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

Abstract

We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-ε}u &&\quad \text{in } Ω\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial Ω, \end{aligned} \right. $$ where $0\inΩ$ and $Ω$ 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 $μ=μ_0ε^α$ with $μ_0>0$ fixed and $α>\frac{N-4}{N-2}$ the existence of nodal solutions that blow up, as $ε\to0^+$, 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 $Ω\cap(\mathbb{R}^2\times\{0\})$ around the origin provided a certain function $f_k:\mathbb{R}^+\times\mathbb{R}^+\times I\to\mathbb{R}$ has stable critical points; here $I=\{t>0:(t,0,\dots,0)\inΩ\}$. If $Ω=B(0,1)\subset\mathbb{R}^N$ 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 $Ω=B(0,1)$ 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 $Ω=B(0,1)$ 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.