𝐿^{𝑝} estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities

Vassilev, Dimiter

doi:10.1090/S0002-9947-2010-04850-0

$L^p$ estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities
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by Dimiter Vassilev

Trans. Amer. Math. Soc. 363 (2011), 37-62

DOI: https://doi.org/10.1090/S0002-9947-2010-04850-0

Published electronically: August 31, 2010

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Abstract:

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-Laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb {R}^n$, which imply asymptotic behavior of the solutions at infinity. In addition, we find the best constant and extremals in the case of the considered $L^2$ Hardy-Sobolev inequality.

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