An elliptic problem involving a potential with exponential growth

We study the nonlinear weighted elliptic problem $$-\nabla . (w_{\beta }(x)|\nabla u|^{N-2} \nabla u)+V(x)|u|^{N-2}u = f(x,u), \quad u\in W_{0}^{1,N}(B,w_{\beta }) , $$ where $B$ is the unit ball of $\mathbb R^{N}$, $N \gt 2$, $w_{\beta }(x)=(1-\log |x|)^{\beta (N-1)}$, $\beta \in [0,1)$, is the singular logarithmic weight with the limiting exponent $N-1$ in the Trudinger–Moser embedding, and $V$ is a continuous positive potential. The nonlinearities critical or subcritical growth in view of Trudinger–Moser inequalities. We prove the existence of nontrivial solutions via 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.