On a biharmonic equation involving nearly critical exponent

This paper is concerned with a biharmonic equation under the Navier boundary condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${(P_{\mp\varepsilon}): \Delta^{2}u = u^{\frac{n+4}{n-4}\mp\varepsilon}}$$ \end{document} , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{R}^n}$$ \end{document}, n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x0 ∈Ω as ε → 0, moreover x0 is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x0 of the Robin’s function, there exist solutions of (P−ε) concentrating around x0 as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P−ε), the supercritical problem (P+ε) has no solutions which concentrate around a point of Ω as ε → 0.