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)].


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
Full Text: EMIS