Singular limits in higher order Lioville-type equations

TL;DR

This paper studies the singular limit behavior of solutions to higher order Liouville-type equations, showing existence of solutions concentrating around prescribed points as a small parameter tends to zero.

Contribution

It provides explicit conditions for the existence of solutions with prescribed concentration profiles in higher order Liouville equations, extending previous results to more complex boundary conditions and potentials.

Findings

Solutions concentrate around critical points of a finite-dimensional functional.

Existence of solutions with prescribed bubble profiles as parameter approaches zero.

Conditions depend on the potential and boundary conditions.

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 $V\neq0$ 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.