| Summary: | We consider the singular semilinear elliptic equation $$ -Delta u-frac{mu }{| x| ^2}u-lambda u=f(x)| u| ^{2^{ast }-1} $$ in $Omega $, $u=0$ on $partial Omega $, where $Omega $ is a smooth bounded domain, in $mathbb{R}^N$, $Ngeq 4$, $2^{ast }:=frac{2N}{N-2}$ is the critical Sobolev exponent, $f:mathbb{R} ^No mathbb{R}$ is a continuous function, $0<lambda <lambda _1$, where $lambda _1$ is the first Dirichlet eigenvalue of $-Delta -frac{mu }{| x| ^2}$ in $Omega $ and $0<mu < overline{mu }:=(frac{N-2}{2})^2$. We show that if $Omega $ and $f$ are invariant under a subgroup of $O(N)$, the effect of the equivariant topology of $Omega $ will give many symmetric nodal solutions, which extends previous results of Guo and Niu [8].
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