Infinitely many sign-changing solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms

TL;DR

This paper proves the existence of infinitely many sign-changing solutions for a class of elliptic problems involving double critical Hardy-Sobolev-Maz'ya terms, using Morse index estimates under certain geometric conditions.

Contribution

It introduces a new approach to establish infinitely many solutions for elliptic equations with double critical Hardy-Sobolev-Maz'ya terms, extending previous results to sign-changing solutions.

Findings

Existence of infinitely many sign-changing solutions under specified conditions.

Application of Morse index estimates to analyze nodal solutions.

Solutions exist when the dimension exceeds certain thresholds related to parameters.

Abstract

In this paper, we investigate the following elliptic problem involving double critical Hardy-Sobolev-Maz'ya terms: $$ \left\{\begin{array}{ll} -\Delta u = \mu\frac{|u|^{2^*(t)-2}u}{|y|^t} + \frac{|u|^{2^*(s)-2}u}{|y|^s} + a(x) u, & {\rm in}\ \Omega,\\ \quad u = 0, \,\, &{\rm on}\ \partial \Omega, \end{array} \right. $$ where $\mu\geq0$, $a(x)>0$, $2^*(t)=\frac{2(N-t)}{N-2}$, $2^*(s) = \frac{2(N-s)}{N-2}$, $0\leq t<s<2$, $x = (y,z)\in \mathbb{R}^k\times \mathbb{R}^{N-k}$, $2\leq k<N$, $(0,z^*) \in \bar{\Omega}$ and $\Omega$ is an bounded domain in $\mathbb{R}^N$. Applying an abstract theorem in \cite{sz}, we prove that if $N>6+t$ when $\mu>0,$ and $N>6+s$ when $\mu=0,$ and $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate Morse indices of these nodal solution.