Positive radial solutions to classes of p-Laplacian systems on the exterior of a ball with nonlinear boundary conditions

Abstract An n × n degenerate elliptic system associated with the p -Laplacian − Δ p i u i = λ K i ( | x | ) f i ( u i + 1 )    in  Ω , κ i ∂ u i ∂ n + g i ( λ , u 1 , u 2 , … , u n ) u i = 0    on  ∂ Ω , u i → 0    as  | x | → ∞ , i = 1 , 2 , … , n is studied in this paper in an exterior domain Ω ≔ { x ∈ R N ∣ | x | > r 0 > 0 } , where 1 p i N , λ > 0 , κ i ≥ 0 , and the functions K i > 0 , g i > 0 , and f i are continuous. This type of nonlinear problems with a nonlinear boundary condition arise in applications such as chemical kinetics and population distribution. Through a fixed-point theorem of the Krasnoselskii type, we prove, in different situations, the existence, multiplicity, or nonexistence of positive radial solutions according to the distinct behaviors of f i near 0 and near ∞ . The new traits of the underlying system include relaxation of the restriction n = 2 to any general n ≥ 2 , different p -Laplacian operators for component solutions, the mixed boundary condition, and possible singularities at 0 possessed by and negativity near 0 of the functions f i .

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