Estimates of the extremal solution for the bilaplacian with general nonlinearity

Let λ⁎>0λ⁎>0 denote the supremum possible value of λ   such that {Δ2u=λf(u)inB1,u=∂u∂n=0on∂B1} has a classical solution, where B1B1 is the unit ball in RNRN, n   is the exterior unit normal vector, and f∈C1(R)f∈C1(R) is nondecreasing and satisfies f(0)>0f(0)>0 and f(t)/t→+∞f(t)/t→+∞ as t→+∞t→+∞. For λ=λ⁎λ=λ⁎ this problem possesses a weak solution u⁎u⁎, the so-called extremal solution. We establish the regularity of this extremal solution for N≤10N≤10. For N≥11N≥11 we establish that limr→0⁡rN−82(u⁎)′(r)=limr→0⁡rN−102u⁎(r)=0 for N≤19N≤19 and limr→0⁡rN−92(u⁎)′(r)=limr→0⁡rN−112u⁎(r)=0 for N≥20N≥20. Our regularity results do not depend on the specific nonlinearity f.