Higher order conformally invariant equations in <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb R}^3 $\end{document}</tex-math></inline-formula> with prescribed volume

Abstract

In this paper we study the following conformally invariant polyharmonic equation ∆ m u = −u 3+2m 3−2m in R 3 , u > 0, with m = 2, 3. We prove the existence of positive smooth radial solutions with prescribed volume R 3 u 6 3−2m dx. We show that the set of all possible values of the volume is a bounded interval (0, Λ * ] for m = 2, and it is (0, ∞) for m = 3.

Nondegeneracy of the bubble solutions for critical equations involving the polyharmonic operator

“…For m > 1, (1.1) arises in the Q-curvature problem. Indeed, if u is a positive solution of (1.1), then the Q-curvature of the conformal metric g = u 4 N-2m |dx| 2 (|dx| 2 is the standard Euclidean metric on R N ) is constant (see [4][5][6]). Swanson [7] showed that the function…”