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PAPER: Sharp constant for a 2D anisotropic Sobolev inequality with critical nonlinearity

REF. DATE:15 July 2010 CREATED/MODIFIED: 14 January 2016 373 views No Comment

Sharp constant for a 2D anisotropic Sobolev inequality with critical nonlinearity (with J. Chen), Journal of Mathematical Analysis and Applications 367:2(2010), 685-692.


ASBTRACT: For the 2-dimensional anisotropic Sobolev inequality of the form

    \[\int_{\bkR^2} |u|^6dxdy \leq \alpha \bigg(\int_{\bkR^2} u_x^2dxdy\bigg)^2\int_{\mathbb{R}^2} |D_x^{-1}u_y|^2dxdy,\]

where D_x^{-1} is defined as

    \[D_x^{-1}h(x,y) = \int_{-\infty}^xh(s,y)ds,\]

it is proved that the sharp (smallest) positive constant \alpha is exactly given by 3(\int_{\mathbb{R}^2}\phi_x^2dxdy)^{-2}, where \phi is a minimal action solution of the equation

    \[(u_{xx} + |u|^4u)_x = D_x^{-1} u_{yy}.\]


Download: Preprint version (RIA | download) / Publisher version (doi:10.1016/j.jmaa.2010.02.020)
Extra info: (P073/2010-07-15)

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