Compactness of solutions to some geometric fourth-order equations

By A. Malchiodi. 2004

96 citations recorded by the source corpus.

Abstract We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant Q-curvature. As a byproduct of our method, we also obtain compactness of such metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0fa36bcf8db261d50c8449df00b7c2bf9cec5e99

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In Applied Mathematics

1. C. B. Ndiaye
2. A. Malchiodi
3. Michael Struwe
4. Z. Djadli
5. F. Robert
6. Adimurthi
7. L. Martinazzi
8. Guofang Wang

In Geometry and Topology

1. C. B. Ndiaye
2. Z. Djadli
3. A. Malchiodi
4. Michael Struwe
5. Adimurthi
6. F. Robert
7. L. Martinazzi
8. Tianling Jin

Citations in the corpus

• Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three (2013, unknown): “, [11, 15, 16, 18, 30], or to the higher order Liouville equation (−∆)u = V e in Ω ⊂ R, V ∈ L∞(Ω), (14) see, e.”
• Morse theory and a scalar field equation on compact surfaces (2008, methodology): “As already mentioned, when qP 6= 8kπ, k ∈ N (and if Pg has no non-trivial kernel), from a theorem in [41] the equation is compact (see also [26]), so it still makes sense to define the Leray-Schauder degree of the equation.”
• Constant Q-curvature metrics in arbitrary dimension (2007, methodology): “where ‐xi stands for the Dirac delta at xi 2 M, and the ti’s are nonnegative numbers such that Pk i=1 ti = 1. The set of formal convex combinations of Dirac deltas at points of M (with a fixed number k of sumands) is known in the literature as the set of formal barycenters of M (see [26], [ 42 ] ) a”
• Blow-up analysis of a nonlocal Liouville-type equation (2015, unknown): “[17], [29] and [33].”
• Existence of conformal metrics with constant Q-curvature (2004, methodology): “The following Lemma is proved in [35] using a modification of the arguments in [16], which in turn extend to the Paneitz operator some previous embeddings due to Adams involving the operator ∆ m in flat domains.”
• Some existence results for the Toda system on closed surfaces (2007, background): “on a four-dimensional manifold M under a similar conformal change of metric, and Pg is the Paneitz operator associated to (M,g), see for example [15], [ 24 ] and the references therein.”
• CONCENTRATION PHENOMENA FOR LIOUVILLE'S EQUATION IN DIMENSION FOUR (2006, background): “In the case when M = S4, Malchiodi-Struwe [14] have shown that any such sequence (gk) of metrics when Vk → 1 uniformly either is relatively compact or blows up at a single concentration-point where a round spherical metric forms after rescaling.”