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Supercritical biharmonic equations with power-type nonlinarity

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

Series: 2007-42, Preprints

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

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 $p\in((n+4)/(n-4),p_c)$.

extremal solution, singular solution, supercritical regime, oscillatory behaviour

This paper was published in:
Ann. Mat. Pura Appl. 188, 171 - 185 (2009).