Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

where λ and L are positive parameters, f ∈ C[0,∞)∩ C2(0,∞), and f (u) > 0 for 0 < u < L. We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f ′′(u) > 0 and uf ′(u)≥ f (u) + 2uf ′′(u) for 0 < u < L. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.


Introduction
In this work, we study the bifurcation curves and exact multiplicity of positive solutions of the quasilinear two-point boundary value problem which plays a role in differential geometry and in the theory of relativity.
Recently, both (1.1) and (1.2), or even more general problems, have been widely investigated in order to assure the existence, as well as the multiplicity, of solutions (see, e.g., [1-7, 10, 11] and the references therein). In [3], the existence and multiplicity of positive solutions for the one-dimensional Minkowski-curvature equation have been proved under the assumption that f is an L p -Carathéodory function by using variational or topological methods; here f is not required to be positive. Furthermore, Coelho et al. [4] proved the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball.
Bereanu et al. [1] proved the existence of classical positive radial solutions of (1.2) by employing Leray-Schauder degree arguments, critical point theory, and lower semicontinu- In particular, if f (|x|, u) = μ(|x|)u p , p > 1 and μ : [0, ∞) → R is continuous, strictly positive on (0, ∞), Bereanu et al. [2] obtained that there exists > 0 such that problem (1.2) has zero, at least one, or at least two positive radial solutions according to λ ∈ (0, ), λ = , or λ > . The proof of this result is based on the method of lower and upper solutions and Leray-Schauder degree arguments. By applying the unilateral global bifurcation theory and some preliminary results on the superior limit of a sequence of connected components, Ma et al. [11] and Dai [5] proved the existence, nonexistence, and multiplicity of radial positive solution of problem (1.2) corresponding to asymptotically linear, sublinear, and superlinear nonlinearities f at zero, respectively, which generalized and improved the results in the literature [1,2].
Nevertheless, it is worth noting that the above references mainly studied the existence but not the exactness of positive solutions. Recently, the exact number of the positive solutions have been considered by Zhang and Feng [12] and Huang [8,9]. In [12], the authors obtained the main results as follows: Theorem A Assume that f satisfies the following conditions: (1) f ∈ C([0, ∞), R) and f (u) > 0 for every u > 0; (2) f ∈ C 1 ([0, ∞), R) and f (u)u ≤ f (u) for every u > 0. Theorem B Assume that f (u) = u p (p > 1). Then there exists λ * > 0 such that problem (1.1) has zero, exactly one, or exactly two positive solutions according to λ ∈ (0, λ * ), λ = λ * , or λ > λ * .
We organized the paper as follows. In Sect. 2, we introduce and give some properties of the time map. Section 3 is devoted to proving the main results. Section 4 contains some examples.

Time map
In this section, we shall make a detailed analysis of time maps for the one-dimensional Minkowski-curvature equation (1.1) and give some properties.
Proof Letting u = rτ , we get It is easy to see that T (r) is continuous with respect to r by Lemma 2.1. Combining with (2.2) and (2.3), we obtain ds.

Proof of the main results
Proof of Theorem 1.1 According to the definition of the time map, problem (1.1) is equivalent to finding r ∈ (0, L) such that Therefore, the number of solutions of (1.1) is precisely the number of solutions of (3.1). From Lemma 2.3 and Lemma 2.5, lim r→0 + T(r) = +∞, lim r→0 + T (r) = -∞. By calculation, for a given r 0 > 0, which implies that there exists r 0 ∈ (0, L) such that lim λ→+∞ T(r 0 ) < L. By Lemma 2.6, T(r) has exactly one critical point in the interval r ∈ (0, L) for λ ≥ λ * . Thus, combining with Lemma 2.2, we have obtained a precise description of the graph of T(r) for λ ≥ λ * , see Fig. 2.