Blowing up solutions for a biharmonic equation with critical nonlinearity

In this paper we consider the following biharmonic equation with critical exponent $\[$(P_{\varepsilon})\dvt\Delta^{2}u=Ku^{\frac{n+4}{n-4}-\varepsilon}$$ , u>0 in Ω and u=Δu=0 on $\[$\curpartial \varOmega $$ , where Ω is a smooth bounded domain in $\[$\mathbb{R}^{n}$$ , n≥5, ε is a small positive parameter, and K is a smooth positive function in $\[$\overline{ \varOmega }$$ . We construct solutions of (P_ε) which blow up and concentrate at strict local maximum of K either at the boundary or in the interior of Ω. We also construct solutions of (P_ε) concentrating at an interior strict local minimum point of K. Finally, we prove a nonexistence result for the corresponding supercritical problem which is in sharp contrast to what happened for (P_ε).