On the nonlinear product of Laplacian related to the biharmonic equation

Abstract

In this paper, we study the solution of nonlinear equation of the form, where the operator Δk is the Laplacian defined by, n is the dimension of the Euclidean space R{double-struck}n, x = (x1, x2,..., xn) ε R{double-struck}0, k is a positive integer, u(x) is an unknown and f is a given function. It is found that the existence of the solution u(x) of such equation depending on the condition of f and Δk-1(Δ+m2)ku(x) and moreover such solution u(x) related to the Laplacian depending on the conditions of k.