Regularity of the solutions for nonlinear biharmonic equations in ℝN

The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation {Δ2u+a(x)u=g(x, u),uɛH2(ℝN),where the condition u ∈ H2(ℝN) plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.