FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 4, PAGES 133-147

Existence of solutions of certain quasilinear elliptic equations in RN without conditions at infinity

G. I. Laptev

Abstract

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The paper deals with conditions for the existence of solutions of the equations

- å i=1nDiAi(x,u,Du) + A0(x,u) = f(x),    x Î Rn,

considered in the whole space Rn, n ³ 2. The functions Ai(x,u, x), i = 1,...,n, A0(x,u), and f(x) can arbitrarily grow as |x| → ¥. These functions satisfy generalized conditions of the monotone operator theory in the arguments u Î R and x Î Rn. We prove the existence theorem for a solution u Î Wloc1,p(Rn) under the condition p > n.

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