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Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. (English) Zbl 0729.35014

The problem \[ (1)\quad -\Delta u=n(n-2)u^{p-\epsilon}\text{ in } \Omega;\quad u>0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega \] is considered, where \(\Omega\) is a smooth bounded domain in \({\mathbb{R}}^ n\) (n\(\geq 3)\), \(p=(n+2)/(n-2)\), and \(\epsilon\geq 0\). If \(S_ n\) is the best Sobolev constant in \({\mathbb{R}}^ n\) and \(u_{\epsilon}\) is the solution of (1), under the assumption \[ \lim_{\epsilon \to 0}(\int_{\Omega}| Du_{\epsilon}|^ 2 dx/\| u_{\epsilon}\|^ 2_{L^{p+1-\epsilon}(\Omega)})=S_ n \] the author proves (after passing to a subsequence):
(i) there exists \(x_ 0\in \Omega\) sucht that \(u_{\epsilon}\to 0\) in \(C^ 1(\Omega \setminus \{x_ 0\}),\)
\(| Du_{\epsilon}|^ 2\to n(n-2)(S_ n/(n(n- 2)))^{n/2}\delta_{x_ 0}\) in \({\mathcal D}'(\Omega);\)
(ii) \(\phi '(x_ 0)=0;\)
(iii) \(\lim_{\epsilon \to 0} \epsilon \| u_{\epsilon}\|^ 2_{L^{\infty}(\Omega)}=2\sigma_ n(n(n-2)/S_ n)^{n/2}| \phi (x_ 0)|;\)
(iv) \(\lim_{\epsilon \to 0} u_{\epsilon}(x)/\sqrt{\epsilon}=(n(n- 2)/S_ n)^{n/4}((n-2)G(x,x_ 0)/\sqrt{2| \phi (x_ 0)|})\forall x\neq x_ 0;\)
where \(\sigma_ n\) is the area of the unit sphere in \({\mathbb{R}}^ n\), G(x,y) is the Green function, g(x,y) is the regular part of the Green function, and \(\phi (x)=g(x,x).\)
Similar results for the problem \[ -\Delta u=n(n-2)u^ p+\epsilon u\text{ in } \Omega;\quad u>0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega \] are also proved.
Reviewer: G.Buttazzo (Pisa)

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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