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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References:

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