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

TL;DR

This paper constructs multi-bubble nodal solutions for a slightly subcritical elliptic problem with Hardy potential in symmetric domains, revealing complex blow-up behaviors and solutions with multiple positive and negative concentration points.

Contribution

It extends previous work by establishing the existence of solutions with more than two blow-up points, specifically with symmetric arrangements of positive and negative blow-ups, under certain conditions.

Findings

Existence of solutions with 2 or 3 negative blow-up points in symmetric domains.

Non-existence of solutions with 4 negative blow-up points arranged on a square.

Existence of solutions with 4 alternating positive and negative blow-up points on a square.

Abstract

We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial \Omega, \end{aligned} \right. $$ where $0\in\Omega$ and $\Omega$ is invariant under the subgroup $SO(2)\times\{\pm E_{N-2}\}\subset O(N)$; here $E_n$ denots the $n\times n$ identity matrix. If $\mu=\mu_0\epsilon^\alpha$ with $\mu_0>0$ fixed and $\alpha>\frac{N-4}{N-2}$ the existence of nodal solutions that blow up, as $\epsilon\to0^+$, positively at the origin and negatively at a different point in a general bounded domain has been proved in \cite{BarGuo-ANS}. Solutions with more than two blow-up points have not been found so far. In the present paper we obtain the existence of nodal solutions with a positive blow-up point at the origin and $k=2$ or $k=3$ negative blow-up points placed symmetrically in $\Omega\cap(\mathbb{R}^2\times\{0\})$ around the origin provided a certain function $f_k:\mathbb{R}^+\times\mathbb{R}^+\times I\to\mathbb{R}$ has stable critical points; here $I=\{t>0:(t,0,\dots,0)\in\Omega\}$. If $\Omega=B(0,1)\subset\mathbb{R}^N$ is the unit ball centered at the origin we obtain two solutions for $k=2$ and $N\ge7$, or $k=3$ and $N$ large. The result is optimal in the sense that for $\Omega=B(0,1)$ there cannot exist solutions with a positive blow-up point at the origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly there do exist solutions on $\Omega=B(0,1)$ with a positive blow-up point at the origin and four blow-up points on the vertices of a square with alternating positive and negative signs. The results of our paper show that the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well understood.