Trudinger-Moser inequalities involving fast growth and weights with strong vanishing at zero

In this paper we study some weighted Trudinger-Moser type problems, namely s F , h = sup u ∈ H , ‖ u ‖ H = 1 ∫ B F ( u ) h ( | x | ) d x , \begin{equation*} \displaystyle {s_{F,h} = \sup _{u \in H, \, \| u\|_H =1 } \int _{B} F(u) h(|x|) dx}, \end{equation*} where B ⊂ R 2 B \subset {\mathbb R}^2 represents the open unit ball centered at zero in R 2 {\mathbb R}^2 and H H stands either for H 0 , rad 1 ( B ) H^1_{0, \textrm {rad}}(B) or H rad 1 ( B ) H^1_{\textrm {rad}}(B) . We present the precise balance between h ( r ) h(r) and F ( t ) F(t) that guarantees s F , h s_{F,h} to be finite. We prove that s F , h s_{F,h} is attained up to the h ( r ) h(r) -radially critical case. In particular, we solve two open problems in the critical growth case.