Solvability in the Sense of Sequences for Some non Fredholm Operators with the Bi-Laplacian

Abstract

We study solvability of some linear nonhomogeneous elliptic problems and prove that under reasonable technical conditions the convergence in \(L^{2}({\mathbb R}^{d})\) of their right sides implies the existence and the convergence in \(H^{4}({\mathbb R}^{d})\) of the solutions. The equations involve the fourth order non Fredholm differential operators and we use the methods of spectral and scattering theory for Schrödinger type operators similarly to our preceding work (Volpert and Vougalter, Electron. J Differ Equ 2013(160):16pp., 2013)[].