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Multiplicity of solutions to the supercritical Bahri–Coron’s problem in pierced domains. (English) Zbl 1166.35333

Summary: We consider the supercritical Dirichlet problem
\[ -\Delta u = u^{{\frac{N+2}{N-2}} +\varepsilon} \quad \text{in}\, \Omega, u>0 \quad \text{in}\, \Omega, u=0 \quad \text{on} \quad \partial\Omega,\tag{\(P_{\varepsilon}\)} \]
where \(N \geq 3, \varepsilon > 0\) and \(\Omega \subset \mathbb{R}^{N}\) is a smooth bounded domain with a small hole of radius \(d\). When \(\Omega\) has some symmetries, we show that \((P_{\varepsilon})\) has an arbitrary number of solutions for \(\varepsilon\) and \(d\) small enough. When \(\Omega\) has no symmetries, we prove the existence of solutions blowing up at two or three points cLose to the hole as \(\varepsilon\) goes to zero for \(d\) small enough.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
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