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

Abstract

We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\varepsilon}u &&\quad \text{in } Ω, \\\ u &= 0&&\quad \text{on } \partialΩ, \end{aligned} \right. \] in a bounded domain $\Omega \subset {\mathbb{R}}^N(N\ge 7)$ with $0\in \Omega$, as $\mu ,\epsilon \to 0^{+}$. In \cite{BarGuo-ANS}, we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as $\mu =O({\epsilon }^{\alpha })$ with $\alpha >\frac{N-4}{N-2}$, $\epsilon \to 0^{+}$. Here we prove the existence of nodal bubble tower solutions, i.e.\ superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as $\mu =O(\epsilon )$, $\epsilon \to 0^{+}$.

18 pages. arXiv admin note: text overlap with arXiv:1508.04007

Cited by in corpus (1)

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