Singular limits in higher order Lioville-type equations

Abstract

In this paper we consider the higher order Lioville-type equation $(-\Delta {)}^{m}u={\rho }^{2m}V(x)e^u$ in $\Omega \subseteq {\mathbb{R}}^{2m}$ with $V\ne 0$ a given smooth potential, $\rho \in {\mathbb{R}}^{+}$ a small parameter which tends to zero from above and where we prescribe the boundary conditions to be either Navier or Dirichlet. We find sufficient conditions under which, as $\rho$ 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 $\Omega$, 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.

24 pages. arXiv admin note: substantial text overlap with arXiv:0709.2878 by other authors

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