Existence of two solutions for the Bahri-Coron problem in an annular domain with a thin hole (Q647630)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\), \(N\geq 3\), with smooth boundary. As a consequence of Pohožaev's identity, it has been known for a long time that the problem \[ -\Delta u = |u|^{2^*-2}u \quad\text{in } \Omega, \qquad u=0 \quad\text{on } \partial\Omega, \leqno(*) \] has only the trivial solution \(u\equiv 0\) if \(\Omega\) is star-shaped. Here \(2^*=2N/(N-2)\). In contrast, \textit{J.-M.~Coron} [C. R. Acad. Sci., Paris, Sér. I 299, 209--212 (1984; Zbl 0569.35032)] showed that if \(\Omega\) contains \(\{ x\in \mathbb{R}^N: R_1 < |x| < R_2 \}\), \(\{ x\in \mathbb{R}^N: |x|< R_1 \} \backslash \overline\Omega \neq \emptyset\), and \(R_2/R_1\) is large enough, then \((*)\) has at least one positive solution. Subsequently \textit{A.~Bahri} and \textit{J.-M.~Coron} [Commun. Pure Appl. Math. 41, No. 3, 253--294 (1988; Zbl 0649.35033)] showed that if the homology group \(H_d(\Omega;\mathbb{Z}_2)\neq 0\) for some positive integer \(d\), then \((*)\) has at least one positive solution (this is the case if \(N=3\) and \(\Omega\) is not contractible). Recently, \textit{M.~Clapp} and \textit{T.~Weth} [Commun. Contemp. Math. 10, No. 1, 81--101 (2008; Zbl 1157.35048)] showed that under the same assumptions as in the result of Coron, \((*)\) has at least two pairs of nontrivial solutions if \(R_2/R_1\) is large enough. Later \textit{E. N.~Dancer} [Bull. Lond. Math. Soc. 20, No. 6, 600--602 (1988; Zbl 0646.35027)], \textit{W. Ding} [J. Partial Differ. Equations 2, No. 4, 83--88 (1989; Zbl 0694.35067)] and \textit{D.~Passaseo} [Manuscr. Math. 65, No. 2, 147--165 (1989; Zbl 0701.35068)] showed that there are contractible domains for which \((*)\) has a solution. Here the authors prove an analogous generalization of the result of Chapp and Weth: there exist at least two pairs of nontrivial solutions of \((*)\) for domains of the type considered by Dancer, Ding and Passaseo. The proof uses a topological tool, the ``fixed point transfer'' of \textit{A.~Dold} [Math. Z. 148, 215--244 (1976; Zbl 0329.55007)].