Hölder continuity up to the boundary of solutions to nonlinear fourth-order elliptic equations with natural growth terms

Abstract

In a bounded open set Ω ⊂ Rn , n 3 , we consider the nonlinear fourth-order partial differential equation ∑|α|=1,2 (−1)|α|Dα Aα(x,u,Du,Du)+B(x,u,Du,Du) = 0. It is assumed that the principal coefficients {Aα}|α|=1,2 satisfy the growth and coercivity conditions suitable for the energy space W̊ 1,q 2,p (Ω) = W̊ 1,q(Ω)∩W̊ 2,p(Ω) , 1 < p< n/2 , 2p < q < n . The lower-order term B(x,u,Du,D2u) behaves as b(u) {|Du|q + |D2u|p}+g(x) where g ∈ Lτ (Ω) , τ > n/q . We establish the Hölder continuity up to the boundary of any solution u∈ W̊ 1,q 2,p (Ω)∩L∞(Ω) by using the measure density condition on ∂Ω , an interior local result and a modified Moser method with special test function.