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 $$\{\begin{array}{rlrl}-\Delta u-\mu \frac{u}{|x{|}^2}& =|u{|}^{2^{\ast }-2-\epsilon }u& & \text{in }\Omega \subset {\mathbb{R}}^N,\\ u& =0& & \text{on }\partial \Omega ,\end{array}$$ 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 $\epsilon \to 0^{+}$, 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 $N\ge 7$, 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.

22 pages

References in corpus (1)

• Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms