Abstract
We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the n-sphere involving an operator of order 2s>n. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of s and settle the question of validity of a corresponding inequality in all dimensions n≥2.
Highlights
Introduction and main resultsWe are interested in sharp constants in conformally invariant Sobolev inequalities.The classical version of this inequality concerns powers (−Δ)s of the Laplacian in Rn with a real parameter
The work of Dou and Zhu [22] spiked a lot of interest in reversed Hardy–Littlewood– Sobolev (HLS) inequalities
For open questions in a non-conformally invariant case motivated by aggregationdiffusion equations, see [11, 12]
Summary
We are interested in sharp constants in conformally invariant Sobolev inequalities. The classical version of this inequality concerns powers (−Δ)s of the Laplacian in Rn with a real parameter. Our goal in this paper is to investigate the range n s> Note that in this case the is negative, and we will restrict ourselves to functions that are positive almost everywhere. It is because of this sign change that we call the inequalities in this paper ‘reverse’ Sobolev inequalities. In our proof we apply a result of Li [37] and to do so, we use a relation between the Euler–Lagrange equations corresponding to (10) and (6). The latter paper contains a characterization of solutions to the Euler–Lagrange equation corresponding to (9) and, just as in [32] this will be a major ingredient in our proof of Theorem 1. Our Theorem 1 plays the same role for the fractional order problems in [14, 13, 15] as the results in [48, 33] do in the Q-curvature problem on three-dimensional manifolds
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