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 $$R^N 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

considered in the whole space $$Rⁿ, $n$³ 2. The functions $A$_i(x,u, x), $i\; =\; 1,...,n$, $A$₀(x,u), and $f(x)$ can arbitrarily grow as $|x|\; \to$¥. These functions satisfy generalized conditions of the monotone operator theory in the arguments $u$Î R and $$x Î Rⁿ. We prove the existence theorem for a solution $u$Î W_loc^1,p(Rⁿ) under the condition $p\; >\; n$.

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Last modified: February 17, 2007