Quantization effects for a fourth-order equation of exponential growth in dimension $4$

Abstract

We investigate the asymptotic behavior as k → +∞ of sequences (u k ) k∈N ∈ C 4 (Ω) of solutions of the equations ∆ 2 u k = V k e 4u k on Ω, where Ω is a bounded domain of R 4 and lim k→+∞ V k = 1 in C 0 loc (Ω). The corresponding 2-dimensional problem was studied by Brézis-Merle and Li-Shafrir who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Struwe and the author [1], such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on ∆u k , we recover such a quantization as in dimension 2.

Concentration phenomena for Liouville's equation in dimension four

“…As shown by Robert [18], the same result holds if for some open subset ∅ = ω ⊂ we have the a priori bounds…”

“…Also in the radially symmetric case there is a complete description of the possible concentration patterns; see [18].…”

Conformal metrics on $\mathbb R^{2 m}$ with constant $Q$-curvature and large volume

“…[BM] and [LS]), on S 2 (see [Str4]) and to the prescribed Q-curvature equation in dimension 2m (see e.g. [DR], [Mal], [MS], [Ndi], [Rob1], [Rob2], [Mar3], [Mar4]). For instance consider the following model problem.…”

Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition

“…Indeed, with (3.4), we get that û k L 1 (Ω) ≤ C for all k ∈ N. It then follows from Theorem 1.2 of [35] that there existsû ∈ C 4 (Ω) such that lim k→+∞ûk =û in C 3 loc (Ω). Therefore S ⊂ ∂Ω.…”

“…It follows from Theorem 1.2 of [35] that there exists S 1 ⊂ Ω, where S 1 is at most finite, such thatû k ≤ C(ω) uniformly in ω for ω ⊂⊂ Ω\S 1 . Therefore, with (3.4), we get that (α k ) cannot go to −∞ when k → +∞.…”