Multiplicity Results for Classes of Infinite Positone Problems

Abstract

We study positive solutions to the singular boundary value problem    −∆ p u = λ f (u) u β in Ω u = 0 on ∂Ω, where ∆ p u = div (|∇u| p−2 ∇u), p > 1, λ > 0, β ∈ (0, 1) and Ω is a bounded domain in R N , N ≥ 1. Here f : [0, ∞) → (0, ∞) is a continuous nondecreasing function such that lim u→∞ f (u) u β+p−1 = 0. We establish the existence of multiple positive solutions for certain range of λ when f satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is f (u) = e αu α+u for α ≫ 1. We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-supersolutions.

A uniqueness result for a singular nonlinear eigenvalue problem

“…Also, this solution belongs to C 2 (Ω) ∩ C 1 (Ω) by regularity results discussed in [4,6]. In [7], under additional hypotheses on the behaviour of f , the authors established the existence of multiple positive solutions for a finite range of λ. In this paper, we study the uniqueness of the positive solution when λ 1.…”

“…Note that lim u→0 f (u)/u β = ∞ and, hence, (P) is a singular boundary-value problem. Since lim u→∞ f (u)/u β+1 = 0, it follows that (P) has a positive solution for all λ > 0 (see [7,9]). Also, this solution belongs to C 2 (Ω) ∩ C 1 (Ω) by regularity results discussed in [4,6].…”

Analysis of positive solutions for classes of quasilinear singular problems on exterior domains

“…In [13], Ko, Lee and Shivaji, using sub-and super solutions, studied the existence of positive solutions of −∆ p u = λ f(u) u δ in Ω, u = on ∂Ω, where λ is a positive parameter and Ω is a bounded domain. Again using sub-and supersolutions, when Ω is a ball in ℝ N and for classes of K(x) = K(|x|), the same authors in [14] analyzed the positive radial solution of (1.1) by reducing it to a two-point boundary value problem via radial and Kelvin transformations.…”

“…Let u and u be a subsolution and a supersolution of (1.1), respectively, such that u ≤ u in Ω e . Then, there exists u ∈ C(Ω e ) ∩ C (Ω e ), with u ≤ u ≤ u in Ω e , satisfying Following the approach used in [13,14], we construct a subsolution u and a supersolution u. Let u > be a solution of (2.1).…”

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

“…We use the method of sub and supersolution combined with a fix point theorem due to Amann to achieve the objective. For the construction of the barrier functions, we have taken some ideas from [28]. The salient feature of this work is the presence of the singular term u −q which is a primary hindrance in making the operator monotone.…”