Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

Abstract

We consider the fourth order problem Δ2u = f(u) on a general bounded domain Ωin Rn with the Navier boundary condition u = Δu = 0 on ∂Ω. Here, is a positive parameter and f : [0; af ) → R+ (0 < af ∞) is a smooth, increasing, convex nonlinearity such that f(0) > 0 and which blows up at af . Let 0 < - := lim inf t→af f(t)f0″(t)/ f″(t)2 r+ := lim af f(t)f0(t)f(t)2 2: We show that if um is a sequence of semistable solutions correspond to m satisfy the stability inequality √m /Ω √f″(um)Φ2dx for all Φ ∈ H10 Ω; then supm //um//L∞(Ω) < af for n < 4α(2-r+)+2r+/r+ max{1,r+}, where α is the largest root of the equation (2-r-)2α4 -8(2-r+)α2 + 4(4-3r+)α-4(1-r+) = 0: In particular, if T-= T+:=T, then supm //um//L∞(Ω) < af for n 12 when T 1, and for n 7 when T 1:57863. These estimates lead to the regularity of the corresponding extremal solution u(x) = lim↑u∗(x); where ∗ is the extremal parameter of the eigenvalue problem.