Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Mazya type equation

Abstract

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -Δu - μ\frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x=(y,z)\in {\mathbb{R}}^h\times {\mathbb{R}}^k={\mathbb{R}}^n$, with $n\ge 5$, $2<k<n$, and $t=n-\frac{(n-2)q}{2}$. We establish the existence of a bi-radial sign-changing solution under the assumptions $0\le \mu <\frac{(k-2{)}^2}{4},2<q<2^{\ast }=\frac{2(n-k+1)}{n-k-1}$. We approach the problem by lifting it to the hyperbolic setting, leading to the equation: $-{\Delta }_{{\mathbb{B}}^N}u-\lambda u=|u{|}^{p-1}u,u\in H^1({\mathbb{B}}^N)$, ${\mathbb{B}}^N$ is the hyperbolic ball model. We study the existence of a sign-changing solution with suitable symmetry by constructing an appropriate invariant subspace of $H^1({\mathbb{B}}^N)$ and applying the concentration compactness principle, and the corresponding solution of the Hardy-Sobolev-Maz'ya type equation becomes bi-radial under the corresponding isometry.