Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of Rn
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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 h 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.
CitationToumi, F., Zeddini, N. (2005). Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of . Electronic Journal of Differential Equations, 2005(143), pp. 1-14.
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