In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers, see Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] or Martinazzi [34]. We answer all of them, proving in particular that blow up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will have a sequel, giving existence results of critical points of the associated functional at all energy levels via degree theory arguments, in the spirit of what had been done for the Liouville equation in the beautiful work of Chen-Lin [8].
The Adimurthi-Druet [1] inequality is an improvement of the standard Moser-Trudinger inequality by adding a L 2 -type perturbation, quantified by α ∈ [0, λ 1 ), where λ 1 is the first Dirichlet eigenvalue of ∆ on a smooth bounded domain. It is known [3,9,13,18] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi-Druet inequality does not admit any extremal, when the perturbation parameter α approaches λ 1 . Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler-Lagrange equation, which take into account the fact that the problem becomes singular as α → λ 1 .
We discuss existence of solutions, compactness and stability properties for Kirchhoff-type systems in closed [Formula: see text]-manifolds [Formula: see text], [Formula: see text]. The Kirchhoff systems we consider are written as [Formula: see text] for all [Formula: see text], where [Formula: see text] is the Laplace–Beltrami operator, [Formula: see text] is a [Formula: see text]-map from [Formula: see text] into the space [Formula: see text] of symmetric [Formula: see text] matrices with real entries, the [Formula: see text]’s are the components of [Formula: see text], [Formula: see text], [Formula: see text] is the Euclidean norm of [Formula: see text], [Formula: see text] is the critical Sobolev exponent, and we require that [Formula: see text] in [Formula: see text] for all [Formula: see text].
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