Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms

TL;DR

This paper constructs multi-bubble nodal solutions for a slightly subcritical elliptic problem with Hardy potential, demonstrating complex blow-up behaviors and solutions with multiple bubbles of different signs in high dimensions.

Contribution

It introduces new multi-bubble nodal solutions with detailed asymptotic profiles for elliptic problems involving Hardy terms, including solutions with up to five bubbles and bubble tower configurations.

Findings

Existence of multi-bubble solutions blowing up at different points.

Construction of solutions with up to five bubbles of different signs.

Development of bubble tower solutions with varying blow-up orders.

Abstract

The paper is concerned with the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega, \\\ u &= 0&&\quad \text{on } \partial\Omega, \end{aligned} \right. \] in a bounded domain $\Omega\subset\mathbb{R}^N$ with $0\in\Omega$, in dimensions $N\ge7$. We prove the existence of multi-bubble nodal solutions that blow up positively at the origin and negatively at a different point as $\epsilon\to0$ and $\mu=\epsilon^\alpha$ with $\alpha>\frac{N-4}{N-2}$. In the case of $\Omega$ being a ball centered at the origin we can obtain solutions with up to $5$ bubbles of different signs. We also obtain nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order. The asymptotic shape of the solutions is determined in detail.