Abstract
We consider the two-dimensional eigenvalue problem for the Laplacian with the Neumann boundary condition involving the critical Hardy potential. We prove the existence of the second eigenfunction and study its asymptotic behavior around the origin. A key tool is the Sobolev type inequality with a logarithmic weight, which is shown in this paper as an application of the weighted nonlinear potential theory.
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Acknowledgements
The authors would like to thank the anonymous referee(s) for their careful reading of the manuscript and for constructive comments. The first author (M.S.) was supported by JSPS KAKENHI Early-Career Scientists, No. 23K13001. The second author (F.T.) was supported by JSPS Grant-in-Aid for Scientific Research (B), No. 23H01084. This work was partly supported by Osaka Central University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).
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Sano, M., Takahashi, F. On Eigenvalue Problems Involving the Critical Hardy Potential and Sobolev Type Inequalities with Logarithmic Weights in Two Dimensions. J Geom Anal 34, 112 (2024). https://doi.org/10.1007/s12220-024-01559-z
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DOI: https://doi.org/10.1007/s12220-024-01559-z