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

TL;DR

This paper proves the existence of a bi-radial sign-changing solution for a Hardy-Sobolev-Maz'ya type equation by transforming it into a hyperbolic setting and applying concentration compactness methods.

Contribution

It introduces a novel approach by lifting the problem to hyperbolic space and constructs invariant subspaces to find sign-changing solutions with symmetry.

Findings

Existence of bi-radial sign-changing solutions under specified conditions.

Application of concentration compactness principle in hyperbolic setting.

Solution exhibits bi-radial symmetry after isometric transformation.

Abstract

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -\Delta u - \mu \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 $n \geq 5$, $2 < k <n$, and $t = n - \frac{(n-2)q}{2}$. We establish the existence of a bi-radial sign-changing solution under the assumptions $0 \leq \mu < \frac{(k-2)^2}{4}, \, 2 < q <2^* = \frac{2(n-k+1)}{n-k-1}$. 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.