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The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems. (English) Zbl 0727.35055

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 A. Bahri and J. M. Coron [Commun. Pure Appl. Math. 41, No. 3, 253–294 (1988; Zbl 0649.35033)], H. Brezis and L. Nirenberg [ibid. 36, 437–477 (1983; Zbl 0541.35029)], E. N. Dancer [J. Differ. Equations 74, No. 1, 120–156 (1988; Zbl 0662.34025)], D. Passaseo [Manuscr. Math. 65, No. 2, 147–165 (1989; Zbl 0701.35068)], and others.

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
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