The nonlinear product of the Bessel Laplace operator and the Bessel Helmholtz operator

Abstract

In this paper, we study the solution of nonlinear equation ΔkB(ΔB + m2)k u(x) = f(x,Δk-1B (ΔB + m2)ku(x)) where the operator ΔkB is the Bessel Laplace operator iterated k-times defined by ΔkB = (Bx1 + Bx2 + · · · + Bxn)k n is the dimension of the space R+n, x = (x1, x2,..., xn) E R+n, 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-1B (ΔB+m2)ku(x). Moreover such solution u(x) related to the nonhomogeneous Bessel biharmonic equation depend on the conditions of k.