On bounded solutions of exterior boundary value problems for linear and quasilinear elliptic differential equations

Abstract

Since a general study of Giraud [6] a number of investigations have been made by various authors concerning the exterior boundary value problems for second order linear elliptic partial differential equations. In this connection we must first of all mention the excellent work by Meyers and Serrin [14] in which several surprizing aspects of the exterior boundary value problems of Dirichlet type as well as of non-Dirichlet type are clarified. We also refer to the recent papers of Oskolkov [18-20] dealing successfully with solutions decaying rapidly at distant points of space. It seems to us, however, that little is known about the solution of non linear exterior boundary value problems. This paper proposes to make a small contribution to this subject. Thus we shall be concerned for the most part with the solvability of the typical exterior boundary value problems for a class of quasilinear elliptic equations of the form