On a quasilinear mean field equation with an exponential nonlinearity

Abstract

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.

A classification result for the quasi-linear Liouville equation

“…In the non-compact situation the nonlinearity concentrates at the blow-up points as a sum of Dirac measures, whose masses likely belong to cnωnN thanks to (1.4). Such a quantization for the concentration masses has been proved [25] for n = 2 and extended [17] to n ≥ 2 by requiring an additional boundary assumption. Very refined asymptotic properties have been later established [2,11,23].…”

A-priori bounds for quasilinear problems in critical dimension

“…This, together with the boundary analysis [12], yields the desired a-priori bound for convex domains. The results in [5] have been later on extended in the quasilinear setting by Aguilar Crespo and Peral Alonso in [1] and we refer also to [15] for concentration compactness issues in this direction.…”

Isolated singularities for the n-Liouville equation

“…Theorem 1.3 gives information on the concentration mass of such Dirac measures at a singular blow-up point, which is expected bo te a super-position of several masses c n ω n carried by multiple sharp collapsing peaks governed by (1.12) γ =0 and possibly the mass (1.13) of a sharp peak described by (1.12). In the regular case such quantization property on the concentration masses has been proved [14] for n = 2 and extended [12] to n ≥ 2 by requiring an additional boundary assumption, while the singular case has been addressed in [2,21] for n = 2. For Theorem 1.4 a similar comment is in order.…”

“…Section 3 is devoted to establish Theorems 1.3-1.4 via Pohozaev identities: going back to an idea of Y.Y. Li and N. Wolanski for n = 2, the Pohozaev identities have revealed to be a fundamental tool to derive information on the mass of a singularity (see for example [2,12,18]). In Sect.…”