Abstract
We study the existence, multiplicity, and stability of positive solutions to: u 00 (x) = f (u(x)) for x 2 ( 1; 1); > 0; u( 1) = 0 = u(1); where f : (0;1) ! R is semipositone (f(0) 0, we obtain the exact number of positive solutions as a function of f(t)=t and establish how the positive solution curves to the above problem change. Also, we give examples where our results apply. This work extends the work in (1) by giving a complete classication of positive solutions for concave-convex type nonlinearities. (t) f(t) t2 and (tf 0 (t) f(t)) 0 = tf 00 (t) with f(0) 0 for t 2 (0; t1) ( (t2;1) and (f(t)=t) 0 < 0 for t 2 (t1; t2)
Highlights
We study the positive solutions to the two point boundary value problem: (1.1) (1.2)
We first observe that any positive solution of (1.1)-(1.2) must be symmetric about the origin
We prove some elementary properties of positive solutions of (1.1)-(1.2) (and of (2.1)-(2.3) for some ρ > 0)
Summary
We study the positive solutions to the two point boundary value problem:. −u (x) = λf (u(x)) for x ∈ (−1, 1), λ > 0, u(−1) = 0 = u(1), where f : [0, ∞) → R is a twice differentiable function such that:. (1) If f satisfies (1.3)-(1.5) and (1.5), there exists λ∗ with 0 < λ∗ < ∞ such that (1.1)-(1.2) has no positive solutions for λ > λ∗ and has a unique positive solution for λ ∈ (0, λ∗] (see Fig. 1). Ρ ≡ ρλ is a decreasing function of λ with ρλ : (0, λ∗] → [θ, ∞) such that ρλ∗ = θ and lim ρλ = +∞. If λ ∈ (λ2, λ∗] (1.1)-(1.2) has exactly one positive solution (see Fig. 2A). For λ = λ2 the problem (1.1)-(1.2) has exactly one positive solution (see Fig. 2B).
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