Abstract
In the present work, we consider the prescribed Q-curvature problem on the unit sphere $S^{n}$ , $n\geq 5$ . Under the hypothesis that the prescribed function satisfies a flatness condition of order $\beta=n$ , we give a complete description of the lack of compactness of the problem and we provide an existence result in terms of an Euler-Hopf index.
Highlights
Introduction and the main resultOn a smooth compact manifold (Mn, g ) of dimension n ≥, the Paneitz operator is defined by Pgn u = g u divg du nQng u, where Sg denotes the scalar curvature of (Mn, g ), Ricg denotes the Ricci curvature of (Mn, g ) and (n – ) + an = (n )(n
Sn → R be a given smooth function, we look for solutions u on Sn satisfying the nonlinear problem involving the critical exponent
As we show in Corollary . , we get a new type of critical points at infinity in the space of variation which is different from those of [ ] and [ ]
Summary
On the Sobolev space H (Sn), the operator Pgn is coercive and has the following expression: This problem is quite delicate and had drown a lot of attention from mathematicians because the equation stands for a critical case which generates blow-up and lack of compactness, that the standard analytic machinery cannot apply. There is a serious problem of divergence of the integrals when β = n To overcome this challenging problem, we perform a local analysis to give precise estimates to the gradient of the Euler Lagrange functional associated to our problem and identify the critical points at infinity. In Section , we study the asymptotic behavior of the gradient flow lines of the Euler-Lagrange functional and we characterize the critical points at infinity.
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