Analysis of positive radial solutions for singular superlinear p-Laplacian systems on the exterior of a ball. (March 2020) - Son, Byungjae; Wang, Peiyong
Abstract: In this paper, we study some well-posedness properties of a positive radial solution pair ( u 1, u 2 ) to a singular boundary value problem in the form: − Δ p u i = λ K i ( | x | ) f i ( u j ) in R N ∖ B r 0, a i ∂ u i ∂ n + c ̃ i ( λ, u j, u i ) u i = 0 on ∂ B r 0, u i → 0 as | x | → ∞, where i, j ∈ 1, 2 , i ≠ j, Δ p stands for the singularp -Laplacian Δ p w ≔ div ( | ∇ w | p − 2 ∇ w ) with 1 < p < N, λ > 0, a i ≥ 0, r 0 > 0, K i ∈ C ( r 0, ∞ ), ( 0, ∞ ), and c ̃ i ∈ C ( 0, ∞ ) × [ 0, ∞ ) × [ 0, ∞ ), ( 0, ∞ ) . Here the functions f i ∈ C ( 0, ∞ ), R, i = 1, 2 satisfy a combined p -superlinear growth at ∞ and possibly have singularities at 0. We establish an existence result for the case λ ≈ 0, multiplicity results of positive radial solutions for certain ranges of λ, and nonexistence results of a positive radial solution for the case λ ≫ 1 .