Existence of Positive Solutions to Weighted Linear Elliptic Equations Under Double Exponential Nonlinearity Growth

In this article, a weighted problem under boundary Dirichlet condition in the unit ball of $${\mathbb {R}}^{2}$$ is considered. The non-linearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. It is proved that there is a nontrivial positive weak solution to this equation. In the critical case, the compactness condition is not satisfied but a suitable asymptotic condition is used to avoid the non-compactness levels to the energy functional.