Standing waves for quasilinear Schrödinger equations involving double exponential growth

Abstract

<abstract><p>We will focus on the existence of nontrivial, nonnegative solutions to the following quasilinear Schrödinger equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>