Stability and intersection properties of solutions to the nonlinear biharmonic equation

We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation ${\mathrm{\Delta }}^2\varphi ={\varphi }^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions are linearly unstable if $p<p_c$ and linearly stable if $p\ge p_c$. Then, we focus on the supercritical case $p\ge p_c$ and we show that the graphs of no two solutions intersect one another.