $ g $ behaves like $ {\rm exp}(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.
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