Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions

By G. Csató, Prosenjit Roy. 2014

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The Moser-Trudinger embedding has been generalized by Adimurthi and Sandeep to the following weighted version: if $$\Omega \subset \mathbb {R}^2$$Ω⊂R2 is bounded, $$\alpha >0$$α>0 and $$\beta \in [0,2)$$β∈[0,2) are such that $$\begin{aligned} \frac{\alpha }{4\pi }+\frac{\beta }{2}\le 1, \end{aligned}$$α4π+β2≤1,then $$\begin{aligned} \sup _{\begin{array}{c} v\in W^{1,2}_0(\Omega ) \\ \Vert \nabla v\Vert _{L^2}\le 1 \end{array}} \int _{\Omega }\frac{e^{\alpha v^2}-1}{|x|^{\beta }}\le C. \end{aligned}$$supv∈W01,2(Ω)‖∇v‖L2≤1∫Ωeαv2-1|x|β≤C.We prove that the supremum is attained, generalizing a well-known result by Flucher, who has proved the case $$\beta =0$$β=0.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf829c9b9d54cb54935c5960daa757a5a61d8225

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