Ground state solution for a weighted elliptic problem under double exponential nonlinear growth

Abstract

This work is concerned with the existence of a positive ground state solution for the following weighted problem:

where $B$ is the unit ball of ${\mathbb{R}}^N$, $\sigma (x)=(\log (e/|x|){)}^{N-1}$, the singular logarithm weight in the Trudinger–Moser embedding. The nonlinearity is assumed to have exponential growth in view of Trudinger–Moser type inequalities. We introduce the Nehari manifold associated to the energy and use minimax techniques to prove the existence of a positive ground state solution. In the critical case, the energy loses compactness at a certain level. We provide a new condition for growth and we stress its importance to check compactness level.