COMMUNICATIONS ON doi:10.3934/cpaa.2020131 PURE AND APPLIED ANALYSIS Volume 19, Number 9, Spetember 2020 pp. 4655–4666 EXISTENCE AND NONEXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR A CLASS OF p-LAPLACIAN SUPERLINEAR PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS Trad Alotaibi and D. D. Hai Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762, USA R. Shivaji ∗ Department of Mathematics and Statisitics University of North Carolina at Greensboro Greensboro, NC 27402, USA (Communicated by Wenxiong Chen) Abstract. We prove the existence of positive radial solutions to the problem    −Δpu = λK(|x|)f (u) in |x| >r 0 , ∂u ∂n +˜ c(u)u = 0 on |x| = r 0 , u(x) → 0 as |x|→∞, where Δpu = div(|∇u| p-2 ∇u),N>p> 1, Ω= {x ∈ R N : |x| >r 0 > 0}, f : (0, ∞) → R is p-superlinear at ∞ with possible singularity at 0, and λ is a small positive parameter. A nonexistence result is also established when f has semipositone structure at 0. 1. Introduction. In this paper, we study the existence and nonexistence of posi- tive radial solutions to the problem        −Δ p u = λK(|x|)f (u) in Ω, ∂u ∂n +˜ c(u)u = 0 on |x| = r 0 , u(x) → 0 as |x|→∞, (1.1) where Δ p u = div(|∇u| p−2 ∇u),N>p> 1, Ω= {x ∈ R N : |x| >r 0 > 0}, and K :[r 0 , ∞) → (0, ∞), ˜ c : [0, ∞) → (0, ∞),f : (0, ∞) → R are continuous with r N+μ K(r) bounded for some μ> 0, and λ is a positive param- eter. Here n denotes the outer unit normal vector on ∂ Ω. Let r = |x| and t =(r/r 0 ) p-N p-1 , then problem (1.1) becomes the ODE problem  −(φ(u ′ )) ′ = λh(t)f (u),t ∈ (0, 1), u(0) = 0,u ′ (1) + c(u(1))u(1) = 0, (1.2) 2020 Mathematics Subject Classification. Primary: 35J66, 35J92; Secondary: 35J75. Key words and phrases. p-Laplace, positive radial solutions, nonlinear boundary conditions. * Corresponding author. 4655