07-42
Supercritical biharmonic equations with power-type nonlinarity
by Ferrero, A.; Grunau, H.-Ch.; Karageorgis, P.
Preprint series: 07-42, Preprints
The paper is published: Ann. Mat. Pura Appl. 188, 171 - 185 (2009).
- MSC:
- 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
- 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
Notes: http://arxiv.org/abs/0711.2202,
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