The mean field equation involving the N -Laplace operator and an exponential nonlinearity is considered in dimension N ≥ 2 on bounded domains with homogeneous Dirichlet boundary condition. By a detailed asymptotic analysis we derive a quantization property in the non-compact case, yielding to the compactness of the solutions set in the so-called non-resonant regime. In such a regime, an existence result is then provided by a variational approach.© 2015 Elsevier Masson SAS. All rights reserved.r é s u m é L'équation de champ moyen avec l'opérateur de N -Laplace et une non-linéarité exponentielle est considérée en dimension N ≥ 2 sur des domaines bornés avec condition Dirichlet homogène au bord. Par une analyse asymptotique fine on dérive une propriété de quantification dans le cas non-compact, en produisant la compacité de l'ensemble des solutions dans le régime non-résonnant. Dans ce régime, un résultat d'existence est alors établi par une approche variationnelle.

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In this paper we consider the following higher order boundary value problemwhere Ω is a smooth bounded domain in R 2m with m ∈ N, V (x) = 0 is a smooth function positive somewhere in Ω and ρ is a positive small parameter. Here, the operator Bj stands for either Navier or Dirichlet boundary conditions. We find sufficient conditions under which, as ρ approaches 0, there exists an explicit class of solutions which admit a concentration behavior with a prescribed bubble profile around some given k-points in Ω, for any given integer k.These are the so-called singular limits. The candidate k-points of concentration must be critical points of a suitable finite dimensional functional explicitly defined in terms of the potential V and the higher order Green's function with respect to the imposed boundary conditions.

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