Blowing up Solutions for a Biharmonic Equation with Critical Nonlinearity

Abstract

In this paper we consider the following biharmonic equation with critical exponent $P_{\epsilon }$ : ${\Delta }^{2}u=K{u}^{(n+4)/(n-4)-\epsilon },u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial \Omega$, where $\Omega$ is a domain in $R^n$, $n\ge 5$, $\epsilon$ is a small positive parameter and $K$ is smooth positive function. We construct solutions of $P_{\epsilon }$ which blow up and concentrate at strict local maximum of $K$ either at the boundary or in the interior of $\Omega$. We also construct solutions of $P_{\epsilon }$ concentrating at an interior strict local minimum of $K$. Finally, we prove a nonexistense result for the corresponding supercritical problem which is in sharp contrast with what happened for $P_{\epsilon }$.

34 pages

References in corpus (1)

• The Paneitz Curvature Problem on Lower Dimensional Spheres