The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems
This interesting paper provides answers to the title question for the semilinear elliptic problem \[ (1)\quad -\Delta u+\lambda u=u^{p- 1}\text{ in } \Omega,\quad u>0\text{ in } \Omega,\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is a smooth bounded domain in \({\mathbb R}^ N\), \(N\geq 3\), \(\lambda\geq 0\), \(2<p<2^*=2N/(N-2)\). Let cat \(\Omega\) denote the Lyusternik-Schnirelman category of \({\bar \Omega}\) in itself. Theorem 1. There exists a function f: (2,2\({}^*)\to [0,\infty)\) such that (1) has at least (cat \(\Omega\)) distinct solutions for every \(\lambda\geq f(p).\) Theorem 2. There exists a function g: \([0,\infty)\to (2,2^*)\) such that (1) has at least (cat \(\Omega\)) distinct solutions for every \(p\in [g(\lambda),2^*).\) Previous theorems linking the topology of \(\Omega\) with the multiplicity of solutions of certain cases of (1) were obtained by \textit{A. Bahri} and \textit{J. M. Coron} [Commun. Pure Appl. Math. 41, No.~3, 253--294 (1988; Zbl 0649.35033)], \textit{H. Brezis} and \textit{L. Nirenberg} [ibid. 36, 437--477 (1983; Zbl 0541.35029)], \textit{E. N. Dancer} [J. Differ. Equations 74, No. 1, 120--156 (1988; Zbl 0662.34025)], \textit{D. Passaseo} [Manuscr. Math. 65, No. 2, 147--165 (1989; Zbl 0701.35068)], and others.
scientific article; zbMATH DE number 4200738
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