Existence of Positive Solutions to Singular Boundary Value Problems Involving φ-Laplacian

Abstract

This paper is concerned with the existence of positive solutions to singular Dirichlet boundary value problems involving φ -Laplacian. For non-negative nonlinearity f = f ( t , s ) satisfying f ( t , 0 ) ¬ ≡ 0 , the existence of an unbounded solution component is shown. By investigating the shape of the component depending on the behavior of f at ∞ , the existence, nonexistence and multiplicity of positive solutions are studied.

Existence, Nonexistence and Multiplicity of Positive Solutions for Singular Boundary Value Problems Involving φ-Laplacian

“…Let λ ∈ (0, R 2 (m)) and v ∈ ∂K m be fixed. Then f (v(t)) ≤ 1 R2(m) ϕ( m A2 ) for t ∈ [0,1]. By the same argument as in the proof of Lemma 3.2, it follows that H(λ, v) ∞ < v for all v ∈ ∂K m and (3.3) holds for any λ ∈ (0, R 2 (m)).…”

“…We notice that, although σ = σ(g) is not necessarily unique, the right hand side of the equality in (2.1) does not depend on a particular choice of σ. In other words, T (g) is independent of the choice of σ ∈ [σ 1 g , σ 2 g ] (see, e.g., [1] or [2]). For g ∈ H ϕ , consider the following problem…”

“…For convenience, we denote by H ξ the set {g ∈ C((0, 1), R + ) : where ξ : R + → R + is an increasing homeomorphism, and we make the following notations: It is well known that ϕ −1 (x)ψ −1 2 (y) ≤ ϕ −1 (xy) ≤ ϕ −1 (x)ψ −1 1 (y) for all x, y ∈ R + (1. 2) and L 1 (0, 1) ∩ C(0, 1) H ψ1 ⊆ H ϕ ⊆ H ψ2 (see, e.g., [1,Remark 1]).…”

“…Problem (1.1) arises naturally in studying radial solutions to quasilinear elliptic equations defined on an annular domain (see, e.g., [1,2]). For ϕ(s) = |s| p−2 s with p > 1, problem (1.1) has been extensively studied in the literature (see [3,4,5,6,7,8,9] for p = 2 and [10,11,12,13,14,15,16] for p > 1).…”

“…For more general ϕ which does not satisfy (A), when c ≡ d ≡ 1 and 0 ≤ h ∈ L 1 (0, 1) with h ≡ 0, Kaufmann and Milne [18] proved the existence of positive solution to problem (1.1) for all λ > 0 under the assumptions on f which induces the sublinear nonlinearity provided ϕ(s) = |s| p−1 s with p > 1. For other interesting results, we refer the reader to [19,1,20] and the references therein.…”