Qualitative properties of entire radial solutions for a biharmonic equation with supercritical nonlinearity

Abstract:

We study some qualitative properties of entire positive radial solutions of the supercritical semilinear biharmonic equation: \[ \Delta ^2 u=u^p \;\; \text {in $\mathbb {R}^n$}, \;\; n \geq 5, \;\; p>\frac {n+4}{n-4}. \tag {*} \] It is known from a paper by Gazzola and Grunau that there is a critical value $p_c>(n+4)/(n-4)$ of $(*)$ for $n \geq 13$ and that $(*)$ has a singular solution $u_s (r)= K_0^{1/(p-1)} r^{-4/(p-1)}$. We show that for $5 \leq n \leq 12$ or $n \geq 13$ and $p<p_c$, any regular positive radial entire solution $u$ of $(*)$ intersects with $u_s (r)$ infinitely many times. On the other hand, if $n \geq 13$ and $p \geq p_c$, then $u(r)<u_s (r)$ for all $r>0$. Moreover, the solutions are strictly ordered with respect to the initial value $a=u(0)$.