Remarks on two fourth order elliptic problems in whole space

Abstract

We are interested in entire solutions for the semilinear biharmonic equation ${\Delta }^{2}u=f(u)$ in ${\mathbb{R}}^N$, where $f(u)=e^u$ or $-u^{-p}(p>0)$. For the exponential case, we prove that any classical entire solution verifies $-\Delta u>0$ without any restriction, which completes the results in \cite{Dupaigne, xu-wei} and yields a nonexistence result in ${\mathbb{R}}^2$ ; we obtain also a refined asymptotic expansion of radial separatrix solution for $N=3$, which answers a question in \cite{Berchio}. For the negative power case, we show the nonexistence of the classical entire solution for any $0<p\le 1$.

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