Bifurcation curve for the Minkowski-curvature equation with concave or geometrically concave nonlinearity

Abstract

We study the bifurcation curve and exact multiplicity of positive solutions in the space $C^{2}\left ( (-L,L)\right ) \cap C\left ( [-L,L]\right )$ for the Minkowski-curvature equation

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda f(u),\text{\ \ }-L< x< L, \\ u(-L)=u(L)=0.\end{array}\displaystyle \right . $$

where $\lambda >0$ is a bifurcation parameter, $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies $f(u)>0$ for $u>0$ and f is either concave or geometrically concave on $(0,\infty )$. If f is a concave function, we prove that the bifurcation curve is monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane. If f is a geometrically concave function, we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane under a mild condition. Some interesting applications are given.

1 Introduction

In this paper, we study the bifurcation curve and exact multiplicity of positive solutions in the space $C^{2}\left ( (-L,L)\right ) \cap C\left ( [-L,L]\right )$ for the Minkowski-curvature equation

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda f(u),\text{\ \ }-L< x< L, \\ u(-L)=u(L)=0.\end{array}\displaystyle \right . $$

(1)

where $\lambda >0$ is a bifurcation parameter. We assume that nonlinearity $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies the basic hypothesis (H1):

• (H1) $f(0)\geq 0$ and $f(u)>0$ for $u>0$.

(1) is the one-dimensional version of the Dirichlet problem associated with the mean curvature equation in Lorentz–Minkowski space (also known as Minkowski space)

$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div}\left ( \dfrac{\nabla u}{\sqrt{1-\left \vert \nabla u\right \vert ^{2}}} \right ) =\lambda f(x,u)\text{\ \ in }\Omega , \\ u|_{\partial \Omega }=0,\end{array}\displaystyle \right . $$

(2)

where Ω is a smooth open bounded set in $\mathbb{R} ^{N}$ with boundary ∂Ω of class $C^{2}$, and $f:\Omega \times \mathbb{R} \rightarrow \mathbb{R}$ is a Carathéodory function. The initial finding related to the Dirichlet problem linked with (2) was made by Flaherty [11], inspired by the foundational work of Cheng and Yau [9]. The mean curvature equation in Lorentz–Minkowski space is an important geometric partial differential equation that describes the mean curvature of a hypersurface in this space. It is noted that a spacelike hypersurface M in Minkowski space $L^{N+1}$, represented by the graph of the function u is an extremum of the Minkowski N-measure

$$ A(M)=\int _{\Omega }\sqrt{1-\left \vert \nabla u\right \vert ^{2}} $$

with a prescribed boundary data, if and only if $\operatorname{div}\left ( \dfrac{\nabla u}{\sqrt{1-\left \vert \nabla u\right \vert ^{2}}}\right ) =0$.

The Minkowski-curvature equation has also been applied to investigate the dynamic aspects of general relativity and the structure of singularities within the solution space of Einstein’s equations, see, e.g., [1, 2, 21] and references therein. Moreover, it also appears in the nonlinear theory of electromagnetism, where it is typically referred to as the Born-Infeld operator, see, e.g., [3–5] and references therein.

In recent years, the ability to solve associated boundary value problems (2) has garnered significant interest in both the ODE and PDE contexts, see, e.g., [6, 10, 13, 19, 20] and references therein. However, it is important to note that the aforementioned references primarily focused on the existence of positive solutions, rather than their exactness. Recently, the exact number of positive solutions has been explored in several studies, see, e.g., Gao and Xu [12], Huang [14, 15], Zhang and Feng [23]. Problem (1) with general nonlinearity $f(u)$ or with many different types nonlinearities, like $u^{p}$ ($p>0$), $u^{p}+u^{q}$ ($0< p\leq q<\infty$), $u^{p}-u^{q}$ ($p,q>0$ and $p\neq q$), $\exp (u)$ and $\exp \left ( \frac{au}{a+u}\right )$ ($a>0$) have been investigated intensively.

In this paper, we mainly consider f in (1) is either concave or geometrically concave on $(0,\infty )$. First, we recall the following definitions and give a proposition. Proposition 1(ii) follows from [16, Proposition 2.3].

Definition 1

Let $I\subset (0,\infty )$ be an interval.

• (i) A function $f:I\rightarrow \mathbb{R}$ is called a concave function on I if

  $$ f(\frac{a+b}{2})\geq \frac{f(a)+f(b)}{2} $$

  for all $a,b\in I$.

• (ii) A function $f:I\rightarrow (0,\infty )$ is called a geometrically concave function on I if

  $$ f(\sqrt{ab})\geq \sqrt{f(a)f(b)} $$

  for all $a,b\in I$.

Proposition 1

Assume $f\in C[0,\infty )\cap C^{2}(0,\infty )$. Then

• (i) f is a concave function on $(0,\infty )$ if and only if $f^{\prime \prime }(u)\leq 0$ on $(0,\infty )$.

• (ii) f is a geometrically concave function on $(0,\infty )$ if and only if $\left [ \frac{uf^{\prime }(u)}{f(u)}\right ] ^{\prime }\leq 0$ on $(0,\infty )$.

We define the bifurcation diagram $C_{L}$ of (1) by

$$\begin{aligned} C_{L}&\equiv \big\{ (\lambda ,\left \Vert u_{\lambda }\right \Vert _{ \infty }):\lambda >0\text{ and }u_{\lambda }\in C^{2}\left ( (-L,L) \right ) \cap C\left ( [-L,L]\right )\\ &\quad \text{is a positive solution of (1)}\big\} . \end{aligned}$$

If $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies (H1) and f is concave on $(0,\infty )$, we prove that the bifurcation curve $C_{L}$ is monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane in Theorem 4 stated below.

As far as we are aware, there is an absence of literature on investigating the precise number of positive solutions for the Minkowski-curvature equation (1) when f is a geometrically concave function. Consequently, the examination of the exact multiplicity of positive solutions for this problem is both intriguing and vital.

When f is a geometrically concave function, we set this condition as hypothesis (H2):

• (H2) f is a geometrically concave function on $(0,\infty )$; that is, $\left [ \frac{uf^{\prime }(u)}{f(u)}\right ] ^{\prime }\leq 0$ on $(0,\infty )$.

We also need a mild hypothesis (H3):

• (H3) $\lim _{u\rightarrow \infty }\frac{uf^{\prime }(u)}{f(u)}\geq \frac{-1}{3}$.

If $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies (H1)–(H3), we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane in Theorem 5 stated below. We say the bifurcation diagram $C_{L}$ is ⊂-shaped on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane if there exists $\lambda ^{\ast }>0$ such that $C_{L}$ consists of a continuous curve with exactly one turning point at some point $(\lambda ^{\ast },\left \Vert u_{\lambda ^{\ast }}\right \Vert _{ \infty })$, where the bifurcation curve $C_{L}$ turns to the right.

When f is a concave function on $(0,\infty )$, the most basic and important examples are $f(u)=u^{p}$ ($0< p<1$) and $f(u)=u^{p}+u^{q}$ ($0< p< q\leq 1$), Zhang and Feng [23] proved that (1) has exactly one positive solution for all $\lambda >0$. See also Huang [14]. In this paper we extend the result to

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda \left ( \sum \limits _{i=1}^{n}a_{i}u^{p_{i}} \right ) \text{,}\; \;-L< x< L\text{, \ }u(-L)=u(L)=0\text{,} \\ 0\leq p_{1}< p_{2}< \cdots < p_{n}\leq 1\text{, }n\geq 2\text{, \ }a_{i}>0 \text{ \ for }i=1,2,\ldots ,n\text{.}\end{array}\displaystyle \right . $$

(3)

We prove that the bifurcation curve $C_{L}$ is monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane. See Theorem 6 stated below and Fig. 1(i) depicted below for the details.

Figure 1

Another common concave nonlinearity is arctanu, which often appears in boundary value problems of differential equations, see, e.g., [7] and [8]. In this paper, we consider

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda \arctan u,\text{\ \ }-L< x< L, \\ u(-L)=u(L)=0.\end{array}\displaystyle \right . $$

(4)

We prove that the bifurcation curve $C_{L}$ is monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane. See Theorem 7 stated below and Fig. 1(ii), depicted below, for the details.

In [18], Hung and Wang investigated the classification and evolution of bifurcation curves of positive solutions for the m-Laplacian Dirichlet problem

$$ \left \{ \textstyle\begin{array}{l} \left ( \varphi _{m}(u^{\prime }(x))\right ) ^{\prime }+\lambda \frac{u^{p}}{1+u^{q}}=0\text{,}\; \;-1< x< 1\text{,} \\ u(-1)=u(1)=0\text{,}\; \;p,q>0\text{.}\end{array}\displaystyle \right . $$

(5)

where $m>1$, $\varphi _{m}(y)=\left \vert y\right \vert ^{m-2}y$, $(\varphi _{m}(u^{\prime }))^{\prime }$ is the one-dimensional m-Laplacian, and $\lambda >0$ is a bifurcation parameter and $p>0$ is an evolution parameter. They gave a classification of a total of five qualitatively different bifurcation curves of (5) in the following theorem.

Theorem 2

([18, Theorem 1.1])

Let $m>1$. Consider (5). Then

• (i) (See Fig. 1(i)) If $p\in \left ( 0,m-1\right )$, then (5) has exactly one positive solution $u_{\lambda }$ for all $\lambda >0$. Moreover, $\lim _{\lambda \rightarrow 0^{+}}\left \Vert u_{\lambda }\right \Vert _{\infty }=0$ and $\lim _{\lambda \rightarrow \infty }\left \Vert u_{\lambda }\right \Vert _{\infty }=\infty$.

• (ii) (See Fig. 1(ii)) If $p=m-1$, then there exists $\hat{\lambda}>0$ such that (5) has exactly one positive solution $u_{\lambda }$ for $\lambda >\hat{\lambda}$, and no positive solution for $0<\lambda \leq \hat{\lambda}$. Moreover, $\lim _{\lambda \rightarrow \hat{\lambda}^{+}}\left \Vert u_{\lambda }\right \Vert _{\infty }=0$ and $\lim _{\lambda \rightarrow \infty }\left \Vert u_{\lambda }\right \Vert _{\infty }=\infty$.

• (iii) (See Fig. 1(iii)) If $p\in \left ( m-1,q+m-1\right )$, then there exists $\lambda ^{\ast }>0$ such that (5) has exactly two positive solutions $u_{\lambda }$, $v_{\lambda }$ with $u_{\lambda }< v_{\lambda }$ for $\lambda >\lambda ^{\ast }$, exactly one positive solution $u_{\lambda ^{\ast }}$ for $\lambda =\lambda ^{\ast }$, and no positive solution for $0<\lambda <\lambda ^{\ast }$. Moreover, $\lim \nolimits _{\lambda \rightarrow \infty }\left \Vert u_{\lambda } \right \Vert _{\infty }=0$ and $\lim \nolimits _{\lambda \rightarrow \infty }\left \Vert v_{\lambda } \right \Vert _{\infty }=\infty$.

• (iv) (See Fig. 1(iv)) If $p=q+m-1$, then there exists $\tilde{\lambda}>0$ such that (5) has exactly one positive solution $u_{\lambda }$ for $\lambda >\tilde{\lambda}$, and no positive solution for $0<\lambda \leq \tilde{\lambda}$. Moreover, $\lim _{\lambda \rightarrow \tilde{\lambda}^{+}}\left \Vert u_{\lambda }\right \Vert _{\infty }=\infty$ and $\lim _{\lambda \rightarrow \infty }\left \Vert u_{\lambda }\right \Vert _{\infty }=0$.

• (v) (See Fig. 1(v)) If $p\in \left ( q+m-1,\infty \right )$, then (5) has exactly one positive solution $u_{\lambda }$ for all $\lambda >0$. Moreover, $\lim _{\lambda \rightarrow 0^{+}}\left \Vert u_{\lambda }\right \Vert _{\infty }=\infty$ and $\lim _{\lambda \rightarrow \infty }\left \Vert u_{\lambda }\right \Vert _{\infty }=0$.

In [6], Boscaggin and Feltrin explored the existence, non-existence, multiplicity of positive periodic solutions for the Minkowski-curvature equation

$$ -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda a(t)\frac{u^{p}}{1+u^{q}} \text{\ \ with }p>1\text{ and }0< q< p, $$

(6)

where $\lambda >0$ is a bifurcation parameter, $a(t)$ is a T-periodic and locally integrable weight function. They proved the following two-solution theorem.

Theorem 3

([6, Theorem 1.1])

Consider (6). Let $a(t)$ be a sign-changing, continuous and T-periodic function, having a finite number of zeros in $[0,T]$, and satisfying

$$ \int _{0}^{T}a(t)dt< 0. $$

Then, there exist $0<\lambda _{\ast }\leq \lambda ^{\ast }<\infty$ such that (6) have no positive T-periodic solution for $0<\lambda <\lambda _{\ast }$, and two positive T-periodic solutions for $\lambda >\lambda ^{\ast }$.

In (6), when $p=3$ and $a(t)=\cos (t-\pi /4)-\sqrt{2}/2$, Boscaggin and Feltrin [6] gave numerical simulations to show that the bifurcation diagram of 2π-periodic solutions is ⊂-shaped on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane.

In this paper, we consider

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda \frac{u^{p}}{1+u^{q}}\text{,}\; \;-L< x< L \text{,} \\ u(-L)=u(L)=0\text{, }0< q\leq p\text{.}\end{array}\displaystyle \right . $$

(7)

In Theorem 8 stated below, we prove that $f(u)=\frac{u^{p}}{1+u^{q}}$ is a geometrically concave function on $(0,\infty )$. Moreover, we obtained results that are somewhat different from those in Theorem 2. That is, we give a classification of totally three qualitatively different bifurcation curves of (7). See Theorem 8 stated below and Fig. 1(i)–(iii) depicted above for the details.

The rest of the paper is organized as follows. Section 2 contains statements of main results. Section 3 contains several lemmas needed to prove the main results. Section 4 contains the proofs of the main results.

2 Main results

The main results in this paper are next Theorems 4 and 5, in which we study the bifurcation curve of positive solutions for Minkowski-curvature equation with concave or geometrically concave nonlinearity. We also give some applications, including $f(u)=\sum \limits _{i=1}^{n}a_{i}u^{p_{i}}$ satisfying (3) (Theorem 6), $f(u)=\arctan u$ (Theorem 7), $f(u)=\frac{u^{p}}{1+u^{q}}$ satisfying (7) (Theorem 8).

Theorem 4

Consider (1). Assume $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies (H1) and f is concave on $(0,\infty )$. Then for any $\lambda ,L>0$, problem (1) has at most one positive solution. Moreover, the following statements (I) and (II) are valid:

• (I) (See Fig. 1(i).) If one of the following conditions holds:

  • (i) $f(0)>0$.

  • (ii) $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\infty$.

  Then the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, (1) has exactly one positive solution for all $\lambda >0$.

• (II) (See Fig. 1(ii).) If $f(0)=0$ and $0<\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}<\infty$, then the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, there exists $\hat{\lambda}>0$ such that (1) has exactly one positive solution for $\lambda >\hat{\lambda}$ and no positive solution for $0<\lambda \leq \hat{\lambda}$.

Remark 1

In Theorem 4, consider $f\in C[0,\infty )\cap C^{2}(0,\infty )$ is concave on $(0,\infty )$, $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=0$, it can be proved that $f(u)<0$ for $u>0$; that is, this case can not occur if f satisfies hypothesis (H1).

Theorem 5

Consider (1). Assume $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies (H1)–(H3). Then for any $\lambda ,L>0$, problem (1) has at most two positive solutions. Moreover, the following statements (I)–(III) are all valid:

• (I) (See Fig. 1(i).) If one of the following conditions holds:

  • (i) $f(0)>0$.

  • (ii) $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\infty$.

  Then the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, (1) has exactly one positive solution for all $\lambda >0$.

• (II) (See Fig. 1(ii).) If $f(0)=0$ and $0<\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}<\infty$, then the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, there exists $\hat{\lambda}>0$ such that (1) has exactly one positive solution for $\lambda >\hat{\lambda}$ and no positive solution for $0<\lambda \leq \hat{\lambda}$.

• (III) (See Fig. 1(iii).) If $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=0$, then the bifurcation curve $C_{L}$ is ⊂-shaped for all $L>0$. More precisely, there exists $\lambda ^{\ast }>0$ such that (1) has exactly two positive solutions for $\lambda >\lambda ^{\ast }$, exactly one positive solution for $\lambda =\lambda ^{\ast }$, and no positive solution for $0<\lambda <\lambda ^{\ast }$.

Remark 2

We conjecture that hypothesis (H3) is not necessary for the results in Theorem 5.

Theorem 6

(See Fig. 1(i))

Consider (3).

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda \left ( \sum \limits _{i=1}^{n}a_{i}u^{p_{i}} \right ) \textit{,}\; \;-L< x< L\textit{, \ }u(-L)=u(L)=0\textit{,} \\ 0\leq p_{1}< p_{2}< \cdots < p_{n}\leq 1\textit{, }n\geq 2\textit{, \ }a_{i}>0 \textit{ \ for }i=1,2,\ldots ,n\textit{.}\end{array}\displaystyle \right . $$

Then $f(u)=\sum \limits _{i=1}^{n}a_{i}u^{p_{i}}$ is a concave function on $(0,\infty )$. Moreover, the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, (3) has exactly one positive solution for all $\lambda >0$.

Theorem 7

(See Fig. 1(ii))

Consider (4).

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda \arctan u,\textit{\ \ }-L< x< L, \\ u(-L)=u(L)=0.\end{array}\displaystyle \right . $$

Then $f(u)=\arctan u$ is a concave function on $(0,\infty )$. Moreover, the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, there exists $\hat{\lambda}>0$ such that (4) has exactly one positive solution for $\lambda >\hat{\lambda}$ and no positive solution for $0<\lambda \leq \hat{\lambda}$.

Theorem 8

Consider (7).

$$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda \frac{u^{p}}{1+u^{q}}\textit{,}\; \;-L< x< L \textit{,} \\ u(-L)=u(L)=0\textit{, }0< q\leq p\textit{.}\end{array}\displaystyle \right . $$

Then $f(u)=\frac{u^{p}}{1+u^{q}}$ is a geometrically concave function on $(0,\infty )$. Moreover, the following statements (i)–(iii) are all valid:

• (i) (See Fig. 1(i)) If $0< q\leq p<1$, then the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, (7) has exactly one positive solution for all $\lambda >0$.

• (ii) (See Fig. 1(ii)) If $0< q\leq p=1$, then the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, there exists $\hat{\lambda}>0$ such that (7) has exactly one positive solution for $\lambda >\hat{\lambda}$ and no positive solution for $0<\lambda \leq \hat{\lambda}$.

• (iii) (See Fig. 1(iii)) If $p>1$ and $0< q\leq p$, then the bifurcation curve $C_{L}$ is ⊂-shaped for all $L>0$. More precisely, there exists $\lambda ^{\ast }>0$ such that (7) has exactly two positive solutions for $\lambda >\lambda ^{\ast }$, exactly one positive solution for $\lambda =\lambda ^{\ast }$, and no positive solution for $0<\lambda <\lambda ^{\ast }$.

Remark 3

If hypothesis (H3) can be removed in Theorem 5, then there is no need to restrict $q\leq p$ in (7). Moreover, the following statements (i)–(iii) are valid:

• (i) If $0< p<1$, then all results in Theorem 8(i) hold.

• (ii) If $p=1$, then all results in Theorem 8(ii) hold.

• (iii) If $p>1$, then all results in Theorem 8(iii) hold.

Further investigations are needed.

3 Lemmas

In this section, in the next Lemmas 9–14, we develop new time-map techniques to prove our main results. In particular, Lemma 11 is a key lemma in the proof of Theorem 5. Assume $f\in C[0,\infty )$ satisfies (H1), then the time map formula which we apply to study the Minkowski-curvature problem (1) takes the form as follows:

$$ T_{\lambda }(r)=\int _{0}^{r} \frac{1+\lambda F(r)-\lambda F(u)}{\sqrt{\left [ 1+\lambda F(r)-\lambda F(u)\right ] ^{2}-1}}du,\quad r\in (0,\infty ), $$

(8)

where $F(u)\equiv \int \nolimits _{0}^{u}f(t)dt$. See [14, (4.1)].

Observe that positive solutions $u_{\lambda }$ for (1) correspond to

$$ ||u_{\lambda }||_{\infty }=r\text{ \ and \ }T_{\lambda }(r)=L\text{.} $$

(9)

See, e.g., [12] and [23] for the derivation of (9). Thus, studying of the exact number of positive solutions of (1) for any fixed $\lambda >0$ is equivalent to studying the shape of the time map $T_{\lambda }(r)$ on $(0,\infty )$. Note that it can be proved that $T_{\lambda }(r)\in C(0,\infty )$ (resp. $C^{1}(0,\infty )$, $C^{2}(0,\infty )$) if $f\in C(0,\infty )$ (resp. $C^{1}(0,\infty )$, $C^{2}(0,\infty )$), see [22, Lemmas 3.2 and 3.6]. The proof can be completed by using the continuity rule and the differentiation rule for parameter-dependent improper integrals.

First, we determine the limits of $T_{\lambda }(r)$ on $(0,\infty )$ in next lemma which follows from [15, Lemma 3.1] and [23, Lemmas 2.3 and 2.4].

Lemma 9

Consider (8). Assume $f\in C[0,\infty )$ satisfies (H1), then the following assertions (i)–(iii) hold:

• (i) If $f(0)>0$, then $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)=0$ for any fixed $\lambda >0$.

• (ii) If $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=A\in \lbrack 0,\infty ]$,

  • (a) If $A=\infty$, then $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)=0$ for any fixed $\lambda >0$.

  • (b) If $A\in (0,\infty )$, then $f^{\prime }(0)=A$ and $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)= \frac{\pi }{2\sqrt{\lambda A}}$ for any fixed $\lambda >0$.

  • (c) If $A=0$, then $f^{\prime }(0)=0$ and $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)=\infty$ for any fixed $\lambda >0$.

• (iii) $\lim _{r\rightarrow \infty }T_{\lambda }(r)=\infty$ for any fixed $\lambda >0$.

By (8), we compute

$$\begin{aligned} \begin{aligned} & T_{\lambda }^{\prime }(r)\\ & \quad =\frac{\lambda }{r}\int _{0}^{r}\frac{\left [ F(r)-F(u)\right ] \left [ 1+\lambda F(r)-\lambda F(u)\right ] \left [ 2+\lambda F(r)-\lambda F(u)\right ] -\left [ rf(r)-uf(u)\right ] }{\left [ \left [ 1+\lambda F(r)-\lambda F(u)\right ] ^{2}-1\right ] ^{3/2}}du, \end{aligned} \end{aligned}$$

(10)

and

$$\begin{aligned} \begin{aligned} &T_{\lambda }^{\prime \prime }(r) \\ ={}&\frac{\lambda ^{2}}{r^{2}}\int _{0}^{r} \frac{3\left [ 1+\lambda F(r)-\lambda F(u)\right ] \left [ rf(r)-uf(u)\right ] ^{2}}{\left [ \left [ 1+\lambda F(r)-\lambda F(u)\right ] ^{2}-1\right ] ^{5/2}}du \\ &-\frac{\lambda ^{2}}{r^{2}}\int _{0}^{r} \frac{\left [ F(r)-F(u)\right ] \left [ 2+\lambda F(r)-\lambda F(u)\right ] \left \{ 2\left [ rf(r)-uf(u)\right ] +\left [ r^{2}f^{\prime }(r)-u^{2}f^{\prime }(u)\right ] \right \} }{\left [ \left [ 1+\lambda F(r)-\lambda F(u)\right ] ^{2}-1\right ] ^{5/2}}du. \end{aligned} \end{aligned}$$

(11)

The following lemma give a condition for monotonicity of $T_{\lambda }(r)$ on $(0,\infty )$, which follows from [23, Lemmas 2.6].

Lemma 10

Consider (8). Assume $f\in C[0,\infty )$ satisfies (H1) and

$$ f(u)-uf^{\prime }(u)\geq 0\textit{ \ for }u>0\textit{,} $$

then $T_{\lambda }^{\prime }(r)>0$ on $(0,\infty )$ for any fixed $\lambda >0$.

We set a key function

$$ K_{f}(u)\equiv \frac{uf^{\prime }(u)}{f(u)}\text{ \ for }u\in (0, \infty ). $$

By Proposition 1, $f\in C[0,\infty )\cap C^{2}(0,\infty )$ is a geometrically concave function on $(0,\infty )$ if and only if $K_{f}(u)$ is decreasing on $(0,\infty )$.

Lemma 11

Consider (8). Assume $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies (H1)–(H3). Then

$$ T_{\lambda }^{\prime \prime }(r)+\frac{K_{f}(r)+1}{2r}T_{\lambda }^{ \prime }(r)>0\textit{ \ for }r\in (0,\infty ). $$

(12)

Moreover, $T_{\lambda }(r)$ has at most one critical point, a local minimum, on $(0,\infty )$ for any fixed $\lambda >0$.

To prove Lemma 11, we need next technical Lemmas 12 and 13. For any $0\leq u< r<\infty$, let

$$\begin{aligned}& \Delta F=F(r)-F(u), \end{aligned}$$

(13)

$$\begin{aligned}& \Delta \tilde{f}=rf(r)-uf(u), \end{aligned}$$

(14)

$$\begin{aligned}& \Delta \hat{f}^{\prime }=r^{2}f^{\prime }(r)-u^{2}f^{\prime }(u). \end{aligned}$$

(15)

Lemma 12

Suppose that $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies (H1) and (H2), then

$$ \inf _{0\leq u< r}\dfrac{\Delta \tilde{f}}{\Delta F}=K_{f}(r)+1 \textit{ \ for any }r\in (0,\infty ), $$

where ΔF and Δf̃ are defined in (13) and (14).

Proof of Lemma 12.

For any fixed $r\in (0,\infty )$, by applying Cauchy’s mean value theorem and (H2), there exists $r_{0}\in (0,r)$ such that

$$ \left . \dfrac{\Delta \tilde{f}}{\Delta F}\right \vert _{u=0}= \frac{rf(r)-0f(0)}{F(r)-F(0)}= \frac{f(r_{0})+r_{0}f^{\prime }(r_{0})}{f(r_{0})}=K_{f}(r_{0})+1\geq K_{f}(r)+1. $$

On the other hand,

$$ \lim _{u\rightarrow r^{-}}\dfrac{\Delta \tilde{f}}{\Delta F}=\lim _{u\rightarrow r^{-}}\frac{rf(r)-uf(u)}{F(r)-F(u)}= \frac{f(r)+rf^{\prime }(r)}{f(r)}=K_{f}(r)+1 $$

by L’Hôpital’s rule. Thus, we have

$$ \inf _{0\leq u< r}\dfrac{\Delta \tilde{f}}{\Delta F}\leq K_{f}(r)+1. $$

Suppose that $\inf _{0\leq u< r}\dfrac{\Delta \tilde{f}}{\Delta F}< K_{f}(r)+1$, then the minimum of $\Delta \tilde{f}/\Delta F$ occurs at an interior point $u_{0}$ on $(0,r)$, hence $\left . \left ( \Delta \tilde{f}/\Delta F\right ) ^{\prime }\right \vert _{u=u_{0}}=0$, which implies that

$$ f(u_{0})\left [ rf(r)-u_{0}f(u_{0})\right ] -\left [ F(r)-F(u_{0}) \right ] \left [ f(u_{0})+u_{0}f^{\prime }(u_{0})\right ] =0. $$

So

$$ \left . \dfrac{\Delta \tilde{f}}{\Delta F}\right \vert _{u=u_{0}}= \frac{rf(r)-u_{0}f(u_{0})}{F(r)-F(u_{0})}= \frac{f(u_{0})+u_{0}f^{\prime }(u_{0})}{f(u_{0})}=K_{f}(u_{0})+1. $$

Thus

$$ \inf _{0\leq u< r}\dfrac{\Delta \tilde{f}}{\Delta F}=\min _{0\leq u< r} \dfrac{\Delta \tilde{f}}{\Delta F}=\left . \dfrac{\Delta \tilde{f}}{\Delta F}\right \vert _{u=u_{0}}=K_{f}(u_{0})+1 \geq K_{f}(r)+1 $$

by (H2). This contradicts our assumption that $\inf _{0\leq u< r}\dfrac{\Delta \tilde{f}}{\Delta F}< K_{f}(r)+1$.

Therefore, for any $r\in (0,\infty )$, we obtain that

$$ \inf _{0\leq u< r}\dfrac{\Delta \tilde{f}}{\Delta F}=K_{f}(r)+1. $$

The proof of Lemma 12 is complete. □

Lemma 13

Suppose that $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies (H1)–(H3), then

$$ \max _{0\leq u< r}\dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}=K_{f}(r)\textit{ \ for any }r\in (0,\infty ), $$

where Δf̃ and $\Delta \hat{f}^{\prime }$ are defined in (14) and (15).

Proof of Lemma 13.

By (H1)–(H3), $\left [ rf(r)\right ] ^{\prime }=f(r)+rf^{\prime }(r)=f(r)\left [ 1+ \frac{rf^{\prime }(r)}{f(r)}\right ] >0$ on $(0,\infty )$. So $rf(r)-uf(u)>0$ for $0\leq u< r<\infty$, that is,

$$ \Delta \tilde{f}>0\text{ \ for\ }0\leq u< r< \infty . $$

(16)

For any fixed $r\in (0,\infty )$ and $0< u< r$, by (16) and (H2), it can be computed that

$$\begin{aligned} \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}-\left ( \left . \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}\right \vert _{u=0}\right ) =&\frac{r^{2}f^{\prime }(r)-u^{2}f^{\prime }(u)}{rf(r)-uf(u)}- \frac{r^{2}f^{\prime }(r)}{rf(r)} \\ =&\frac{uf(u)\left [ K_{f}(r)-K_{f}(u)\right ] }{rf(r)-uf(u)}\leq 0. \end{aligned}$$

(17)

In addition, by L’Hôpital’s rule,

$$ \lim _{u\rightarrow r^{-}} \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}=\lim _{u\rightarrow r^{-}} \frac{r^{2}f^{\prime }(r)-u^{2}f^{\prime }(u)}{rf(r)-uf(u)}= \frac{2rf^{\prime }(r)+r^{2}f^{\prime \prime }(r)}{f(r)+rf^{\prime }(r)}. $$

By (H2), it can be computed that

$$\begin{aligned} \lim _{u\rightarrow r^{-}} \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}-\left ( \left . \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}} \right \vert _{u=0}\right ) =& \frac{2rf^{\prime }(r)+r^{2}f^{\prime \prime }(r)}{f(r)+rf^{\prime }(r)}-\frac{r^{2}f^{\prime }(r)}{rf(r)} \\ =&\frac{rf(r)}{f(r)+rf^{\prime }(r)}K_{f}^{\prime }(r)\leq 0. \end{aligned}$$

(18)

Therefore, for any $r\in (0,\infty )$, the maximum of $\Delta \hat{f}^{\prime }/\Delta \tilde{f}$ occurs at $u=0$ by (17) and (18). Moreover,

$$ \max _{0\leq u< r}\dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}= \left . \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}\right \vert _{u=0}=\frac{rf^{\prime }(r)}{f(r)}=K_{f}(r). $$

The proof of Lemma 13 is complete. □

We are now in a position to prove Lemma 11.

Proof of Lemma 11.

By (13)–(15), (10) and (11) can be written as

$$\begin{aligned}& T_{\lambda }^{\prime }(r)=\frac{\lambda }{r}\int _{0}^{r} \frac{\left ( \Delta F\right ) \left ( 1+\lambda \Delta F\right ) \left ( 2+\lambda \Delta F\right ) -\Delta \tilde{f}}{\left [ \left ( \lambda \Delta F\right ) \left ( 2+\lambda \Delta F\right ) \right ] ^{3/2}}du \text{.}\\& T_{\lambda }^{\prime \prime }(r)=\frac{\lambda ^{2}}{r^{2}}\int _{0}^{r} \frac{3\left ( 1+\lambda \Delta F\right ) \left ( \Delta \tilde{f}\right ) ^{2}-\left ( \Delta F\right ) \left ( 2+\lambda \Delta F\right ) \left [ 2\left ( \Delta \tilde{f}\right ) +\left ( \Delta \hat{f}^{\prime }\right ) \right ] }{\left [ \left ( \lambda \Delta F\right ) \left ( 2+\lambda \Delta F\right ) \right ] ^{5/2}}du\text{.} \end{aligned}$$

For any fixed $r\in (0,\infty )$, let $M=M(r)\equiv K_{f}(r)+1$, we compute that

$$ T_{\lambda }^{\prime \prime }(r)+\frac{M}{2r}T_{\lambda }^{\prime }(r)= \frac{\lambda ^{2}}{2r^{2}}\int _{0}^{r} \frac{Q_{1}+Q_{2}}{\left [ \left ( \lambda \Delta F\right ) \left ( 2+\lambda \Delta F\right ) \right ] ^{5/2}}du \text{,} $$

(19)

where

$$ Q_{1}=\left ( 2+\lambda \Delta F\right ) \left [ 3\left ( \Delta \tilde{f}\right ) ^{2}-\left ( 4+M\right ) \left ( \Delta F\right ) \left ( \Delta \tilde{f}\right ) -2\left ( \Delta F\right ) \left ( \Delta \hat{f}^{\prime } \right ) +2M\left ( \Delta F\right ) ^{2}\right ] , $$

(20)

and

$$ Q_{2}=3(\lambda \Delta F)(\Delta \tilde{f})^{2}+\lambda M(\Delta F)^{3} \left ( 2+\lambda \Delta F\right ) \left ( 3+\lambda \Delta F\right ) \text{.} $$

We claim that $Q_{1}\geq 0$, $Q_{2}>0$ for any $r\in (0,\infty )$, $0\leq r< u$.

For any $0\leq u< r<\infty$, let $\rho =\rho (r,u)\equiv \Delta \tilde{f}/\Delta F$. By (H2), (H3), Lemmas 12 and 13,

$$ \rho =\frac{\Delta \tilde{f}}{\Delta F}\geq \inf _{0\leq u< r} \dfrac{\Delta \tilde{f}}{\Delta F}=K_{f}(r)+1\geq \frac{2}{3}. $$

(21)

Moreover,

$$\begin{aligned}& \rho \geq K_{f}(r)+1=M, \end{aligned}$$

(22)

$$\begin{aligned}& \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}\leq \max _{0\leq u< r}\dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}=K_{f}(r)=M-1. \end{aligned}$$

(23)

Now by (20)–(23), we obtain that

$$\begin{aligned} Q_{1} =&\left ( 2+\lambda \Delta F\right ) \left ( \Delta F\right ) ^{2} \left [ 3\left ( \dfrac{\Delta \tilde{f}}{\Delta F}\right ) ^{2}- \left ( 4+M\right ) \left ( \dfrac{\Delta \tilde{f}}{\Delta F}\right ) -2\left ( \dfrac{\Delta \hat{f}^{\prime }}{\Delta \tilde{f}}\right ) \left ( \dfrac{\Delta \tilde{f}}{\Delta F}\right ) +2M\right ] \\ \geq &\left ( 2+\lambda \Delta F\right ) \left ( \Delta F\right ) ^{2} \left [ 3\rho ^{2}-\left ( 4+M\right ) \rho -2\left ( M-1\right ) \rho +2M\right ] \\ =&\left ( 2+\lambda \Delta F\right ) \left ( \Delta F\right ) ^{2} \left ( 3\rho -2\right ) \left ( \rho -M\right ) \\ \geq &0\text{ \ for }0\leq u< r< \infty . \end{aligned}$$

(24)

On the other hand, since $0<\Delta F<1$ and $M=K_{f}(r)+1>0$, we have

$$ Q_{2}=3(\lambda \Delta F)(\Delta \tilde{f})^{2}+\lambda M(\Delta F)^{3} \left ( 2+\lambda \Delta F\right ) \left ( 3+\lambda \Delta F\right ) >0\text{ \ for }0\leq u< r< \infty . $$

(25)

So by (19), (24) and (25), we obtain that

$$ T_{\lambda }^{\prime \prime }(r)+\frac{K_{f}(r)+1}{2r}T_{\lambda }^{ \prime }(r)=\frac{\lambda ^{2}}{2r^{2}}\int _{0}^{r} \frac{Q_{1}+Q_{2}}{\left [ \left ( \lambda \Delta F\right ) \left ( 2+\lambda \Delta F\right ) \right ] ^{5/2}}du>0\text{ \ for }r\in (0,\infty )\text{.} $$

Thus, if $r^{\ast }$ is a critical point of $T_{\lambda }(r)$ on $(0,\infty )$, then $T_{\lambda }^{\prime \prime }(r^{\ast })>0$ and hence $T_{\lambda }(r^{\ast })$ is a local minimum.

From the above analyzes, we conclude that $T_{\lambda }(r)$ has at most one critical point, a local minimum, on $(0,\infty )$.

This completes the proof of Lemma 11. □

In the following Lemma 14, we study the behaviors of $T_{\lambda }(r)$ as λ varies. Lemma 14 mainly follows from [14, Lemma 4.2 and (4.19)] and [22, Lemmas 3.2 and 3.7], and hence we omit the proof.

Lemma 14

Consider (8). Assume $f\in C[0,\infty )$ satisfies (H1). For any fixed $r>0$, $T_{\lambda }(r)$ is a continuous, strictly decreasing function of λ on $(0,\infty )$. Moreover,

$$ \lim _{\lambda \rightarrow 0^{+}}T_{\lambda }(r)=\infty \ \ \textit{and \ }\lim _{\lambda \rightarrow \infty }T_{\lambda }(r)=r. $$

4 Proof of main results

Proof of Theorem 4.

• (I) First, we study the shape and asymptotic behaviors of $T_{\lambda }(r)$. Assume one of the following conditions holds:

  • (i) $f(0)>0$.

  • (ii) $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\infty$.

  By Lemma 9, we obtain:

  (1) $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)=0$ and $\lim _{r\rightarrow \infty }T_{\lambda }(r)=\infty$ for all $\lambda >0$.

  Let $\Phi (u)=2F(u)-uf(u)$ for $u\geq 0$, then $\Phi (0)=0$ and

  $$ \Phi ^{\prime }(u)=\left . \left \{ \textstyle\begin{array}{l@{\quad}l} f(0) & \text{for }u=0, \\ f(u)-uf^{\prime }(u) & \text{for }u>0.\end{array}\displaystyle \right . \right . $$

  (26)

  Since f is concave on $(0,\infty )$,

  $$ \Phi ^{\prime \prime }(u)=-uf^{\prime \prime }(u)\geq 0\text{ on }(0, \infty ). $$

  (27)

  By (H1), (26) and (27), we obtain that $\Phi ^{\prime }(0)=f(0)\geq 0$ and $\Phi ^{\prime }(u)=f(u)-uf^{\prime }(u)\geq 0$ for all $u>0$. By Lemma 10, $T_{\lambda }^{\prime }(r)>0$ on $(0,\infty )$ for all $\lambda >0$. Therefore, we obtain:

  (2) For all $\lambda >0$, $T_{\lambda }(r)$ is strictly increasing on $(0,\infty )$.

  Assume $L>0$ is fixed. For every $\lambda >0$, there exists a unique $r_{\lambda }>0$ such that $T_{\lambda }(r_{\lambda })=L$ by properties (1) and (2) above. By (9), we obtain:

  (3) For every $\lambda >0$, there exists a unique positive solution $u_{\lambda }$ of (1) corresponding to $||u_{\lambda }||_{\infty }=r_{\lambda }$.

  For any fixed $\lambda _{2}>\lambda _{1}>0$, there exist $r_{\lambda _{1}},r_{\lambda _{2}}>0$ and positive solutions $u_{\lambda _{1}}$, $u_{\lambda _{2}}$ such that $T_{\lambda _{1}}(r_{\lambda _{1}})=T_{\lambda _{2}}(r_{\lambda _{2}})=L$, $||u_{\lambda _{1}}||_{\infty }=r_{\lambda _{1}}$, and $||u_{\lambda _{2}}||_{\infty }=r_{\lambda _{2}}$. By Lemma 14,

  $$ T_{\lambda _{2}}(r_{\lambda _{1}})< T_{\lambda _{1}}(r_{\lambda _{1}})=L. $$

  Thus, $0< r_{\lambda _{1}}< r_{\lambda _{2}}<\infty$ by properties (1) and (2) above. We obtain:

  (4) $||u_{\lambda _{1}}||_{\infty }=r_{\lambda _{1}}< r_{\lambda _{2}}=||u_{ \lambda _{2}}||_{\infty }$ for $\lambda _{2}>\lambda _{1}>0$.

  By properties (3) and (4) above, we immediately obtain the exact multiplicity result of positive solutions of (1). Moreover, the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$.

• (II) First, we study the shape and asymptotic behaviors of $T_{\lambda }(r)$. Assume $f(0)=0$ and $0<\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=A<\infty$. By Lemma 9, we obtain:

  (1)$\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)= \frac{\pi }{2\sqrt{\lambda A}}$ and $\lim _{r\rightarrow \infty }T_{\lambda }(r)=\infty$ for all $\lambda >0$.

  Applying similar arguments as in the proofs of part (I), we have $T_{\lambda }^{\prime }(r)>0$ on $(0,\infty )$ for all $\lambda >0$. Therefore, we obtain:

  (2) For all $\lambda >0$, $T_{\lambda }(r)$ is strictly increasing on $(0,\infty )$.

  Assume $L>0$ is fixed. For every $0<\lambda \leq \hat{\lambda}\equiv \frac{\pi ^{2}}{4AL^{2}}$, $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)= \frac{\pi }{2\sqrt{\lambda A}}\geq L$ and $T_{\lambda }(r)>L$ on $(0,\infty )$ by properties (1) and (2) above. By (9), we obtain:

  (3) For every $0<\lambda \leq \hat{\lambda}=\frac{\pi ^{2}}{4AL^{2}}$, there exists no positive solution of (1).

  For every $\lambda >\hat{\lambda}=\frac{\pi ^{2}}{4AL^{2}}$, $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)= \frac{\pi }{2\sqrt{\lambda A}}< L$ and there exists a unique $r_{\lambda }>0$ such that $T_{\lambda }(r_{\lambda })=L$ by properties (1) and (2) above. By (9), we obtain:

  (4) For every $\lambda >\hat{\lambda}=\frac{\pi ^{2}}{4AL^{2}}$, there exists a unique positive solution $u_{\lambda }$ of (1) corresponding to $||u_{\lambda }||_{\infty }=r_{\lambda }$.

  For any fixed $\lambda _{2}>\lambda _{1}>\hat{\lambda}$, applying similar arguments as in the proofs of part (I), there exist $r_{\lambda _{1}},r_{\lambda _{2}}>0$ and positive solutions $u_{\lambda _{1}}$, $u_{\lambda _{2}}$ such that:

  (5) $||u_{\lambda _{1}}||_{\infty }=r_{\lambda _{1}}< r_{\lambda _{2}}=||u_{ \lambda _{2}}||_{\infty }$ for $\lambda _{2}>\lambda _{1}>\hat{\lambda}$.

  By properties (3)–(5) above, we immediately obtain the exact multiplicity result of positive solutions of (1). Moreover, the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$.

The proof of Theorem 4 is now complete. □

Proof of Theorem 5.

• (I) First, we study the shape and asymptotic behaviors of $T_{\lambda }(r)$. Assume one of the following conditions holds:

  • (i) $f(0)>0$.

  • (ii) $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\infty$.

  By Lemma 9, we obtain:

  (1) $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)=0$ and $\lim _{r\rightarrow \infty }T_{\lambda }(r)=\infty$ for all $\lambda >0$.

  We claim that $T_{\lambda }^{\prime }(r)>0$ on $(0,\infty )$ for all $\lambda >0$. Assume that $T_{\tilde{\lambda}}(r)$ is not strictly increasing on $(0,\infty )$ for some $\tilde{\lambda}>0$. Then there exist $r_{1},r_{2}\in (0,\infty )$ such that $r_{1}< r_{2}$ are critical points of $T_{\tilde{\lambda}}(r)$, $r_{1}$ is a local maximum, and $r_{2}$ is a local minimum. Thus, $T_{\tilde{\lambda}}^{\prime \prime }(r_{1})\leq 0$ and $T_{\tilde{\lambda}}^{\prime \prime }(r_{2})\geq 0$. This contradicts to (12) in Lemma 11. So $T_{\lambda }^{\prime }(r)>0$ on $(0,\infty )$ for all $\lambda >0$. Therefore, we obtain:

  (2) For all $\lambda >0$, $T_{\lambda }(r)$ is strictly increasing on $(0,\infty )$.

  Applying similar arguments as in the proofs of part (I) of Theorem 4, we have the following properties (3) and (4):

  (3) For every $\lambda >0$, there exists a unique $r_{\lambda }>0$ and a unique positive solution $u_{\lambda }$ of (1) corresponding to $||u_{\lambda }||_{\infty }=r_{\lambda }$.

  For any fixed $\lambda _{2}>\lambda _{1}>0$, there exist $r_{\lambda _{1}},r_{\lambda _{2}}>0$ and positive solutions $u_{\lambda _{1}}$, $u_{\lambda _{2}}$ such that:

  (4) $||u_{\lambda _{1}}||_{\infty }=r_{\lambda _{1}}< r_{\lambda _{2}}=||u_{ \lambda _{2}}||_{\infty }$ for $\lambda _{2}>\lambda _{1}>0$.

  By properties (3) and (4) above, we immediately obtain the exact multiplicity result of positive solutions of (1). Moreover, the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$.

• (II) First, we study the shape and asymptotic behaviors of $T_{\lambda }(r)$. Assume $f(0)=0$ and $0<\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=A<\infty$. By Lemma 9, we obtain:

  (1) $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)= \frac{\pi }{2\sqrt{\lambda A}}$ and $\lim _{r\rightarrow \infty }T_{\lambda }(r)=\infty$ for all $\lambda >0$.

  We claim that $f(u)-uf^{\prime }(u)\geq 0$ for all $u>0$. Assume that $f(\tilde{u})-\tilde{u}f^{\prime }(\tilde{u})<0$ for some $\tilde{u}>0$, then $\frac{\tilde{u}f^{\prime }(\tilde{u})}{f(\tilde{u})}>1$. By (H2), $\lim _{u\rightarrow 0^{+}}\frac{uf^{\prime }(u)}{f(u)}$ exists and $B\equiv \lim _{u\rightarrow 0^{+}}\frac{uf^{\prime }(u)}{f(u)}\geq \frac{\tilde{u}f^{\prime }(\tilde{u})}{f(\tilde{u})}>1$. So

  $$ \lim _{u\rightarrow 0^{+}}f^{\prime }(u)=\lim _{u\rightarrow 0^{+}} \frac{uf^{\prime }(u)}{f(u)}\cdot \lim _{u\rightarrow 0^{+}} \frac{f(u)}{u}=Bf^{\prime }(0)>f^{\prime }(0)=A\in (0,\infty ). $$

  We obtain that $f^{\prime }(u)$ has a jump discontinuity at $u=0$, this contradicts to Darboux’s theorem. So $f(u)-uf^{\prime }(u)\geq 0$ for all $u>0$. By Lemma 10, $T_{\lambda }^{\prime }(r)>0$ on $(0,\infty )$ for all $\lambda >0$. Therefore, we obtain:

  (2) For all $\lambda >0$, $T_{\lambda }(r)$ is strictly increasing on $(0,\infty )$.

  Applying similar arguments as in the proofs of part (II) of Theorem 4, we have the following properties (3)–(5):

  (3) For every $0<\lambda \leq \hat{\lambda}=\frac{\pi ^{2}}{4AL^{2}}$, there exists no positive solution of (1).

  (4) For every $\lambda >\hat{\lambda}=\frac{\pi ^{2}}{4AL^{2}}$, there exists a unique $r_{\lambda }>0$ and a unique positive solution $u_{\lambda }$ of (1) corresponding to $||u_{\lambda }||_{\infty }=r_{\lambda }$.

  For any fixed $\lambda _{2}>\lambda _{1}>\hat{\lambda}$, there exist $r_{\lambda _{1}},r_{\lambda _{2}}>0$ and positive solutions $u_{\lambda _{1}}$, $u_{\lambda _{2}}$ such that:

  (5) $||u_{\lambda _{1}}||_{\infty }=r_{\lambda _{1}}< r_{\lambda _{2}}=||u_{ \lambda _{2}}||_{\infty }$ for $\lambda _{2}>\lambda _{1}>\hat{\lambda}$.

  By properties (3)–(5) above, we immediately obtain the exact multiplicity result of positive solutions of (1). Moreover, the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$.

• (III) First, we study the shape and asymptotic behaviors of $T_{\lambda }(r)$. Assume $f(0)=0$ and $\lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=0$. By Lemma 9, we obtain:

  (1) $\lim _{r\rightarrow 0^{+}}T_{\lambda }(r)=\lim _{r \rightarrow \infty }T_{\lambda }(r)=\infty$ for all $\lambda >0$.

  By Lemma 11, we obtain:

  (2) $T_{\lambda }(r)$ has exactly one critical point, a local minimum, on $(0,\infty )$ for any fixed $\lambda >0$.

  Let $\eta (\lambda )\equiv \inf \{T_{\lambda }(r):r\in (0,\infty )\}$. By modifying the arguments used in the proof of [17, Lemma 3.7(i)], we are able to prove that:

  (3) $\eta (\lambda )$ is a continuous, strictly decreasing function of λ on $(0,\infty )$, $\lim _{\lambda \rightarrow 0^{+}}\eta (\lambda )=\infty$ and $\lim _{\lambda \rightarrow \infty }\eta (\lambda )=0$.

  We omit the proof of property (3).

  Assume $L>0$ is fixed. Then there exists a unique $\lambda ^{\ast }>0$ and a unique $\bar{r}_{\lambda ^{\ast }}>0$ such that $\eta (\lambda ^{\ast })=\inf \{T_{\lambda ^{\ast }}(r):r\in (0, \infty )\}=T_{\lambda ^{\ast }}(\bar{r}_{\lambda ^{\ast }})=L$ by properties (1)–(3) above. By (9), we obtain:

  (4) For $\lambda =\lambda ^{\ast }$, there exists a unique $\bar{r}_{\lambda ^{\ast }}>0$ and a unique positive solution $u_{\lambda ^{\ast }}$ of (1) corresponding to $||u_{\lambda ^{\ast }}||_{\infty }=\bar{r}_{\lambda ^{\ast }}$.

  For every $0<\lambda <\lambda ^{\ast }$, $\eta (\lambda )=\inf \{T_{\lambda }(r):r\in (0,\infty )\}>L$ by properties (1)–(4) above. By (9), we obtain:

  (5) For every $0<\lambda <\lambda ^{\ast }$, there exists no positive solution of (1).

  For every $\lambda >\lambda ^{\ast }$, $\eta (\lambda )=\inf \{T_{\lambda }(r):r\in (0,\infty )\}< L$, and there exists a unique $\hat{r}_{\lambda }\in (0,\bar{r}_{\lambda ^{\ast }})$ and a unique $\check{r}_{\lambda }\in (\bar{r}_{\lambda ^{\ast }},\infty )$ such that $T_{\lambda }(\hat{r}_{\lambda })=T_{\lambda }(\check{r}_{\lambda })=L$ by properties (1)–(4) above. (Observe that $T_{\lambda }(\bar{r}_{\lambda ^{\ast }})< T_{\lambda ^{\ast }}(\bar{r}_{\lambda ^{\ast }})=L$ by Lemma 14.) By (9), we obtain:

  (6) For every $\lambda >\lambda ^{\ast }$, there exist two positive solutions $u_{\lambda }$, $v_{\lambda }$ of (1) corresponding to $u_{\lambda }< v_{\lambda }$, $||u_{\lambda }||_{\infty }=\hat{r}_{\lambda }$ and $||v_{\lambda }||_{\infty }=\check{r}_{\lambda }$.

  For any fixed $\lambda _{2}>\lambda _{1}>\lambda ^{\ast }$, there exist $\hat{r}_{\lambda _{1}},\hat{r}_{\lambda _{2}}\in (0,\bar{r}_{\lambda ^{ \ast }})$ and positive solutions $u_{\lambda _{1}}$, $u_{\lambda _{2}}$ such that $T_{\lambda _{1}}(\hat{r}_{\lambda _{1}})=T_{\lambda _{2}}(\hat{r}_{ \lambda _{2}})=L$, $||u_{\lambda _{1}}||_{\infty }=\hat{r}_{\lambda _{1}}$, and $||u_{\lambda _{2}}||_{\infty }=\hat{r}_{\lambda _{2}}$. By Lemma 14,

  $$ T_{\lambda _{2}}(\hat{r}_{\lambda _{2}})< T_{\lambda _{1}}(\hat{r}_{ \lambda _{1}})=L. $$

  Thus, $0<\hat{r}_{\lambda _{2}}<\hat{r}_{\lambda _{1}}<\bar{r}_{\lambda ^{ \ast }}$ by properties (1)–(4) and (6) above. We obtain:

  (7) $||u_{\lambda _{1}}||_{\infty }=\hat{r}_{\lambda _{1}}>\hat{r}_{\lambda _{2}}=||u_{\lambda _{2}}||_{\infty }$ for $\lambda _{2}>\lambda _{1}>\lambda ^{\ast }$.

  For any fixed $\lambda _{2}>\lambda _{1}>\lambda ^{\ast }$, there exist $\check{r}_{\lambda _{1}},\check{r}_{\lambda _{2}}\in (\bar{r}_{ \lambda ^{\ast }},\infty )$ and positive solutions $v_{\lambda _{1}}$, $v_{\lambda _{2}}$ such that $T_{\lambda _{1}}(\check{r}_{\lambda _{1}})=T_{\lambda _{2}}( \check{r}_{\lambda _{2}})=L$, $||v_{\lambda _{1}}||_{\infty }=\check{r}_{\lambda _{1}}$, and $||v_{\lambda _{2}}||_{\infty }=\check{r}_{\lambda _{2}}$. By Lemma 14,

  $$ T_{\lambda _{2}}(\check{r}_{\lambda _{2}})< T_{\lambda _{1}}( \check{r}_{\lambda _{1}})=L. $$

  Thus, $\bar{r}_{\lambda ^{\ast }}<\check{r}_{\lambda _{1}}<\check{r}_{\lambda _{2}}<\infty$ by properties (1)–(4) and (6) above. We obtain:

  (8) $||v_{\lambda _{1}}||_{\infty }=\check{r}_{\lambda _{1}}<\check{r}_{\lambda _{2}}=||v_{\lambda _{2}}||_{\infty }$ for $\lambda _{2}>\lambda _{1}>\lambda ^{\ast }$.

  By properties (4)–(8) above, we immediately obtain the exact multiplicity result of positive solutions of (1). Moreover, the bifurcation curve $C_{L}$ is ⊂-shaped for all $L>0$.

The proof of Theorem 5 is now complete. □

Proof of Theorem 6.

For $f(u)=\sum \limits _{i=1}^{n}a_{i}u^{p_{i}}$, where $0\leq p_{1}< p_{2}<\cdots <p_{n}\leq 1$, $n\geq 2$, $a_{i}>0$ for $i=1,2,\ldots ,n$. It is known that $f\in C[0,\infty )\cap C^{2}(0,\infty )$, $f(u)>0$ for $u>0$, and

$$ f(0)\left . \left \{ \textstyle\begin{array}{l@{\quad}l} >0 & \text{if }p_{1}=0, \\ =0 & \text{if }p_{1}>0.\end{array}\displaystyle \right . \right . $$

So f satisfies hypothesis (H1). It is easy to compute that $f^{\prime \prime }(u)<0$ on $(0,\infty )$, and hence f is a concave function on $(0,\infty )$. In addition,

$$ \lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\lim _{u\rightarrow 0^{+}} \frac{\sum \limits _{i=1}^{n}a_{i}u^{p_{i}}}{u}=\infty \text{.} $$

By Theorem 4(i), the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, (3) has exactly one positive solution for all $\lambda >0$.

The proof of Theorem 6 is now complete. □

Proof of Theorem 7.

For $f(u)=\arctan u$, it is known that that $f\in C[0,\infty )\cap C^{2}(0,\infty )$, $f(0)=0$ and $f(u)>0$ for $u>0$. So f satisfies hypothesis (H1). We further compute that

$$ f^{\prime \prime }(u)=-\frac{2u}{\left ( 1+u^{2}\right ) ^{2}}< 0 \text{ \ on }(0,\infty ). $$

So f is a concave function on $(0,\infty )$. By L’Hôpital’s rule, we obtain that

$$ \lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\lim _{u\rightarrow 0^{+}} \frac{\arctan u}{u}=\lim _{u\rightarrow 0^{+}}\frac{1}{1+u^{2}}=1\in (0, \infty )\text{.} $$

By Theorem 4(ii), the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, there exists $\hat{\lambda}>0$ such that (4) has exactly one positive solution for $\lambda >\hat{\lambda}$ and no positive solution for $0<\lambda \leq \hat{\lambda}$.

The proof of Theorem 7 is now complete. □

Proof of Theorem 8.

For $f(u)=\frac{u^{p}}{1+u^{q}}$ with $0< q\leq p$, it is known that $f\in C[0,\infty )\cap C^{2}(0,\infty )$, $f(0)=0$ and $f(u)>0$ for $u>0$. So f satisfies hypothesis (H1). We further compute that

$$ \left [ \frac{uf^{\prime }(u)}{f(u)}\right ] ^{\prime }=- \frac{q^{2}u^{q-1}}{\left ( 1+u^{q}\right ) ^{2}}< 0\text{ \ on }(0,\infty ). $$

So f satisfies hypothesis (H2), and hence $f(u)$ is a geometrically concave function on $(0,\infty )$ by Proposition 1(ii). In addition,

$$ \lim _{u\rightarrow \infty }\frac{uf^{\prime }(u)}{f(u)}=\lim _{u \rightarrow \infty }\left ( p-q+\frac{q}{1+u^{q}}\right ) =p-q\geq 0. $$

Thus f satisfies hypothesis (H3).

Case (i) $0< q\leq p<1$. We compute that

$$ \lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\lim _{u\rightarrow 0^{+}} \frac{1}{u^{1-p}(1+u^{q})}=\infty \text{.} $$

By Theorem 5(i), the bifurcation curve $C_{L}$ is monotone increasing for all $L>0$. More precisely, (7) has exactly one positive solution for all $\lambda >0$.

Case (ii) $0< q\leq p=1$. We compute that

$$ \lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\lim _{u\rightarrow 0^{+}} \frac{1}{1+u^{q}}=1\in (0,\infty )\text{.} $$

By Theorem 5(ii), the bifurcation curve $C_{L}$ is monotonically increasing for all $L>0$. More precisely, there exists $\hat{\lambda}>0$ such that (7) has exactly one positive solution for $\lambda >\hat{\lambda}$ and no positive solution for $0<\lambda \leq \hat{\lambda}$.

Case (iii) $p>1$ and $0< q\leq p$. We compute that

$$ \lim _{u\rightarrow 0^{+}}\frac{f(u)}{u}=\lim _{u\rightarrow 0^{+}} \frac{u^{p-1}}{1+u^{q}}=0\text{.} $$

By Theorem 5(iii), the bifurcation curve $C_{L}$ is ⊂-shaped for all $L>0$. More precisely, there exists $\lambda ^{\ast }>0$ such that (7) has exactly two positive solutions for $\lambda >\lambda ^{\ast }$, exactly one positive solution for $\lambda =\lambda ^{\ast }$, and no positive solution for $0<\lambda <\lambda ^{\ast }$.

The proof of Theorem 8 is now complete. □