Proof of two conjectures of H. Brezis and L. A. Peletier (Q920287)

Let \(n\geq 3\) and \(p=(n+2)/(n-2)\) the critical exponent. The author studies positive variational solutions \(u_{\epsilon}\) with zero- Dirichlet data of \[ -\Delta u=n(n-2)u^{p-\epsilon}\quad (resp.\quad =n(n-2)u^ p+\epsilon u) \] on a smoothly bounded domain \(\Omega\) of \({\mathbb{R}}^ n\), if \(\epsilon\) tends to zero (hence \(n\geq 4\) for the second case, see [\textit{H. Brezis}, \textit{L. Nirenberg}, Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)]). If \(u_{\epsilon}\) is a minimizing sequence for the Sobolev inequality \[ \| u_{\epsilon}\|^ 2_{H^ 1_ 0}\| u_{\epsilon}\|^{-2}_{p+1}\to S_ n=\pi (n-2)n(\Gamma (n/2)/\Gamma (n))^{2/n}, \] then he calculates the asymptotic behaviour of \(\| u_{\epsilon}\|_{\infty}\). For decisive parts of the proofs the author refers to his earlier paper [\textit{O. Rey}, J. Funct. Anal. 89, 1-52 (1990)]. The reader should not get confused by some ``critical'' misprints (e.g. missing p in (II), missing \(\epsilon\) in (40)).