## 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.

### MSC:

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