Concentration phenomena to a higher order Liouville equation
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Calculus of Variations
Concentration phenomena to a higher order Liouville equation Ali Hyder1 Received: 23 January 2020 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study blow-up and quantization phenomena for a sequence of solutions (u k ) to the prescribed Q-curvature problem n 2nu k 2n (−) u k = Q k e in ⊂ R , e2nu k d x ≤ C,
under natural assumptions on Q k . It is well-known that, up to a subsequence, either (u k ) is bounded in a suitable norm, or there exists βk → ∞ such that u k = βk (ϕ + o(1)) in \ (S1 ∪ Sϕ ) for some non-trivial non-positive n-harmonic function ϕ and for a finite set S1 , where Sϕ is the zero set of ϕ. We prove quantization of the total curvature ˜ Q k e2nu k d x on ˜ ( \ Sϕ ). the region Mathematics Subject Classification 35B44 · 35B33 · 35J91
1 Introduction to the problem Given a bounded domain ⊂ R2n , we will consider a sequence (u k ) of solutions to the prescribed Q-curvature equation (−)n u k = Q k e2nu k in ,
(1)
under the uniform (volume) bound e2nu k d x ≤ C, k = 1, 2, 3, . . .
(2)
and suitable bounds on Q k ≥ 0.
Communicated by A. Chang. The author is supported by the Swiss National Science Foundation projects No. P2BSP2-172064 and P400P2-183866.
B 1
Ali Hyder [email protected] Johns Hopkins University, Baltimore, Maryland 21218, USA 0123456789().: V,-vol
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A. Hyder
Contrary to the two dimensional situation studied by Brézis and Merle [2] and Li and Shafrir [11] (see also [4,13,14,19,21]) where blow up occurs only on a finite set S1 , in dimension 4 and higher it is possible to have blow-up on larger sets. More precisely, for a finite set S ⊂ let us introduce K(, S) := {ϕ ∈ C ∞ ( \ S) : ϕ ≤ 0, ϕ ≡ 0, n ϕ ≡ 0},
(3)
and for a function ϕ ∈ K(, S) set Sϕ := {x ∈ \ S : ϕ(x) = 0}.
(4)
We shall use the notations S∞ and Ssph to denote the set of all blow-up points and the set of all spherical blow-up points respectively, where such points are defined as follows: A point x ∈ is said to be a blow-up point if there exists a sequence of points (x k ) in such that xk → x and u k (xk ) → ∞. A point x ∈ S∞ is said to be a spherical blow-up point if there exists x k → x and rk → 0+ such that for some c ∈ R 2n−1 (R2n ), ηk (x) := u k (xk + rk x) + log rk + c → η(x) in Cloc
where η is a spherical solution to
(−)n u = (2n − 1)!e2nu in R2n , that is, η is of the form
η(x) = log
R2n
e2nu(x) d x < ∞,
2λ 1 + λ2 |x − x0 |2
(5)
,
(6)
for some λ > 0 and x0 ∈ R2n . Theorem A ([1,16]) Let be a bounded domain in R2n , n > 1 and let (u k ) be a sequence of solutions to (1)–(2), where 0 Q k → Q 0 > 0 in Cloc (),
and define the set S1 := x ∈ : lim lim inf r →0+ k→∞
Br (x)
Qk e
2nu k
(7)
1 , 1 := (2n − 1)!|S 2n |. dx ≥ 2
Then up to extracting a subsequence one of the following is true. 2n−1,α (i) For every 0 ≤ α < 1, (u k ) is bounded in Cloc (). (ii) There exists ϕ ∈ K(, S1 ) and a sequence of numbers βk → ∞ such that uk 2n−1,α → ϕ in Cloc ( \
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