Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

Abstract

We show a new example of blow-up behaviour for the prescribed Q-curvature equation in even dimension 6 and higher, namely given a sequence (V k ) ⊂ C 0 (R 2n ) suitably converging we construct for n ≥ 3 a sequence (u k ) of radially symmetric solutions to the equationwith u k blowing up at the origin and on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the 4-dimensional case studied by F. Robert (J. Diff. Eq. 2006). MSC: 35J92, 53A30.Recently the authors together with S. Iula proved a partial converse to Theorem A, which we state in a simplified form.such that S 1 = ∅, (5) holds with S = S ϕ and u k → +∞ locally uniformly on S ϕ .

Concentration phenomena for the fractional Q‐curvature equation in dimension 3 and fractional Poisson formulas

“…In fact, in case ( ii ) of Theorem 1.2, one can prescribe the blow‐up set , in the sense that given any , one can construct a sequence solving () and () with on , as shown in [28]. Moreover in the radial case of dimension 6, it was also shown in [29] that the blow‐up set and can coexist; see also [23, 31, 34] for the case of a closed manifold of even dimension 4 and higher.…”