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

Abstract

The paper is concerned with the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-ε}u &&\quad \text{in } Ω, \\\ u &= 0&&\quad \text{on } \partialΩ, \end{aligned} \right. \] in a bounded domain $\Omega \subset {\mathbb{R}}^N$ with $0\in \Omega$, in dimensions $N\ge 7$. We prove the existence of multi-bubble nodal solutions that blow up positively at the origin and negatively at a different point as $\epsilon \to 0$ 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.

51 pages

References in corpus (1)

• Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries