Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions. (English) Zbl 1327.35002
Summary: The Moser-Trudinger embedding has been generalized by Adimurthi and K. Sandeep [NoDEA, Nonlinear Differ. Equ. Appl. 13, No. 5–6, 585–603 (2007; Zbl 1171.35367)] to the following weighted version: if \(\Omega \subset \mathbb R^2\) is bounded, \(\alpha >0\) and \(\beta \in [0,2)\) are such that
\[
\frac{\alpha }{4\pi }+\frac{\beta }{2}\leq 1,
\]
then
\[
\sup_{\begin{aligned} v\in W^{1,2}_0(\Omega ) \\ \| \nabla v\| _{L^2}\leq 1 \end{aligned}}\int_\Omega\frac{e^{\alpha v^2}-1}{| x|^\beta}\leq C.
\]
We prove that the supremum is attained, generalizing a well-known result by M. Flucher [Comment. Math. Helv. 67, No. 3, 471–497 (1992; Zbl 0763.58008)], who has proved the case \(\beta =0\).
MSC:
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
References:
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