Positive solution for weighted logarithmic problem involving continuous potential under exponential nonlinearities growth

Abstract In this paper, we study the nonlinear weighted elliptic problem [[EQUATION]] where $B$ is the unit ball of $\mathbb{R^{N}}$, $N>2$, $ w(x)=\big(\log \frac{1}{|x|}\big)^{\beta(N-1)}$,$~~\beta\in[0,1)$ the singular logarithm weight with the limiting exponent $N-1$ in the Trudinger-Moser embedding. $V$ is a continuous positive potentiel. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities. We prove the existence of non trivial solutions via the critical point theory. In the critical case, the associated energy functional does not satisfy the compactness condition. We give a new growth condition and we point out its importance for checking the Palais-Smale compactness condition.