A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball

Analysis of positive solutions for a class of semipositone <i>p</i>-Laplacian problems with nonlinear boundary conditions

“…Next we establish the following result by arguments similar to Lemma 2.8 in [15]. 1 there exist positive constants C 1 and C 2 (independent of λ) such that if u is a positive solution of (1) then…”

“…To achieve the uniqueness result, we first adapt ideas in [15] to derive useful growth estimations, derivative estimations and that u(t) ≥ λ Remark 1. A simple example of (1) satisfying our hypotheses is f (s) = s q * − 1, h(t) = 1 t η and c(s) = e s where 0 < q * < p − 1 and 0 < α + η < 1.…”

“…, there exist unique constants β and θ such that 0 < β < θ, f (β) = 0 and F α (θ) = 0. We first recall the results in Lemmas 2.1 -2.3 from [15] which we restate as Lemmas 3.1 -3.2 below:…”

Existence and Nonexistence of Solutions to p-Laplacian Problems on Unbounded Domains

“…For the last several decades, there has been extensive study of Problem (2) with various assumptions for the domain Ω and the nonlinearity g = g(λ, x, u). For example, for p = 2, Ω = (−1, 1), and g(λ, x, u) = |x| l u p , Tanaka ([1]) showed the existence of one positive even solution and two positive non-even solutions to problem (2) when l(p − 1) ≥ 4 and l ≥ 0. Recently, for p ∈ (1, N), Ω = R N \ B R (0) and g(λ, x, u) = λK(|x|) f (u), Shivaji,Sim,and Son ([2]) proved the uniqueness of positive solution to Problem (2) for large λ under suitable additional assumptions on the reaction term f satisfying f (0) < 0.…”

“…Recently, for p ∈ (1, N), Ω = R N \ B R (0) and g(λ, x, u) = λK(|x|) f (u), Shivaji,Sim,and Son ([2]) proved the uniqueness of positive solution to Problem (2) for large λ under suitable additional assumptions on the reaction term f satisfying f (0) < 0. More recently, for p ∈ (1, N), Ω = R N \ B 1 (0) and g(λ, x, u) = λK(x)|u| p−2 u + h(x), Drábek,Ho,and Sarkar ([3]) investigated the Fredholm alternative for Problem (2) and also discussed the striking difference between the exterior domain and the entire space. For more references, we refer the reader to [4][5][6][7][8][9][10][11][12][13] for bounded domains and to [14][15][16][17][18][19][20][21][22][23] for unbounded domains.…”