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

de Figueiredo, Djairo; do Ó, João Marcos; dos Santos, Ederson

doi:10.1090/proc/13114

Trudinger-Moser inequalities involving fast growth and weights with strong vanishing at zero
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by Djairo G. de Figueiredo, João Marcos B. do Ó and Ederson Moreira dos Santos PDF

Proc. Amer. Math. Soc. 144 (2016), 3369-3380 Request permission

Abstract:

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

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