Cao, Daomin; Chabrowski, J. On the number of positive solutions for nonhomogeneous semilinear elliptic problem. (English) Zbl 0851.35039 Adv. Differ. Equ. 1, No. 5, 753-772 (1996). The paper deals with the problem of multiplicity of positive solutions of the Dirichlet problem \[ - \Delta u+ \lambda u= |u|^{p- 2} u+ h\quad \text{in} \quad \Omega,\quad u= 0\quad \text{on} \quad \partial\Omega\tag{1} \] in terms of \(\text{cat } \Omega\) (the Lyusternik-Schnirelman category of \(\overline\Omega\) in itself), where \(\Omega\subset \mathbb{R}^N\), \(N\geq 3\), is a bounded domain, \(\lambda\geq 0\), \(h\in L^2(\Omega)\), \(h\geq 0\) and \(h\not\equiv 0\), and \(2< p< 2^*= 2N/(N- 2)\).It is shown that there exists a function \(\overline \lambda: (2, 2^*)\to [0, + \infty)\) such that for all \(\lambda\geq \overline \lambda(p)\) problem (1) admits at least \(\text{cat } \Omega+ 1\) distinct positive solutions. Moreover, if \(\text{cat } \Omega> 1\), then the number of distinct positive solutions is at least \(\text{cat } \Omega+ 2\).In particular, for an uncontractible domain \(\Omega\), problem (1) has at least four distinct positive solutions. The method of the proof exploits the Lyusternik-Schnirelman theory of critical points which was also used in some earlier papers by Benci, Cerami, Passaseo, Candela, the first author et al. for the case \(h\equiv 0\) in \(\Omega\). Reviewer: S.Migorski (Krakow) MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Keywords:multiplicity of positive solutions × Cite Format Result Cite Review PDF