Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth - Yony Raúl Santaria Leuyacc (Electronic Research Archive)
In this work, we investigated the existence of nontrivial weak solutions for the equation \begin{document}$ -{\rm div}(w(x)\nabla u) \ = \ f(x,u),\qquad x \in \mathbb{R}^2, $\end{document} where $ w(x) $ is a positive radial weight, the nonlinearity $ f(x, s) $ possesses growth at infinity of the type $ {\rm \exp}\big((α_0+h(|x|)\big)|s|^{2/(1-β)}) $, with $ α_0 > 0 $, $ 0 < β < 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.