Publication Date:
2014-09-25
Description:
We study the existence of a positive radial solution to the nonlinear eigenvalue problem − Δ u = λ K 1 ( | x | ) f ( v ) in Ω e , − Δ v = λ K 2 ( | x | ) g ( u ) in Ω e , u ( x ) = v ( x ) = 0 if | x | = r 0 (〉0), u ( x ) → 0 , v ( x ) → 0 as | x | → ∞ , where λ 〉 0 is a parameter, Δ u = div ( ∇ u ) is the Laplace operator, Ω e = { x ∈ R n ∣ | x | 〉 r 0 , n 〉 2 } , and K i ∈ C 1 ( [ r 0 , ∞ ) , ( 0 , ∞ ) ) ; i = 1 , 2 are such that K i ( | x | ) → 0 as | x | → ∞ . Here f , g : [ 0 , ∞ ) → R are C 1 functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for λ ≫ 1 for the single equations case. MSC: 34B16, 34B18.
Print ISSN:
1687-2762
Electronic ISSN:
1687-2770
Topics:
Mathematics
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