Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

Abstract

Let B ⊂ ℝⁿ be the unit ball centered at the origin. The authors consider the following biharmonic equation:

$$\left\{ {\begin{array}{*{20}{c}} {{\Delta ^2}u = \lambda {{\left( {1 + u} \right)}^p}}&{in \mathbb{B},} \\ {u = \frac{{\partial u}}{{\partial \nu }} = 0}&{on\partial \mathbb{B},} \end{array}} \right.$$

where $p > \frac{{n + 4}}{{n - 4}}$ and v is the outward unit normal vector. It is well-known that there exists a λ* > 0 such that the biharmonic equation has a solution for λ ∈ (0, λ*) and has a unique weak solution u* with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies $u* \leqslant {r^{ - \frac{4}{{p - 1}}}} - 1$ on the unit ball, which actually solve a part of the open problem left in [Dàvila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193].