## Multiple solutions for a singular semilinear elliptic problems with critical exponent and symmetries.(English)Zbl 1198.35113

Summary: We consider the singular semilinear elliptic equation $$-\Delta u-\frac{\mu}{|x|^2}u-\lambda u=f(x)|u|^{2^*-1}$$ in $$\Omega$$, $$u=0$$ on $$\partial\Omega$$, where $$\Omega$$ is a smooth bounded domain, in $$\mathbb R^N$$, $$N\geq 4$$, $$2^*:=\frac{2N}{N-2}$$ is the critical Sobolev exponent, $$f:\mathbb R^N\to\mathbb R$$ 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 $$\Omega$$ gives many symmetric nodal solutions, which extends previous results of Q. Guo and P. Niu [J. Differ. Equations 245, No. 12, 3974–3985 (2008; Zbl 1159.35031)].

### MSC:

 35J75 Singular elliptic equations 35J61 Semilinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35J20 Variational methods for second-order elliptic equations 49J52 Nonsmooth analysis 58E35 Variational inequalities (global problems) in infinite-dimensional spaces

Zbl 1159.35031
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