Nonlinear elliptic equations of second order with almost-periodic coefficients

Abstract

Elliptic linear (and more general) operators with almost-periodic coefficients have been investigated in detail in [i, 2]. Certain results about nonlinear equations have also been obtained in [3]. In [4], divergent almost-periodic operators have been investigated from the point of view of the monotonicity method. An existence theorem for classical bounded solutions of almost-periodic nonlinear elliptic equations has been obtained by topological methods in [5]. However, it is not clear whether these solutions have the properties of almost periodicity in some sense. In the present article we construct classical bounded Besicovitch almost-periodic solutions for second-order equations. This is done with the help of the usual method of successive approximations in combination with monotonicity and extension to the Bohr compactification. In this connection, we note the article [6], where monotonic successive approximations also play an important role. Let Cb(~ n) be the space of the bounded continuous functions on ~n, and CAP(~ ~) be the space of the almost-periodic functions (all functions are supposed to be real-valued). The latter space is canonically identified with the space C(N~) of the continuous functions on the Bohr compactification N~ of the additive group Nn [2]. Let us set C~(Nn)_~{u~Cb(N~)laauE Cb ( ~ ) , ~ E ~ } , CAP ~ ( ~ ) ----{u E CAP ( ~ 0 10~u E CAP ( ~ ) , ~ E ~ } = CAP ( ~ ) N C~ (~ ) . Here 0----(O/OXl, , O/dx~), O ~ : Ol~l/Ox7 ~ . . . . . % --0X~ , ~ = ( ~ t . . . . . ~n)~ Z + , and I ~ l = ~ ~ . . . ~ . By d e f i n i t i o n , t h e s p a c e Bo(Nn), l ~ p ~ , o f t h e B e s i c o v i t c h a l m o s t p e r i o d i c f u n c t i o n s i s L~(N~,d~) , w h e r e d~ i s t h e H a a r m e a s u r e on N~. I n t h e s e q u e l , a l l t h e a l m o s t p e r i o d i c f u n c t i o n s a r e a s s u m e d , w i t h o u t a n y r e s e r v a t i o n s , t o b e e x t e n d e d t o N~. Let