Positive radial solutions to classes of p-Laplacian systems on the exterior of a ball with nonlinear boundary conditions. (January 2022) - Son, Byungjae; Wang, Peiyong
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 .