ON SOLVABILITY OF THE BOUNDARY VALUE PROBLEMS FOR THE INHOMOGENEOUS ELLIPTIC EQUATIONS ON NONCOMPACT RIEMANNIAN MANIFOLDS

Abstract

We study questions of existence and belonging to a given functional class of solutions of the inhomogeneous elliptic equations ∆u - c(x)u = g(x), where c(x) >(=)0, g(x) are H¨older fuctions on a noncompact Riemannian manifold M without boundary. In this work we develop an approach to evaluation of solutions to boundary-value problems for linear and quasilinear equations of the elliptic type on arbitrary noncompact Riemannian manifolds. Our technique is essentially based on an approach from the papers by E. A. Mazepa and S. A. Korol’kov connected with an introduction of equivalency classes of functions and representations. We investigate the relationship between the existence of solutions of this equation on M and outside some compact set B ⊂ M with the same growth "at infinity".

Keywords

• Riemannian manifold
• nonhomogeneous elliptic equations
• boundary-value problems