Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball.

We study positive radial solutions to: − Δ u = λ K ( | x | ) f ( u ) ; x ∈ Ω e , where λ > 0 is a parameter, Ω e = { x ∈ R N : | x | > r 0 , r 0 > 0 , N > 2 } , Δ is the Laplacian operator, K ∈ C ( [ r 0 , ∞ ) , ( 0 , ∞ ) ) satisfies K ( r ) ≤ 1 r N + μ ; μ > 0 for r ≫ 1 and f ∈ C 1 ( [ 0 , ∞ ) , R ) is a concave increasing function satisfying lim s → ∞ ⁡ f ( s ) s = 0 and f ( 0 ) < 0 (semipositone). We are interested in solutions u such that u → 0 as | x | → ∞ and satisfy the nonlinear boundary condition ∂ u ∂ η + c ˜ ( u ) u = 0 if | x | = r 0 where ∂ ∂ η is the outward normal derivative and c ˜ ∈ C ( [ 0 , ∞ ) , ( 0 , ∞ ) ) is an increasing function. We will establish the uniqueness of positive radial solutions for large values of the parameter λ . [ABSTRACT FROM AUTHOR]