Solvability in the sense of sequences for some fourth order non-Fredholm operators

Abstract

We study solvability of some linear nonhomogeneous elliptic problems and establish that under reasonable technical conditions the convergence in L2(Rd) of their right sides implies the existence and the convergence in H4(Rd) of the solutions. The problems contain the squares of the sums of second order non-Fredholm differential operators and we use the methods of the spectral and scattering theory for Schrödinger type operators. We especially emphasize that here we deal with the fourth order operators in contrast to the second order operators in [29] and investigate the dependence of the solvability conditions on the dimension of our problem when the constant a=0. We also consider the case of solvability with a single potential in an arbitrary dimension.