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

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.