Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on the Unit Ball

Abstract

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = f(|x|,\,u,\,|\nabla u|)\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$

where $\Omega =\{x\in \mathbb {R}^N:\;|x|<1\}$, $N\ge 2$, $f:[0,\,1]\times \mathbb {R}^+\times \mathbb {R}^+ \rightarrow \mathbb {R}$ are continuous, $\mathbb {R}^+=[0,\,\infty )$. Under some inequality conditions, the existence results of positive radial solution are obtained. The proofs of the main results are based on the method of lower and upper solutions and truncating function technique.

1 Introduction

In this paper, we discuss the existence of positive radial solution for the elliptic boundary value problem (BVP) with nonlinear gradient term

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = f(|x|,\,u,\,|\nabla u|)\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$

(1.1)

on the unit ball $\Omega =\{x\in \mathbb {R}^N:\;|x|<1\}$ in $\mathbb {R}^N$, where $N\ge 2$, $f:I\times \mathbb {R}^+\times \mathbb {R}^+ \rightarrow \mathbb {R}$ is a nonlinear function, $I=[0,\,1]$.

For the special case of BVP (1.1) that f does not contain gradient term, namely the simply elliptic boundary problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = f(|x|,\,u)\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$

(1.2)

the existence of radial solutions has been considered by many authors with different methods and techniques; see [1,2,3,4] and references therein. When $\Omega$ is an annulus or exterior domain, the existence of radial solutions has also been widely discussed; for the annulus see [5,6,7], and the exterior domain see [8,9,10,11] and references therein. For the annulus and exterior domain, one well-known result is when $f(r,\,\xi )$ is nonnegative, and superlinear growth on $\xi$ at origin and infinity BVP (1.2) has a positive radial solution [8, 11]. But this result is not true for the ball. Grossi [3] has pointed out, when $\Omega$ is the unit ball and $p\ge \frac{N+2}{N-2}$ (supercritical growth case), the elliptic boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = |u|^p\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0 \end{array}\right. \end{aligned}$$

(1.3)

has no positive solution. Corresponding to BVP (1.2), $f(r,\,\xi )=|\xi |^p$ is superlinear growth on $\xi$ at origin and infinity. By [4, Theorem 1], when $\Omega$ is an annulus, BVP (1.3) has at least one positive radial solution. In general, the existence of positive radial solutions to elliptic boundary value problems on the ball is more complicated than one on the annulus and exterior domain.

The elliptic boundary value problems with general gradient term arise in many different areas of applied mathematics, and the existence of the solution is considered by several authors [12,13,14,15]. The purpose of this paper is to obtain existence results of positive radial solutions for BVP (1.1). Because of the influence of the gradient term, BVP (1.1) is more difficult than BVP (1.2). For the case that $\Omega$ is an annulus or exterior domain, the authors of references [12, 13] recently extended the existence results of positive radial solution for BVP (1.2) in [8, 11] to BVP (1.1). They by using the theory of fixed point index proved when nonlinearity $f(r,\,\xi ,\,\eta )$ is nonnegative and superlinear or sublinear growth on $\xi$ and $\eta$ at the origin and infinity, BVP (1.1) has a positive radial solution. When $f(r,\,u,\,\eta )$ is the superlinear growth on $\xi$ and $\eta$, they also assumed that $f(r,\,\xi ,\,\eta )$ satisfies a Nagumo-type growth condition on $\eta$ (see the following Condition (F3) in Theorem 1.1). The Nagumo-type growth condition restricts f is at most quadratic growth on $\eta$. However, for the case of the ball, there is no similar existence result. A typical applicable example of references [12, 13] is the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u =a|u|^p+b|\nabla u|^q\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$

(1.4)

where $a,\,b\,p,\,q$ are constants. When $a>0$, $b\ge 0$, $p>1$ and $1< q\le 2$, the nonlinearity $f(r,\, \xi ,\,\eta )=a|\xi |^p+b|\eta |^q$ is the superlinear growth on $\xi$ and $\eta$ at the origin and infinity and satisfies the Nagumo-type growth condition on $\eta$. By [11, Theorem 1.1], BVP (1.4) has a positive radial solution when $\Omega$ is an annulus. But, when $\Omega$ is the unit ball, since BVP (1.3) is the special case of BVP (1.4) for $a=1$ and $b=0,$ it follows that BVP (1.4) has no positive solution. This example shows that the results of references [12, 13] are no longer valid for BVP (1.1) on the ball.

In this paper, we will use different methods to establish the existence results of positive radial solution for BVP (1.1) on the unit ball under the nonlinearity $f(r,\,\xi ,\,\eta )$ that can be governed, respectively, by a linear function of $\xi$ and $\eta$ at infinity and zero in two cases where $f(r,\,\xi ,\,\eta )$ satisfies or does not satisfy the Nagumo-type growth condition on $\eta$. The governing conditions allow that $f(r,\,u,\,\eta )$ is the downward superlinear growth on $\xi$ and $\eta$ at infinity and zero; see Example (1.7). In the case without the Nagumo-type growth condition, $f(r,\,\xi ,\,\eta )$ may be downward super-quadratic growth; see Example (1.9).

Next, we always assume that $\Omega$ is the unit ball in $\mathbb {R}^N$, and $\lambda _1$ is the main eigenvalue of Laplace operator $-\Delta$ on $\Omega$ with boundary condition $u|_{\partial \Omega }=0$. Our main results are as follows:

Theorem 1.1

Let $f:I\times \mathbb {R}^+\times \mathbb {R}^+\rightarrow \mathbb {R}$ be continuous and satisfy the following conditions:

(F1):

there exist constants $a,\,b\ge 0$ satisfying $a+b<N$ and $H>0$ such that

$$\begin{aligned} f(r, \xi , \eta )\le a \xi +b \eta , \quad (r, \xi , \eta )\in I\times \mathbb {R}^+\times \mathbb {R}^+,\;\;|(\xi ,\eta )|>H, \end{aligned}$$

(F2):

there exists a positive constant $\delta >0$ such that

$$\begin{aligned} f(r, \xi , \eta )\ge \lambda _1\,\xi , \quad \forall \;(r, \xi , \eta )\in I\times \mathbb {R}^+\times \mathbb {R}^+,\;|(\xi ,\eta )|<\delta , \end{aligned}$$

(F3):

for any $M>0$, there exists a positive monotone nondecreasing continuous function $g_M(\eta )$ on $\mathbb {R}^+$ satisfying

$$\begin{aligned} \int _0^{\infty }\frac{\eta \,d \eta }{\;g_M(\eta )}=\infty , \end{aligned}$$

(1.5)

such that

$$\begin{aligned} |f(r, \xi , \eta )|\le g_M(\eta ),\quad \forall \;(r,\xi ,\eta )\in I\times [0,\,M]\times \mathbb {R}^+, \end{aligned}$$

(1.6)

then BVP (1.1) has at least one classical positive radial solution.

In Theorem 1.1, the conditions (F1) and (F2) allow that $f(r,\,u,\,\eta )$ is downward superlinear growth on $\xi$ and $\eta$ at the infinity and origin. The condition (F3) is a Nagumo-type growth condition and restricts f is at most quadratic growth on $\eta$. To show the applicability of Theorem 1.1, we consider a concrete example:

$$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u =\alpha \,u-\beta u^p+\gamma | \nabla u|-\delta \,|\nabla u|^q,\qquad x\in \Omega ,\\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$

(1.7)

where $\alpha ,\,\beta ,\gamma ,\,\delta >0$, $p,\,q>1$, are constants. Corresponding to BVP (1.1), the nonlinearity

$$\begin{aligned} f(r,\,\xi ,\,\eta )=\alpha \,\xi -\beta \,\xi ^p+\gamma \,\eta -\delta \,\eta ^q, \qquad \xi ,\; \eta \ge 0. \end{aligned}$$

(1.8)

Set $C_1=\max \{\alpha \,\xi -\beta \xi ^p\;|\; \xi \ge 0\}$, $C_1=\max \{\gamma \eta -\delta \eta ^q\;|\; \eta \ge 0\}$. When $|(\xi , \eta )|>2(C_1+C_2)$ and $\xi ,\,\eta \ge 0$, by (1.8) we have

$$\begin{aligned} f(r,\,\xi ,\,\eta )\le C_1+C_2\le \frac{1}{2}|(\xi , \eta )|\le \frac{1}{2}\xi +\frac{1}{2}\eta . \end{aligned}$$

Hence, (F1) holds.

Let $\alpha >\lambda _1$, we easily verify that f satisfies Condition (F2). From (1.8), easily see that $g_M(\eta )=C(M)+(\gamma +\delta +1)\,\eta ^q$, where $C(M)= \max \{|\alpha \,\xi -\beta \xi ^p|\;|\; 0\le \xi \le M\}$, and only when $q\le 2$ Condition (F3) holds. By Theorem 1.1, when $\alpha >\lambda _1$ and $1<q\le 2$, BVP (1.7) has a classical positive radial solution.

For the case that f does not satisfy the Nagumo-type growth condition (F3), we have the following result:

Theorem 1.2

Let $f:I\times \mathbb {R}^+\times \mathbb {R}^+\rightarrow \mathbb {R}$ be continuous and satisfy (F1), (F2) and the following condition:

(F4):

for every $r\in I$ and $\eta \ge 0$, $f(r,\,\xi ,\,\eta )$ is increasing on $\xi$ in $\mathbb {R}^+$,

then BVP (1.1) has at least one classical positive radial solution.

Theorem 1.2 allows that $f(r,\,u,\,\eta )$ is downward super-quadratic growth on $\eta$. See the following example:

$$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u =\alpha \,\sqrt{u}+\beta \,|\nabla u|-\gamma \, |\nabla u|^3,\qquad x\in \Omega ,\\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$

(1.9)

where $\alpha ,\,\beta ,\gamma$ are positive constants. Corresponding to BVP (1.1), the nonlinearity

$$\begin{aligned} f(r,\,\xi ,\,\eta )=\alpha \,\sqrt{\xi }+\beta \,\eta -\gamma \,\eta ^3, \qquad \xi ,\; \eta \ge 0. \end{aligned}$$

(1.10)

From (1.10), we can verify that f satisfies the conditions (F1), (F2) and (F4). Hence by Theorem 1.2, BVP (1.9) has at least one classical positive radial solution.

The proofs of Theorem 1.1 and 1.2 are based on the method of lower and upper solutions and truncating function technique, which will be given in Sect. 3. Some preliminaries to prove our main results are presented in Sect. 2.

2 Preliminaries

Let $u=u(|x|)$ be a radially symmetric solution of BVP (1.1), writing $r = |x|$, a direct calculation shows that u(r) satisfies the equation

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)= f(r,\,u(r),\,|u'(r)|),\quad r\in I,\\ u'(0)=0,\quad u(1)=0. \end{array}\right. \end{aligned}$$

(2.1)

Clearly, if $u\in C^2[0,\,1]$ is a solution of BVP (2.1), then $u(|x|)\in C^2(\bar{\Omega })$ is a classical radial solution of BVP (1.1). We discuss BVP (2.1) to obtain positive radial solutions of BVP (1.1).

Let C(I) denote the Banach space of all continuous function u(r) on I with norm $\Vert u\Vert _C=\max _{r\in I}|u(r)|$. Generally, for $n\in \mathbb {N}$, we use $C^n(I)$ to denote the Banach space of all nth-order continuous differentiable function on I with the norm $\Vert u\Vert _{C^n}=\max \{\,\Vert u\Vert _C,\,\Vert u'\Vert _C,\,\ldots ,\,\Vert u^{(n)}\Vert _C\}$. Let $C^+(I)$ denote the cone of nonnegative functions in C(I).

To discuss BVP (2.1), we first consider the corresponding linear boundary value problem (LBVP)

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)= h(r),\quad r\in I,\\ u'(0)=0,\quad u(1)=0, \end{array}\right. \end{aligned}$$

(2.2)

where $h\in C(I)$ is a given function.

Lemma 2.1

For every $h\in C(I)$, LBVP (2.2) has a unique solution $u:=S\,h\in C^2(I)$. Moreover, the solution operator $S: C(I)\rightarrow C^1(I)$ is a completely continuous linear operator and

$$\begin{aligned} \Vert S\Vert _{\mathcal {B}(C(I),\,C(I))}\le \frac{1}{2N},\quad \Vert S\Vert _{\mathcal {B}(C(I),\,C^1(I))}\le \frac{1}{N}. \end{aligned}$$

(2.3)

Proof

LBVP (2.2) can be rewritten the equivalent form of

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1}u'(r))'=r^{N-1}h(r),\quad r\in I,\\ u'(0)=0,\quad u(1)=0. \end{array}\right. \end{aligned}$$

(2.4)

Integrating Eq. (2.4), we obtain LBVP (2.2) has a unique solution given by

$$\begin{aligned} u(r)=\int _r^1\frac{1}{t^{N-1}}\,dt\int _0^t s^{N-1}h(s)ds:=Sh(r). \end{aligned}$$

(2.5)

From this expression, it follows that

$$\begin{aligned} u'(r)= & {} -\frac{1}{r^{N-1}}\int _0^r s^{N-1}h(s)ds, \end{aligned}$$

(2.6)

$$\begin{aligned} u''(r)= & {} -h(r)+\frac{N-1}{r^{N}}\int _0^r s^{N-1}h(s)ds. \end{aligned}$$

(2.7)

Using (2.5)–(2.7) and a direct estimation, we obtain that

$$\begin{aligned} \Vert u\Vert _C\le \frac{1}{2N}\Vert h\Vert _C,\quad \Vert u'\Vert \le \frac{1}{N}\Vert h\Vert _C,\quad \Vert u''\Vert _C\le 2\Vert h\Vert _C. \end{aligned}$$

(2.8)

Hence, the solution operator of LBVP (2.1) $S:C(I)\rightarrow C^2(I)$ is a linear bounded operator. By the compactness of the embedding $C^2(I)\hookrightarrow C^1(I)$, $S:C(I)\rightarrow C^1(I)$ is completely continuous. Notice that $Sh=u$, by (2.8)

$$\begin{aligned} \Vert Sh\Vert _C\le \frac{1}{2N}\Vert h\Vert _C,\quad \Vert Sh\Vert _{C^1}\le \frac{1}{N}\Vert h\Vert _C. \end{aligned}$$

Hence, (2.3) holds. $\square$

Lemma 2.2

Let $a\in [0,\,2N)$ and $h\in C^+(I)$. Then the linear boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)=au(r)+ h(r),\quad r\in I,\\ u'(0)=0,\quad u(1)=0 \end{array}\right. \end{aligned}$$

(2.9)

has a unique solution u and satisfies $u\ge 0$, $u'\le 0$.

Proof

Let $h\in C^+(I)$. By (2.5), the solution of LBVP (2.2) $Sh\in C^+(I)$. Hence, the solution operator of LBVP (2.2) $S: C(I)\rightarrow C(I)$ is a positive linear operator. By the definition of S, LBVP (2.9) is equivalent to the operator equation in C(I)

$$\begin{aligned} ({I}-aS)u=Sh, \end{aligned}$$

(2.10)

where I is the unit operator in C(I). By (2.3), $\Vert aS\Vert _{\mathcal {B}(C(I),\,C(I))}<1$, hence ${I}-aS: C(I)\rightarrow C(I)$ has bounded inverse operator $T=({I}-aS)^{-1}$ and can be expressed by the Nuemann series

$$\begin{aligned} T=({I}-aS)^{-1}=\sum _{n=0}^\infty a^nS^n. \end{aligned}$$

(2.11)

Hence, Eq. (2.10), equivalently LBVP (2.9), has a unique solution $u=T(Sh)$. By (2.11), $T: C(I)\rightarrow C(I)$ is a positive operator, so $u=T(Sh)\in C^+(I)$, namely, $u\ge 0$. Set $h_1(r)=au(r)+h(r)$, then $h_1\in C^+(I)$. By the definition of S, $u=Sh_1$. Hence by (2.6), $u'\le 0$. $\square$

Lemma 2.3

Let $f:I\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ be continuous. If there exist constants $a,\,b\ge 0$ satisfying $a+b<N$ and $c>0$ such that

$$\begin{aligned} |f(r, \xi , \eta )|\le a|\xi |+b |\eta |+c, \quad (r, \xi , \eta )\in I\times \mathbb {R}\times \mathbb {R}, \end{aligned}$$

(2.12)

then the nonlinear boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)= f(r,\,u(r),\,u'(r)),\quad r\in I,\\ u'(0)=0,\quad u(1)=0 \end{array}\right. \end{aligned}$$

(2.13)

has at least one solution $u\in C^2(I)$.

Proof

Define a mapping $F: C^1(I)\rightarrow C(I)$ by

$$\begin{aligned} F(u)(r)=f(r, u(r), u'(r)), \qquad r\in I,\quad u\in C^1(I). \end{aligned}$$

(2.14)

Clearly, $F: C^1(I)\rightarrow C(I)$ is continuous. By (2.12) we have

$$\begin{aligned} \Vert F(u)\Vert _C \le a\Vert u\Vert _C+b\Vert u'\Vert _C+c\le (a+b)\Vert u\Vert _{C^1}+c,\quad u\in C^1(I). \end{aligned}$$

(2.15)

By the complete continuity of $S:C(I)\rightarrow C^1(I)$, the composite mapping $A=S\circ F: C^1(I)\rightarrow C^1(I)$ is completely continuous. By the definition of the solution operator S of LBVP (2.2), the solution of BVP (2.13) is equivalent to the fixed point of A. Choose a constant $R\ge \frac{c}{N-(a+b)}$ and set $D=\{u\in C^1(I)\;|\;\Vert u\Vert _{C^1}\le R\}$. Then D is a bounded convex closed set in $C^1(I)$. For every $u\in D$, by Lemma 2.1 and (2.15), we have

$$\begin{aligned} \Vert Au\Vert _{C^1}&=\Vert S(F(u)\Vert _{C^1} \le \Vert S\Vert _{\mathcal {B}(C(I),\,C^1(I))}\,\Vert F(u)\Vert _C\\&\le \frac{1}{N}\Vert F(u)\Vert _C\le \frac{a+b}{N}\Vert u\Vert _{C^1}+\frac{c}{N}\\&\le \frac{a+b}{N}\,R+\frac{c}{N}\le R. \end{aligned}$$

This means that $Au\in D$. Hence $A(D)\subset D$, and by the Schauder fixed-point theorem, A has a fixed point in D, which is a solution of BVP (2.13). $\square$

Lemma 2.4

Let $a,\,b\ge 0$ and satisfy $a+b<N$, $h\in C^+(I)$. Then the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)=a\,u(r)+b\,|u'(r)| + h(r),\quad r\in I,\\ u'(0)=0,\quad u(1)=0 \end{array}\right. \end{aligned}$$

(2.16)

has a unique solution u and satisfies $u\ge 0$, $u'\le 0$.

Proof

Corresponding to BVP (2.13), the function f on the right is

$$\begin{aligned} f(r, \xi , \eta )=a\xi +b|\eta | + h(r), \quad (r, \xi , \eta )\in I\times \mathbb {R}\times \mathbb {R}. \end{aligned}$$

(2.17)

Clearly, this function satisfies the condition of Lemma 2.3. Hence, BVP (2.16) has at least one solution.

Let $u_1,\,u_2\in C^2(I)$ be two solutions of BVP (2.16), then $u_1$ and $u_2$ are fixed points of $A=S\circ F$, and $u_2-u_1=A u_2-A u_1=S(F(u_2)-F(u_2))$. If $u_1\not =u_2$, then by Lemma 2.1 and (2.17) one gets that

$$\begin{aligned} \Vert u_2-u_1\Vert _{C^1}&=\Vert S(F(u_2)-F(u_1))\Vert _{C^1} \le \Vert S\Vert _{\mathcal {B}(C(I),\,C^1(I))}\,\Vert F(u_2)-F(u_1)\Vert _C\\&\le \frac{1}{N}\,(a\Vert u_2-u_1\Vert _C+b\Vert u_2'-u_1'\Vert _C)\\&\le \frac{a+b}{N}\,\Vert u_2-u_1\Vert _{C^1}\\&< \Vert u_2-u_1\Vert _{C^1}, \end{aligned}$$

which is a contradiction! Hence, $u_1=u_2$. This means that BVP (2.16) has only one solution.

Let $u\in C^2(I)$ be the unique solution of BVP (2.16) and $h_1=b|u'(r)|+h(r)$. Then, $h_1\in C^+(I)$ and u is the solution of LBVP (2.9) for the nonhomogeneous terms $h_1$. By Lemma 2.2, $u\ge 0$ and $u'\le 0$. $\square$

Let $f: I\times \mathbb {R}\times \mathbb {R}^+ \rightarrow \mathbb {R}$ be continuous. If a function $v\in C^2(I)$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} -v''(r)-\frac{N-1}{r}\,v'(r)\le f(r,\,v(r),\,|v'(r)|),\quad r\in I,\\ v'(0)=0,\quad v(1)\le 0, \end{array}\right. \end{aligned}$$

(2.18)

we call it a lower solution of BVP (2.1), and if a function $w\in C^2(I)$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} -w''(r)-\frac{N-1}{r}\,w'(r)\ge f(r,\,w(r),\,|w'(r)|),\quad r\in I,\\ w'(0)=0,\quad w(1)\ge 0, \end{array}\right. \end{aligned}$$

(2.19)

we call it an upper solution of BVP (2.1).

Lemma 2.5

Let $v_0\in C^2(I)$ be a lower solution of BVP (1.1) and $w_0\in C^2(I)$ an upper solution, and ${v_0}'\ge {w_0}'$. Then, $v_0\le w_0$.

Proof

Let $u=w_0-v_0$, then $u'\le 0$. By the definitions of lower and upper solutions, we have

$$\begin{aligned} u(r)=u(1)-\int _r^1 u'(s)ds\ge 0, \quad r\in I. \end{aligned}$$

Hence $u\ge 0$, and $v_0\le w_0$. $\square$

3 Proofs of the Main Results

Proof of Theorem 1.1

We use the method of upper and lower solutions to prove that BVP (2.1) has a positive solution.

Let $a,\,b,\, H$ be the constant in Condition (F1). Set $C_0=\max \{|f(r, \xi , \eta )-(a\xi +b\eta )|\;|\; \xi ,\,\eta \ge 0,\; |(\xi , \eta )|\le H\,\}+1$, then by Condition (F1),

$$\begin{aligned} f(r, \xi , \eta )\le a\xi +b\eta +C_0,\qquad \xi ,\,\eta \ge 0. \end{aligned}$$

(3.1)

By Lemma 2.4, the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)=au(r)+b|u'(r)| + C_0,\quad r\in I,\\ u'(0)=0,\quad u(1)=0 \end{array}\right. \end{aligned}$$

(3.2)

has a unique solution $w_0\in C^2(I)$ which satisfies $w_0\ge 0$ and $w_0'\le 0$. By the equation and (3.1), we easily see that $w_0$ is an upper solution of BVP (2.1).

It is well known that the elliptic eigenvalue problem,

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda \,u,\qquad x\in \Omega ,\\ u|_{\partial \Omega }=0, \end{array}\right. \end{aligned}$$

(3.3)

has a minimum positive real eigenvalue $\lambda _1$, and $\lambda _1$ has a positive unit eigenfunction $\phi _1\in C^2(\overline{\Omega })\cap C^+(\overline{\Omega })$ with $\Vert \phi _1\Vert _C=1$, namely $\phi _1$ satisfies the equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta \phi _1 = \lambda _1\,\phi _1,\qquad x\in \Omega ,\\ \phi _1|_{\partial \Omega }=0. \end{array}\right. \end{aligned}$$

(3.4)

By the well-known symmetry result on the elliptic boundary value problem [16, Theorem 1], $\phi _1$ is radially symmetric. Hence there exists $\psi \in C^2(I)\cap C^+(I)$ with $\Vert \psi \Vert _C=1$ such that $\phi _1(x)=\psi (|x|)$. By Eq. (3.4), $\psi (r)$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} -\psi ''(r)-\frac{N-1}{r}\,\psi '(r)=\lambda _1\,\psi (r),\quad r\in I,\\ \psi '(0)=0,\quad \psi (1)=0. \end{array}\right. \end{aligned}$$

(3.5)

By Lemma 2.2, $\psi '\le 0$.

Set $\delta _0=\min \{\delta /(1+\Vert \psi '\Vert _C^{\;2})^{1/2},\;C_0/\lambda _1\}$, choose $\varepsilon \in (0,\,\delta _0)$ and $v_0=\varepsilon \psi$. Then for every $r\in I$, we have

$$\begin{aligned} v_0(r)\ge 0,\quad v_0'(r)\le 0,\quad |(v_0(r), -v_0'(r))|\le \varepsilon \,(1+\Vert \psi '\Vert _C^{\;2})^{1/2}<\delta . \end{aligned}$$

By Condition (F2) and Eq. (3.5),

$$\begin{aligned} f(r, v_0(r), |v_0'(r)|)\ge \lambda _1 v_0(r)=-v_0''(r)-\frac{N-1}{r}\,v_0'(r). \end{aligned}$$

Hence, $v_0$ is a lower solution of BVP (2.1). We show that

$$\begin{aligned} 0\le v_0\le w_0,\quad w_0'\le v_0'\le 0. \end{aligned}$$

(3.6)

Consider $u=w_0-v_0$. Since $w_0$ satisfies (3.2) and $v_0$ satisfies (3.5), it follows that

$$\begin{aligned} -u''(r)-\frac{N-1}{r}\,u'(r) =aw_0(r)+b|w_0'(r)| + C_0-\lambda _1 v_0(r)\ge C_0-\lambda _1\varepsilon \ge 0. \end{aligned}$$

By Lemma 2.2, $u\ge 0$ and $u'\le 0$. Hence, (3.6) holds.

Define a functions $\sigma : I\times \mathbb {R}\rightarrow \mathbb {R}^+$ by

$$\begin{aligned} \sigma (r,\xi )=\max \{v_0(r),\; \min \{\xi ,\,w_0(r)\}\,\},\quad (r,\,\xi )\in I\times \mathbb {R}. \end{aligned}$$

(3.7)

Then $\sigma : I\times \mathbb {R}\rightarrow \mathbb {R}^+$ are continuous and satisfy

$$\begin{aligned} v_0(r)\le \sigma (r,\xi )\le w_0(r),\quad (r,\,\xi )\in I\times \mathbb {R}. \end{aligned}$$

(3.8)

For $M=\Vert w_0\Vert _C>0$, by Condition (F3), there exists a positive monotone nondecreasing continuous function $g_M(\eta )$ on $\mathbb {R}^+$ satisfying (1.5) such that (1.6) holds. By (1.5), there exists $K_0>0$ such that

$$\begin{aligned} \int _0^{K_0}\frac{\rho d\rho }{g_M(\rho )}>\Vert w_0\Vert _C=M. \end{aligned}$$

(3.9)

Choose a positive constant $K=K_0+\Vert w_0'\Vert _C+1$ and set

$$\begin{aligned}{}[\eta ]_K=\min \{|\eta |,\;K\},\quad \eta \in \mathbb {R}. \end{aligned}$$

(3.10)

Make a truncating function $f^*$ of f by

$$\begin{aligned} f^*(r, \xi ,\eta )= f(r, \sigma (r,\xi ),[\eta ]_K)-\frac{\xi -\sigma (r,\xi )}{{\xi }^2+1},\quad (r, \xi ,\eta )\in I\times \mathbb {R}\times \mathbb {R},\nonumber \\ \end{aligned}$$

(3.11)

and consider the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)=f^*(r, u(r), u'(r)),\quad r\in I,\\ u'(0)=0,\quad u(1)=0. \end{array}\right. \end{aligned}$$

(3.12)

By (3.7), (3.10) and (3.11) , $f^*:I\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ is continuous and bounded. Hence by Lemma 2.3, BVP (3.12) has a solution $u_0\in C^2(I)$. We show that

$$\begin{aligned} v_0\le u_0\le w_0. \end{aligned}$$

(3.13)

In fact, if $v_0\not \le u_0$, then the function,

$$\begin{aligned} \omega (r)=u_0(r)-v_0(r),\quad r \in I, \end{aligned}$$

(3.14)

satisfies $\min _{r\in I} \omega (r)<0$. Since $\omega (1)=0$, there exists $r_0\in [0,\,1)$ such that

$$\begin{aligned} \omega (r_0)=\min _{r\in I}\omega (r)<0,\quad \omega '(r_0)= 0,\quad \omega ''(r_0)\ge 0, \end{aligned}$$

from which and (3.14) it follows that

$$\begin{aligned} u_0(r_0)<v_0(r_0),\quad u_0'(r_0)=v_0'(r_0),\quad u_0''(r_0)\ge v_0''(r_0). \end{aligned}$$

(3.15)

From this and the definitions (3.7) and (3.10), we see that

$$\begin{aligned} \sigma (r_0,\,u_0(r_0))= v_0(r_0),\quad [u_0'(r_0)]_K=|v_0'(r_0)|. \end{aligned}$$

(3.16)

Hence by Eq. (3.12) and the definition of the lower solution $v_0$, we obtain that

$$\begin{aligned} -u_0''(r_0)-\frac{N-1}{r_0}\,u_0'(r_0)&=f^*(r_0,\,u_0(r_0),\,u_0'(r_0))\\&=f(r_0,\,\sigma (r_0,u_0(r_0)),\,[u_0'(r_0)]_K)-\frac{u_0(r_0)-\sigma _0(r_0,u_0(r_0))}{u_0^{\;2}(r_0)+1}\\&=f(r_0,\,v_0(r_0),\,|v_0(r_0)|)-\frac{u_0(r_0)-v_0(r_0)}{u_0^{\;2}(r_0)+1}\\&\ge -v_0''(r_0)-\frac{N-1}{r_0}\,v_0'(r_0) -\frac{u_0(r_0)-v_0(r_0)}{u_0^{\;2}(r_0)+1}\\&> -v_0''(r_0)-\frac{N-1}{r_0}\,v_0'(r_0); \end{aligned}$$

this contradicts (3.15). Hence, $v_0\le u_0$.

With a similar argument, we can show that $u_0\le w_0$, so (3.13) holds.

By (3.13), for every $r\in I$, $0\le v_0(r)\le u_0(r)\le w_0(r)\le \Vert w_0\Vert _C=M$. Hence by (1.6), we have

$$\begin{aligned} |f(r, u(r), |u'(r)|)|\le g_M(|u'(r)|),\quad r\in I. \end{aligned}$$

(3.17)

Next, we show that

$$\begin{aligned} |u_0'(r)|\le K_0,\quad r\in I. \end{aligned}$$

(3.18)

In fact, if (3.18) does not hold, since $u_0'(0)=0$, by the maximum theorem, there exist $s_0\in (0,\,1]$ such that

$$\begin{aligned} \Vert u_0'\Vert _C=\max _{r\in I}|u_0'(r)|=|u'_0(s_0)|>K_0. \end{aligned}$$

So there are two cases: $u_0'(s_0)>K_0$ or $u_0'(s_0)<-K_0$.

Case 1. $u_0'(s_0)>K_0$. Set

$$\begin{aligned} s_1 =\inf \{ r\in (0,\, s_0]\;|\;u_0'(r)>K_0\,\},\quad r_1 =\sup \{ r\in [0,\, s_1),\;u_0'(r)=0\,\}. \end{aligned}$$

Then, $0\le r_1<s_1\le s_0$ and

$$\begin{aligned} 0\le u_0'(r)\le K_0,\quad r\in [r_1,\,s_1];\quad u_0'(r_1)=0,\; u_0'(s_1)=K_0. \end{aligned}$$

(3.19)

Hence, for $r\in [r_1,\,s_1]$, by (3.12), (3.13) and (3.17), we have

$$\begin{aligned} u_0''(r)&\le u_0''(r)+\frac{N-1}{r_0}\,u_0'(r_0)=-f^*(r, u_0(r), u_0'(r))\\&= -f(r,\, u_0(r),\, |u_0'(r)|)\\&\le g_M(u_0'(r)). \end{aligned}$$

From this inequality, it follows that

$$\begin{aligned} \frac{\,u_0'(r)\,u_0''(r)\,}{g_M(u_0'(r))}\le u_0'(r),\quad r\in [r_1, s_1]. \end{aligned}$$

(3.20)

Integrating this inequality on $[r_1, s_1]$ and making the variable transformation $\rho =u_0'(r)$ for the left side, we have

$$\begin{aligned} \int _0^{K_0}\frac{\rho \,\mathrm {d}\rho }{\;g_M(\rho )}\le u_0(s_1)-u_0(r_1)\le u_0(s_1)\le \Vert u_0\Vert _{C}\le \Vert w_0\Vert _{C} =M. \end{aligned}$$

This contradicts (3.9)!

Case 2. $u_0'(s_0)<-K_0$. Set

$$\begin{aligned} s_1 =\inf \{ r\in (0,\, s_0]\;|\;u_0'(r)<-K_0\,\},\quad r_1 =\sup \{ r\in [0,\, s_1),\;u_0'(r)=0\,\}. \end{aligned}$$

Then $0\le r_1<s_1\le s_0$ and

$$\begin{aligned} -K \le u_0'(r)\le 0,\quad r\in [r_1,\,s_1];\quad u_0'(r_1)=0,\; u_0'(s_1)=-K_0. \end{aligned}$$

(3.21)

Hence, for $r\in [r_1,\,s_1]$, by (3.12), (3.13) and (3.17), we have

$$\begin{aligned} -u_0''(r)&\le - u_0''(r)-\frac{N-1}{r_0}\,u_0'(r_0)=f^*(r, u_0(r), u_0'(r))\\&= f(r, u_0(r), |u_0'(r)|)\\&\le g_M(-u_0'(r)). \end{aligned}$$

From this inequality, it follows that

$$\begin{aligned} \frac{\,(-u_0'(r))\,(-u_0''(r))\,}{g_M(-u_0'(r))}\le -u_0'(r),\quad r\in [r_1, s_1]. \end{aligned}$$

(3.22)

Integrating this inequality on $[r_1, s_1]$ and making the variable transformation $\rho =-u_0'(r)$ for the left side, we have

$$\begin{aligned} \int _0^{K_0}\frac{\rho \,\mathrm {d}\rho }{\;g_M(\rho )}\le - u_0(s_1)+u_0(r_1)\le u_0(r_1)\le \Vert u_0\Vert _{C}\le \Vert w_0\Vert _{C} =M. \end{aligned}$$

This contradicts (3.9)!

The contradictions of the two cases indicate (3.18) holds. Hence by (3.13), (3.18) and Eq. (3.6), for every $r\in I$, we have

$$\begin{aligned} -u_0''(r)-\frac{N-1}{r}\,u_0'(r)&=f^*(r,\,u_0(r),\,u_0'(r))\\&=f(r,\,\sigma _0(r, u_0(r)),\,[u'(r)]_K)-\frac{u_0(r)-\sigma _0(r, u_0(r))}{u_0^{\;2}(r)+1}\\&=f(r,\,u_0(r)),\,|u_0(r)|). \end{aligned}$$

This means that $u_0$ is a solution of BVP (2.1). Hence, $u_0(|x|)$ is a classical positive radial solution of BVP (1.1).

The proof of Theorem 1.1 is completed. $\square$

Proof of Theorem 1.2

Let $v_0, w_0\in C^2(I)$ be the functions constructed in the proof of theorem 1.1. Then they satisfy (3.6). Let $\sigma : I\times \mathbb {R}\rightarrow \mathbb {R}$ be the function defined by (3.7) and define $\sigma _1: I\times \mathbb {R}\rightarrow \mathbb {R}$ by

$$\begin{aligned} \sigma _1(r,\eta )=-\max \{w_0'(r),\; \min \{\eta ,\,v_0'(r)\}\,\},\quad (r,\,\eta )\in I\times \mathbb {R}. \end{aligned}$$

(3.23)

Then it is continuous and satisfies

$$\begin{aligned} -v_0'(r)\le \sigma _1(r,\eta )\le -w_0'(r),\quad (r,\eta )\in I\times \mathbb {R}. \end{aligned}$$

(3.24)

Make a truncating function $f^{**}$ of f by

$$\begin{aligned} f^{**}(r, \xi ,\eta )= f(r,\, \sigma (r,\xi ),\,\sigma _1(r,\eta )),\quad (r, \xi ,\eta )\in I\times \mathbb {R}\times \mathbb {R}. \end{aligned}$$

(3.25)

By (3.7), (3.24) and (3.25), $f^{**}:I\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ is continuous and bounded. Hence, by Lemma 2.3 the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(r)-\frac{N-1}{r}\,u'(r)=f^{**}(r, u(r), u'(r)),\quad r\in I,\\ u'(0)=0,\quad u(1)=0 \end{array}\right. \end{aligned}$$

(3.26)

has a solution $u_0\in C^2(I)$. We show that

$$\begin{aligned} w_0'\le u_0'\le v_0'. \end{aligned}$$

(3.27)

In fact, if $w_0'\not \le u_0'$, then the function,

$$\begin{aligned} \theta (r)=u_0'(r)-w_0'(r),\quad r \in I, \end{aligned}$$

(3.28)

satisfies $\min _{r\in I} \theta (r)<0$. Since $\theta (0)=0$, there exists $r_0\in (0,\,1]$ such that

$$\begin{aligned} \theta (r_0)=\min _{r\in I}\theta (r)<0,\quad \theta '(r_0)\le 0. \end{aligned}$$

By this we obtain that

$$\begin{aligned} u_0'(r_0)<w_0'(r_0),\quad u_0''(r_0)\le w_0''(r_0). \end{aligned}$$

(3.29)

So we have

$$\begin{aligned} \sigma _1(r_0,\,u_0'(r_0))= -w_0'(r_0)=|w_0'(r_0)|. \end{aligned}$$

(3.30)

Hence by Eq. (3.26), Condition (F4) and the definition of the upper solution $w_0$, we obtain that

$$\begin{aligned} -u_0''(r_0)-\frac{N-1}{r_0}\,u_0'(r_0)&=f^{**}(r_0,\,u_0(r_0),\,u_0'(r_0))\\&=f(r_0,\,\sigma (r_0,u_0(r_0)),\,\sigma _1(r_0,\,u_0'(r_0)))\\&=f(r_0,\,\sigma (r_0,u_0(r_0)),\,|w_0'(r_0)|)\\&\le f(r_0,\,w_0(r_0)),\,|w_0'(r_0)|)\\&\le -w_0''(r_0)-\frac{N-1}{r_0}\,w_0'(r_0). \end{aligned}$$

This contradicts (3.29). Hence, $w_0'\le u_0'$.

With a similar argument, we can show that $u_0'\le v_0'$. Hence (3.27) holds. Now by (3.27) and the boundary condition $u(1)=0$, we easily obtain that

$$\begin{aligned} v_0\le u_0\le w_0. \end{aligned}$$

(3.31)

By (3.31), (3.27) and the definitions of the functions $\sigma$ and $\sigma _1$,

$$\begin{aligned} \sigma (r_0,\,u_0(r_0))=u_0(r_0),\quad \sigma _1(r_0,\,u_0'(r_0))=-u_0'(r_0),\quad r\in I. \end{aligned}$$

Hence by Eq. (3.26), we have

$$\begin{aligned} -u_0''(r)-\frac{N-1}{r}\,u_0'(r)&=f^{**}(r,\,u_0(r),\,u_0'(r))\\&=f(r,\,\sigma _0(r, u_0(r)),\,\sigma _1(r_0,\,u_0'(r_0)))\\&=f(r,\,u_0(r)),\, |u_0'(r)|),\quad r\in I. \end{aligned}$$

Hence $u_0$ is a solution of BVP (2.1), and $u_0(|x|)$ is a classical positive radial solution of BVP (1.1).

The proof of Theorem 1.2 is completed. $\square$