Elliptic problems in \(\mathbb R^N\) with critical and singular discontinuous nonlinearities. (English) Zbl 1347.35099
The authors are concerned with the problem \(-\Delta u=\lambda(u^{2^\ast-1}+\chi_{\{u<a\}}u^{-\delta})\) in \(\Omega\), subject to homogeneous Dirichlet boundary condition. Here \(\Omega\subset\mathbb{R}^N\), \(N\geq 3\), is a smooth and bounded domain, \(\delta\in (0,3)\), \(a>0\), \(\lambda>0\). The main result of the paper establishes that for any \(a>0\), there exists \(\Lambda^a>0\) such that the above problem has no solutions for \(\lambda>\Lambda^a\) while for \(\lambda\in (0,\Lambda^a)\) the problem possesses at least two solutions.
Reviewer: Marius Ghergu (Dublin)
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
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