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

Let B ⊂ ℝ be the unit ball centered at the origin. The authors consider the following biharmonic equation(Formula presented.)where p>n+4n−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*≤r−4p−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].