Supercritical biharmonic equations with power-type nonlinearity

TL;DR

This paper investigates supercritical biharmonic equations with power nonlinearities, analyzing oscillatory behaviors of solutions, existence of singular solutions, and regularity of extremal solutions within certain parameter ranges.

Contribution

It establishes oscillatory behavior of radial solutions near singular solutions and proves existence and regularity results for solutions in supercritical regimes.

Findings

Oscillatory behavior around known singular solutions for certain p.

Existence of at least one singular solution with eigenvalue parameter.

Smoothness of extremal solutions within specific p ranges.

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