Blowing up Solutions for a Biharmonic Equation with Critical Nonlinearity

Details Blowing up Solutions for a Biharmonic Equation with Critical Nonlinearity Blowing up Solutions for a Biharmonic Equation with Critical Nonlinearity

2004-08-25

arxiv.org/abs/math/0408352v1

Mathematics

Analysis of PDEs

34 pages

Scientific paper

In this paper we consider the following biharmonic equation with critical exponent $P_\epsilon$ : $\Delta^2 u= Ku^{(n+4)/(n-4)-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a domain in $R^n$, $n\geq 5$, $\epsilon$ 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 $P_\epsilon$.

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