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.
Categories: math.AP