On S-shaped and reversed S-shaped bifurcation curves for singular problems

We analyze the positive solutions to the singular boundary value problem ( −(|u ′ | p−2 u ′ ) ′ = � g(u) u� ; (0, 1), u(0) = 0 = u(1), where p > 1,� ∈ (0, 1),� > 0 and g : [0, ∞) → R is a C 1 function. In particular, we discuss examples when g(0) > 0 and wheng(0) < 0 that lead to S-shaped and reversed S-shaped bifurcation curves, respectively.

In Case (A) we will prove that for α large, the bifurcation curve of positive solution is at least S-shaped, while in Case (B) for certain ranges of a, b and c, we will prove that the bifurcation curve of positive solution is at least reversed S-shaped.For p = 2 and β = 0, results on S-shaped bifurcation curves have been studied by many authors ([3], [7], [8], [11] and [12]) and results on a reversed Sshaped bifurcation curve have been studied by Castro and Shivaji in ([4]).We will establish the results via the quadrature method which we will describe in Section 2. In Section 3, we will discuss Case (A), and in Section 4 we will discuss Case (B).In Section 5, we provide computational results describing the exact shapes of the two bifurcation curves.

Preliminaries
In this section we give some preliminaries.Let f (u) = g (u)  u β and we rewrite (1.1) as: It follows easily that if u is a strictly positive solution of (2.1), then necessarily u must be symmetric about x = 1 2 , u ′ > 0; (0, 1 2 ) and u ′ < 0; ( 1 2 , 1).To prove our main results, we will first state some lemmas that follow from the quadrature method described in [2] and [10] for the one dimensional p−Laplacian problem for p > 1. See also [3], [4] and [9] for the description of the quadrature method in the case p = 2. Define F : R + → R by F (u) := u 0 f (s) ds and G : D ⊆ R + → R + be defined by where D = {ρ > 0|f (ρ) > 0 and F (ρ) > F (s), ∀ 0 ≤ s < ρ}.
Lemma 2.1.(See [10]) (u, λ) is a positive solution of (2.1) with λ > 0 if and only if λ(ρ) ).Now we also state an important lemma that can be easily deduced from the results in [3] where H(s) = F (s) − 1 p sf (s).We will deduce information on the nature of the bifurcation curve by analyzing the sign of dG(ρ)  dρ .It is clear that dG(ρ) dρ has the same sign as d dρ λ(ρ) a sufficient condition for dG(ρ) dρ to be positive is: and a sufficient condition for dG(ρ) dρ to be negative is: p is a strictly increasing function, i.e. the bifurcation curve is neither S-shaped nor reversed S-shaped.
Hence this will establish that the bifurcation curve is at least S-shaped (see Figure 2).EJQTDE, 2011 No. 31, p. 3 In Section 4, for the Case B, for certain ranges of a, b, c and p we will show that f and F take the following shapes (see Figure 3) and f ′ (s) > 0; s ≥ 0.
Hence this will establish that the bifurcation curve is at least reversed S-shaped (see Figure 5).

Infinite Positone Case A
Here we study the Case A, namely the boundary value problem : where p > 1, α > 0 and 0 < β < 1.We prove: Proof .To prove Theorem 3.1, from our discussion in Section 2 it is enough to show that when α ≫ 1 H has the shape in Figure 1 : namely • there exists ρ 1 > 0 such that H(ρ 1 ) < 0.

Infinite Semipositone Case B
Here we study the Case B, namely the boundary value problem : where p > 1, a, b and c are positive real numbers and 0 < β < 1.We establish: (1) for λ ≤ λ 2 , (4.1) has at least one solution.
Proof.To prove Theorem 4.1 from our discussion in Section 2, it is enough to show that for certain parameter values H has the shape in Figure 4 : namely • H ′ (s) < 0; 0 < s ≤ θ.

Computational Results
Here using Mathematica computations of (2.2), we derive the exact bifurcation curves for the following examples:

Figure 4 :
Figure 4: Function H for the Case B