Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of Rn

Abstract

Let D be an unbounded domain in ℝn (n ≥ 2) with a nonempty compact boundary ∂D. We consider the following nonlinear elliptic problem, in the sense of distributions, Δu = ƒ(., u), u > 0 in D, u|∂D = αφ, lim|x|→+∞ u(x)/h(x) = βλ, where α, β, λ are nonnegative constants with α + β > 0 and φ is a nontrivial nonnegative continuous function on ∂D. The function ƒ is nonnegative and satisfies some appropriate conditions related to a Kato class of functions, and <i>h</i> is a fixed harmonic function in D, continuous on ¯D. Our aim is to prove the existence of positive continuous solutions bounded below by a harmonic function. For this aim we use the Schauder fixed-point argument and a potential theory approach.