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by Martinazzi

Venue: | J. Funct. Anal |

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**15**

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...ove show that they are necessary to some extent (see the first open problem in the last section). Moreover, contrary to [Rob2] and [LS], we do not assume that V0 > 0. In fact, as already discussed in =-=[Mar3]-=-, if V0 has changing sign, one can show using the results of [Mar2] that, if (6) holds, blow-up happens only at points where V0 > 0. We also point out that when m = 2, F. Robert [Rob3] proved a versio...

by
Xuezhang Chen, Li Ma, Xingwang Xu

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...(32) that l = 1 and Q∞(p) = (n − 1)!. Finally, the rest part of the proof of Lemma 3 is the same as the proof of Lemma 3.2 in [6]. Remark 1. We should point out that, one can not apply Theorem 9 in =-=[12]-=- to derive Lemma 3 directly. The assumption in [12]: Qk → Q∞ in C0(Sn) is much stronger than the one in Lemma 3. Similar blow-up analysis as in [12] has also been done by Malchiodi [10]. However those...

by
Luca Martinazzi, Centro De Giorgi, Mircea Petrache
, 2010

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...N ∈ N, contrary to our assumption on λ. As shown in [2], things are more subtle in higher dimension, and we cannot work locally as in [17]. Instead, we can rely on a recent result by the first author =-=[13]-=- specific for closed manifolds (see also [7], [11] and [15]) to obtain compactness for the sequence (uk), unless λk → NΛ1 for some N ∈ N. Roughly speaking, the geometric constant Λ1 enters our problem...

by
Tianling Jin, Ali Maalaoui, Luca Martinazzi, Jingang Xiong
, 2014

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...inger type inequality, see, e.g., [1, 2, 11, 15, 22, 25, 30, 35], or the equation P 2mg u+Qg = Qe 2mu on a manifold (M2m, g) (13) which prescribes the Q-curvature of the manifold (M,e2ug), see, e.g., =-=[12, 18, 19, 21, 34]-=-, or to the higher order Liouville equation (−∆)mu = V e2mu in Ω ⊂ R2m, V ∈ L∞(Ω), (14) see, e.g., [23, 28, 29, 31, 26]. The main idea is that if a sequence {uk} of solutions (or the heat flow) of (12...

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