Persistence of Coron’s solution in nearly critical problem. (English) Zbl 1147.35041

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 2, 331-357 (2007); erratum ibid. 8, No. 1, 207-209 (2009).
Let \(D\) be a bounded domain with smooth boundary in \(\mathbb R^N\) \((N\geq 3)\) and let \(\lambda\) be a real parameter. This paper is devoted to the study of positive solutions of the nonlinear elliptic equations \(-\Delta u=u^{(N+2)/(N-2)+\lambda}\) in \(D\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial D\). The main result of the present paper is the following: let \(\Lambda\in\mathbb R\) be fixed. Then there exists \(\varepsilon_0>0\) such that for any \(\varepsilon\in (0,\varepsilon_0)\) and any \(\Lambda_\varepsilon\in\mathbb R\) with \(\lim_{\varepsilon\rightarrow 0}\Lambda_\varepsilon =\Lambda\), there is a solution \(u_\varepsilon\) of the above problem corresponding to \(\lambda =\Lambda_\varepsilon \varepsilon^{(N-2)/2}\) such that \(|\nabla u_\varepsilon |^2dx\rightharpoonup C_N\delta_0\) in the sense of measures as \(\varepsilon\rightarrow 0\), where \(C_N\) is a positive constant. The proof combines elliptic estimates with a scaling argument and variational tools.


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