by Ferrero, A., Grunau, H.-Ch., Karageorgis, P..

**Series:** 2007-42, Preprints

- MSC:
- 35J65 Nonlinear boundary value problems for linear elliptic equations
- 35J40 Boundary value problems for higher-order elliptic equations

**Abstract:**

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$,

where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole

space $\mathbb{R}^n$ as well as in a modified form with

$\lambda(1+u)^p$ as right-hand-side with

an additional eigenvalue parameter $\lambda>0$

in the unit ball, in the latter case

together with Dirichlet boundary conditions. As for entire regular

radial solutions we prove oscillatory behaviour around the explicitly known

radial {\it singular} solution, provided $p\in((n+4)/(n-4),p_c)$, where

$p_c\in ((n+4)/(n-4),\infty]$ is a further critical exponent, which was

introduced in a recent work by Gazzola and the second author.

The third author proved already that these oscillations do not occur in

the complementing case, where $p\ge p_c$.

Concerning the Dirichlet problem we prove existence of at least one

singular solution with corresponding eigenvalue parameter. Moreover,

for the extremal solution in the bifurcation diagram for this nonlinear

biharmonic eigenvalue problem, we prove smoothness as long as $p\in((n+4)/(n-4),p_c)$.

**Keywords:**

extremal solution, singular solution, supercritical regime, oscillatory behaviour

**This paper was published in:**

Ann. Mat. Pura Appl. 188, 171 - 185 (2009).