Sign-changing bubble tower solutions for a Paneitz-type problem

TL;DR

This paper proves the existence of multiple sign-changing solutions to a biharmonic problem involving a Paneitz-type operator, with solutions concentrating at a small hole in the domain, expanding understanding of nonlinear biharmonic equations.

Contribution

It introduces new methods to construct arbitrarily many sign-changing bubble solutions concentrating at a domain's hole for a Paneitz-type problem.

Findings

Existence of arbitrarily many sign-changing solutions.

Solutions concentrate at the center of the small hole.

Profiles are superpositions of bubbles with alternating signs.

Abstract

This paper is concerned with the following biharmonic problem \begin{equation}\label{ineq} \begin{cases} \Delta^2 u=|u|^{\frac{8}{N-4}}u &\text{ in } \ \Omega\backslash \overline{{B(\xi_0,\varepsilon)}}, u=\Delta u=0 &\text{ on } \ \partial (\Omega \backslash \overline{{B(\xi_0,\varepsilon)}}), \end{cases} \end{equation} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N\geq 5$, and $B(\xi_0,\varepsilon)$ is a ball centered at $\xi_0$ with radius $\varepsilon$, $\varepsilon$ is a small positive parameter. We obtain the existence of solutions for problem (\ref{ineq}), which is an arbitrary large number of sign-changing solutions whose profile is a superposition of bubbles with alternate sign which concentrate at the center of the hole.