Concentration phenomena to a higher order Liouville equation

2020/01/23 by Hyder, Ali
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2001.08334

Abstract

We study blow-up and quantization phenomena for a sequence of solutions (u_k) to the prescribed Q-curvature problem (-Δ)ⁿu_k= Q_ke^2nu_k in Ω⊂ℝ²ⁿ, ∫_Ωe^2nu_kdx≤ 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 β_k→∞ such that u_k=β_k(φ+o(1)) in Ω∖ (S₁∪ S_φ) for some non-trivial non-positive n-harmonic function φ and for a finite set S₁, where S_φ is the zero set of φ. We prove quantization of the total curvature ∫_ΩQ_ke^2nu_kdx on the region Ω\Subset(Ω∖ S_φ). We also consider a non-local case in dimension three.

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