LAUSR

In this paper, we study a Trudinger-Moser inequality with a vanishing weight in the unit disk. Precisely, let $$B$$ B be the unit disk and β ≥ 0 be a real number. Denote $${\cal S} = \left\{{u \in W_0^{1,2}\left(B\right);u} \right.$$ S = { u ∈ W 0 1 , 2 ( B ) ; u is a radially symmetric function and ∥∇ u ∥ 2 ≤ 1 . Suppose a function h ( x ) is radially symmetric, nonnegative, continuous on $$\overline{\mathbb{B}}$$ B ¯ and satisfies h ( x ) > 0 ⇔ z ≠ 0 and lim x →0 h ( x )| x | −2 β = 1. Using the blow-up analysis, we prove that the supremum $$\mathop {\sup}\limits_{u \in {\cal S}} \int_B {{e^{4\pi \left({1 + \beta} \right){u^2}}}h\left(x \right)dx} $$ sup u ∈ S ∫ B e 4 π ( 1 + β ) u 2 h ( x ) d x can be attained by some nonnegative decreasing function $${u_0} \in {\cal S} \cap {C^\infty}\left({\overline{\mathbb{B}}} \right)$$ u 0 ∈ S ∩ C ∞ ( B ¯ ) with ∥∇ u 0 ∥ 2 = 1. This improves the recent result of Yang-Zhu [31] and complements that of de Figueiredo-do ϓ-dosSantos[6]