Singular limits for the bi-laplacian operator with exponential nonlinearity in $\R^4$

TL;DR

This paper investigates the behavior of solutions to a bi-laplacian PDE with exponential nonlinearity in four-dimensional domains, showing solutions can blow up at multiple points as a parameter tends to zero.

Contribution

It establishes the existence of solutions that blow up at multiple points in a 4D domain with certain topological properties, as the parameter approaches zero.

Findings

Solutions blow up at multiple points as parameter tends to zero

Existence of solutions depends on the domain's topological properties

Blow-up points can be prescribed in number and location

Abstract

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{4}$ such that for some integer $d\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {\Delta^{2}u-\rho^{4}k(x)e^{u}=0 & \hbox{in}\Omega, u=\Delta u=0 & \hbox{on}\partial\Omega, has a solution blowing-up, as $\rho\to0$, at $m$ points of $\Omega$, for any given number $m$.