This paper is concerned with the equation \(\Delta ^{2}\phi =\phi ^{p}\) on \(\mathbb{R}^{n}\), where \(n>4\) and \(p\geq (n+4)/(n-4)\). In this case it is known that the equation has positive \(C^{4}\) radially symmetric solutions. Let \(\phi _{0}\;\)be one such solution, and let\(\;Q(\alpha )=\alpha (\alpha +2)(\alpha +2-n)(\alpha +4-n)\). It is shown that, if \(pQ(4/(p-1))>Q((n-4)/2)\), then the linearized operator \(\Delta ^{2}-p\phi _{0}^{p-1}\) has a negative eigenvalue. In the complementary case it is shown that \(\Delta ^{2}-p\phi_{0}^{p-1}\) has no negative spectrum, and that the graphs of distinct positive \(C^{4}\) radially symmetric solutions of \(\Delta ^{2}\phi =\phi ^{p}\) are disjoint.