Radial Positive Solutions for Semilinear Elliptic Problems with Linear Gradient Term in $\mathbb {R}^N$

Abstract

We are concerned with the linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\frac{\kappa }{|x|^2} x\cdot \nabla u =\lambda K(|x|) u, & x\in \mathbb {R}^N,\\ u(x)>0, & x\in \mathbb {R}^N,\\[2ex] u(x)\rightarrow 0, & |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$

where $\lambda$ is a positive parameter, $\kappa \in [0,N-2)$, $N> 2$ and $K:\mathbb {R}^N \rightarrow (0,\infty )$ is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue $\lambda _1^{\text {rad}}$ and the corresponding positive eigenfunction $\varphi _1$ satisfies $\lim \nolimits _{|x|\rightarrow \infty }\varphi _1(|x|)=\frac{c}{|x|^{N-2-\kappa }}$ for some $c>0$. As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques.

1 Introduction

The research concerning the existence of positive (nonnegative) solutions of the elliptic problems

$$\begin{aligned} -\Delta u+b(|x|)x\cdot \nabla u =\lambda K(|x|) f(u) \end{aligned}$$

(1.1)

in exterior domains of $\mathbb {R}^N$ ($N>2$) have been studied by many authors, see Constantin [11, 12], Ehrnström [18, 19], Yin [29] and the references therein. Problem (1.1) arises in many applications e.g. in pseudoplastic fluids [25], reaction-diffusion processes or chemical heterogeneous catalysts [6], heat conduction in electrically conducting materials [14]. One interesting question from a mathematical point of view is the search for solutions that vanish at positive infinity, a phenomenon called evanescence. It is also of great importance to describe the behavior of solutions near infinity.

In this paper, we are interested in under what conditions do the linear second-order elliptic problems

have a radial principal eigenvalue? What is the asymptotic behavior of its principal eigenfunction? As the application of our spectrum result, we also study the existence of positive radial solutions of nonlinear singular semilinear elliptic problems with asymptotically linear reaction terms

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\frac{\kappa }{|x|^2} x\cdot \nabla u=\lambda K(|x|)f(u),\ \ \ & x\in \mathbb {R}^N,\\[2ex] u(x)\rightarrow 0, & |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$

(1.3)

where $K\in L^\infty (\mathbb {R}^N)$ is a continuous function, $f\in C((0,\infty ),\mathbb {R})$ and $\kappa \in [0,N-2)$, $N> 2$.

To the best of our knowledge, for the case $\kappa =0$, the existence of principal eigenvalue and the asymptotic behavior of principal eigenfunction of the linear problem

where K satisfies concern decay conditions, have been extensively studied, see Allegretto [2], Naito [26], Brown et al. [7], Brown and Stavrakakis [8], Edelson and Rumbos [17] and the references therein. Typical results for such linear eigenvalue problem are as follows.

Theorem A

(Naito [26, Theorem 1]) Let

$$\begin{aligned} \int _0^\infty rK(r)dt<\infty \end{aligned}$$

(1.5)

be satisfied. Then there exists $\mu _1^{\text {rad}}\in (0, \infty )$ such that

• (i) if $\mu =\mu _1^{\text {rad}},$ then every nontrivial radial entire solution u(r) of $(1.4)_\mu$ has no zero in $[0, \infty )$ and has the asymptotic behavior that

  $$\begin{aligned}\underset{r\rightarrow \infty }{\lim }\ r^{N-2}u(r)\ \text {exists and is a non-zero finite value}; \end{aligned}$$

• (ii) if $\mu \in (\mu _1^{\text {rad}},\infty ),$ then every nontrivial radial entire solution of $(1.4)_\mu$ has at least one zero in $[0,\infty )$.

We will obtain similar results to Theorem A for problem $(1.2)_\lambda$, so we make the following assumptions

• (H1) $0\le \kappa <N-2$ is a constant.

• (H2) $K\in L^\infty (\mathbb {R}^N)$ is a continuous function and there exist $\bar{c} > 0$, $\alpha \in (0,1)$ and $\sigma >N+(N-2)\alpha$ such that

  $$\begin{aligned} K(|x|) \le \frac{\bar{c}}{|x|^{\sigma }}\quad \text {for}\ |x|\gg 1. \end{aligned}$$

Theorem 1.1

Assume that (H1), (H2) hold. Then there exists $\lambda _1^{\text {rad}}\in (0, \infty )$ such that

• (i) if $\lambda =\lambda _1^{\text {rad}},$ then every nontrivial radial entire solution u(r) of $(1.2)_\lambda$ has no zero in $[0, \infty )$ and has the asymptotic behavior that

  $$\begin{aligned} \underset{r\rightarrow \infty }{\lim }\ r^{N-2-\kappa }u(r)\ \text {exists and is a non-zero finite value}. \end{aligned}$$

• (ii) if $\lambda \in (\lambda _1^{\text {rad}},\infty ),$ then every nontrivial radial entire solution of $(1.2)_\lambda$ has at least one zero in $[0,\infty )$.

Corollary 1.1

Assume that (H1), (H2) hold. And let $\varphi _1$ is the positive eigenfunction corresponding to $\lambda _1^{\text {rad}}$ of $(1.2)_\lambda$. Then there exist $0<c_1<c_2$ such that

$$\begin{aligned} \frac{c_1}{|x|^{N-2-\kappa }} \le \varphi _1(|x|) \le \frac{c_2}{|x|^{N-2-\kappa }}, \quad \forall \ |x| \gg 1. \end{aligned}$$

Remark 1.1

Theorem 1.1 is a natural prolongation of Theorem A where the case $\kappa =0$ was studied.

Remark 1.2

In the proof of Theorem 1.1 we use a strategy inspired by Naito [26] and apply the method of variable transformation. Due to the appearance of linear convective term $\frac{\kappa }{|x|^2} x\cdot \nabla u$, we have to introduce a new variable

$$\begin{aligned}\xi (t)=t^{\frac{1}{N-2-\kappa }},\end{aligned}$$

therefore it is natural to assume $\kappa <N-2$. See Sect. 2 for details.

Based on the spectrum of $(1.4)_\mu$, some interesting results on existence of positive solutions for the following semilinear elliptic problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\mu K(|x|)f(u),\ \ \ & x\in \mathbb {R}^N,\\[2ex] u(x)\rightarrow 0, & |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$

(1.6)

when nonlinearity $f:\mathbb {R}\rightarrow \mathbb {R}^+$ is continuous and $f(0)=0$, have been obtained by many authors via the Schauder–Tychonoff fixed point theorem or the Rabinowitz global bifurcation theorem [27], see Edelson and Rumbos [17, 28], Edelson and Furi [16], Dai et al. [13] and the references therein. It is noted that the above literatures focussed on the positone problem, i.e. $f(0)\ge 0$.

However, few studies have been focussed on the existence of positive solutions for a class of semipositone [$f(0)<0$] problem in whole $\mathbb {R}^N$, see Alves et al.[3], Abrantes Santos et al. [1]. One likely reason is that, for semipositone problem, the existence of a positive solution is not so simple, because the standard arguments via the mountain pass theorem combined with the maximum principle do not directly give a positive solution for the problem, and in this case, a very careful analysis must be done.

Thus, based on our spectrum result (Theorem 1.1 and Corollary 1.1), we will obtain the existence of positive solutions of problem (1.3) which is a semipositone nonlinear problem. Assume

(H3) there exist constants $m,c_3,M>0$ and $\zeta :(0,\infty )\rightarrow \mathbb {R}$ is continuous function such that

$$\begin{aligned} f(s)=ms-\frac{c_3}{s^\alpha }+\zeta (s),\quad |\zeta (s)|\le M,\quad \forall \ s\in (0,\infty ). \end{aligned}$$

Obviously, $\lim \nolimits _{s\rightarrow 0^+}f(s)=-\infty$, this means that nonlinearity f is singular at 0. If f has such a singularity, problem (1.3) is referred as infinite semipositone problem. Moreover, $\lim \nolimits _{s\rightarrow +\infty }f(s)=m$, that is, f is asymptotically linear at infinity.

Define

$$\begin{aligned} E:=\left\{ u\in C([0,+\infty ),\mathbb {R}):\ \sup \limits _{r\in [0,+\infty )}\frac{|u(r)|}{\varphi _1(r)}<\infty \right\} , \end{aligned}$$

where $\varphi _1$ is the eigenfunction which is defined in Corollary 1.1. From [16, Lemma 1], E is a Banach space under the norm

$$\begin{aligned} \Vert u\Vert _{\varphi _1}=\sup \limits _{r\in [0,+\infty )}\frac{u(r)}{\varphi _1(r)}. \end{aligned}$$

Define $\Sigma$ be the closure of the set of positive solutions of (1.3) in $[0,\infty )\times E$ and $\lambda _\infty :=\frac{\lambda _1^{\text {rad}}}{m}$.

Theorem 1.2

Assume that (H1)–(H3) hold. Then $\lambda _\infty$ is a bifurcation point from infinity for positive radial solutions. More precisely, there exists an unbounded closed connected set $\mathscr {C}_\infty \subset \Sigma$ that bifurcates from $(\lambda _\infty ,\infty )$. If

$$\begin{aligned} \liminf \limits _{s\rightarrow +\infty }\ \zeta (s)>0\quad \left( \text {resp.}\ \limsup \limits _{s\rightarrow +\infty }\ \zeta (s)<0\right) , \end{aligned}$$

then there exists $\epsilon >0$ such that

$$\begin{aligned}\text {Proj}_\mathbb {R}\ \mathscr {C}_\infty =[\lambda _\infty -\epsilon ,\lambda _\infty )\subset (0,\infty )\ \quad (\text {resp.}\ \text {Proj}_\mathbb {R}\ \mathscr {C}_\infty =(\lambda _\infty ,\lambda _\infty +\epsilon ]\subset (0,\infty )).\end{aligned}$$

Corollary 1.2

Assume that (H1)–(H3) hold. Then there exists $\epsilon >0$ such that (1.3) has at least one positive radial solution provided either

• (i) $\liminf \nolimits _{s\rightarrow +\infty }\ \zeta (s)>0$ and $\lambda \in [\lambda _\infty -\epsilon ,\lambda _\infty );$ or

• (ii) $\limsup \nolimits _{s\rightarrow +\infty }\ \zeta (s)<0$ and $\lambda \in (\lambda _\infty ,\lambda _\infty +\epsilon ]$.

Remark 1.3

Theorem 1.2 extends the main existence results in Joseph and Sankar [22, Theorem 1.8] and Krishnasamy and Sankar [23, Theorem 1.4] where $\kappa =0$.

Remark 1.4

The proof of Theorem 1.2 is inspired by Ambrosetti and Hess [4], Ambrosetti et al. [5]. However, it is clear that they cannot be easily applied to problems such as (1.3) due to the appearance of linear convective term $\frac{\kappa }{|x|^2} x\cdot \nabla u$ and singularity of f. So we need to show new estimates, e.g, the specific expressions of Green’s functions, compactness of operators, construction of auxiliary problems, asymptotic behavior of solutions and so on. See Sect. 3 for details.

Remark 1.5

Existence of positive solutions to the problems (1.6) on the exterior domain of a ball and $f(0)\in (-\infty , 0)$ been extensively studied by several authors when f may be singular at the origin, see Castro and Shivaji [9, 10], Drábek and Sankar [15], Hai and Shivaji [20], Joseph and Sankar [22], Krishnasamy and Sankar [23], Ma [24] and the references along this line. In fact, there is an essential difference between the problem on the exterior domain of a ball and the problem in $\mathbb {R}^N$.

2 Principal Eigenvalue and Asymptotic Behavior of Eigenfunction

A nontrivial radial entire function u(r) where $r= |x|$, is a solution of $(1.2)_\lambda$ if and only if u(r) is a solution of the equation

satisfying $u(0)=c$ and $u'(0)=0$ for some real number $c\ne 0$. Then, since $\tilde{u}(r)=u(r)/u(0)$ is also a solution of $(2.1)_\lambda$ satisfying $\tilde{u}(0)=1, \tilde{u}'(0)=0$, there is no loss of generality in assuming that u satisfies

$$\begin{aligned} u(0)=1, \quad u'(0)=0. \end{aligned}$$

(2.2)

We denote by $u_\lambda (r)$ the solution of the initial value problem $(2.1)_\lambda , (2.2)$. It can be easily verified that, for every $\lambda > 0$, ${u_\lambda }(r)$ is uniquely defined on $[0,\infty )$ and satisfies

$$\begin{aligned} u_\lambda (r)=1-\frac{\lambda }{N-2-\kappa }\int ^r_0\Big [1-\big (\frac{s}{r}\big )^{N-2-\kappa }\Big ]sK(s){u_\lambda }(s)ds,\quad r\ge 0. \end{aligned}$$

In discussing the properties of $u_\lambda (r)$, the results for solutions of equations of the type

$$\begin{aligned} v''+\frac{N-1-\kappa }{r} v'+q(r)v=0, \quad r\ge r_0 \end{aligned}$$

(2.3)

will be effectively used, where $q\in C[r_0,\infty )$, $r_0>0$ is a constant.

We first introduce a transformation. Let ${r=\xi (t)}$ and

$$\begin{aligned} w(t)=tv({\xi (t)}). \end{aligned}$$

(2.4)

Then (2.3) is equivalent to

$$\begin{aligned} v''(\xi (t))+\frac{N-1-\kappa }{\xi (t)} v'(\xi (t))+q(\xi (t))v(\xi (t))=0. \end{aligned}$$

By (2.4), one has

$$\begin{aligned} w'(t)=v(\xi (t))+tv'(\xi (t))\xi '(t) \end{aligned}$$

and

$$\begin{aligned} w''(t)&=v'(\xi (t))\xi '(t)+v'(\xi (t))\xi '(t)+tv''(\xi (t))[\xi '(t)]^2+tv'(\xi (t))\xi ''(t)\\&=2v'(\xi (t))\xi '(t)+tv'(\xi (t))\xi ''(t)+tv''(\xi (t))[\xi '(t)]^2\\&=\big [2\xi '(t)+t\xi ''(t)\big ]\,v'(\xi (t))+t[\xi '(t)]^2\, v''(\xi (t)). \end{aligned}$$

Let $\xi$ satisfy

$$\begin{aligned}\frac{\big [2\xi '(t)+t\xi ''(t)\big ]}{t[\xi '(t)]^2}=\frac{N-1-\kappa }{\xi (t)}. \end{aligned}$$

Then (2.3) can be equivalently written as

$$\begin{aligned} w''(t)+Q(t)w(t)=0,\quad t\ge t_0, \end{aligned}$$

(2.5)

where

$$\begin{aligned} Q(t)=q(\xi (t))[\xi '(t)]^2,\quad {t_0=(N-2-\kappa )r_0^{N-2-\kappa }}. \end{aligned}$$

Lemma 2.1

If $q \in C[r_0, \infty )$ and

$$\begin{aligned} \int ^\infty _{r_0} r^{-N+4+\kappa }|q(r)|dr<\infty , \quad \kappa \in [0,N-2). \end{aligned}$$

(2.6)

Then Eq. (2.3) has a fundamental system of solutions $\{v_1(r), v_2(r)\}$ such that

$$\begin{aligned}\underset{r\rightarrow \infty }{\lim }v_1(r) = 1 \quad \text {and}\quad \underset{r\rightarrow \infty }{\lim }r^{N-2-\kappa }v_2(r) = 1.\end{aligned}$$

Proof

Let us set

$$\begin{aligned}\xi (t)=t^{\frac{1}{N-2-\kappa }}\quad \text {and}\quad w(t)=tv(\xi (t)).\end{aligned}$$

We find that w(t) is a solution of the equation

$$\begin{aligned} w''+Q(t)w=0, \quad t\ge t_0, \end{aligned}$$

(2.7)

where

$$\begin{aligned}Q(t)=[\xi '(t)]^2q(\xi (t))=\frac{1}{(N-2-\kappa )^2}t^{\frac{6-2N+2\kappa }{N-2-\kappa }}q\left( t^{\frac{1}{N-2-\kappa }}\right) , \end{aligned}$$

and that condition (2.6) is rewritten as

$$\begin{aligned} \int _{{t_0}}^\infty t|Q(t)|dt <\infty . \end{aligned}$$

(2.8)

It is well known that if $Q\in C[t_0, \infty )$ satisfies (2.8), then (2.7) has a fundamental system of solutions $\{w_1(t), w_2(t)\}$ such that

$$\begin{aligned}\underset{t\rightarrow \infty }{\lim }\; \frac{w_1(t)}{t}=1 \quad \text {and}\quad \underset{t\rightarrow \infty }{\lim }\; w_2(t)=1, \end{aligned}$$

(see, for example, Hartman [21, Corollary 9.1, p. 380]). Then

$$\begin{aligned}v_1(r) =r^{-N+2+\kappa }w_1(r^{N-2-\kappa }), \quad v_2(r)=r^{-N+2+\kappa }w_2(r^{N-2-\kappa })\end{aligned}$$

give the desired linearly independent solutions of (2.3). $\square$

From (H2), K satisfies

$$\begin{aligned}\int ^\infty _{r_0} r^{-N+4+\kappa }|K(r)|dr<\infty .\end{aligned}$$

By Lemma 2.1, we can conclude that, for each $\lambda >0$, the solution $u_\lambda (r)$ of $(2.1)_\lambda$, (2.2) has a finite number of zeros in $[0,+\infty )$ and satisfies either

$$\begin{aligned} \lim \limits _{r\rightarrow \infty }u_\lambda (r)\ \ \text {exists and is a non-zero finite value,} \end{aligned}$$

(2.9)

or

$$\begin{aligned} \lim \limits _{r\rightarrow \infty }r^{N-2-\kappa }u_\lambda (r)\ \ \text {exists and is a non-zero finite value.} \end{aligned}$$

(2.10)

Now by the same argument in the proof of [26, Lemmas 2–6], with obvious changes, we may deduce the following desired results.

Lemma 2.2

Let $r_0>0$. Suppose that $q\in C[r_0,\infty )$ and $q(r)\ge 0$ for $r\ge r_0$. If there is a solution v(r) of (2.3) having no zero in $[r_0,\infty ),$ then

$$\begin{aligned} (r^{N-2-\kappa }-r_0^{N-2-\kappa })\int _r^\infty s^{-N+3+\kappa }q(s)ds\le N-2-\kappa ,\quad r\ge r_0. \end{aligned}$$

(2.11)

Lemma 2.3

Let $0<\eta <\lambda$. If $u_\lambda (r)>0$ on $[0,r_\lambda ),$ where $0<r_\lambda \le \infty ,$ then $u_\eta (r)\ge u_\lambda (r)(>0)$ on $[0,r_\lambda )$.

Lemma 2.4

Let $0<\eta <\lambda$.

• (i) If $u_\lambda (r)>0$ for $r\ge 0,$ then $u_\eta (r)\ge u_\lambda (r)$ for $r\ge 0;$ in particular $u_\eta (r)>0$ for $r\ge 0$.

• (ii) If $u_\eta (r)$ has a zero in $[0,\infty ),$ then $u_\lambda (r)$ has a zero in $[0,\infty )$. Let $r_\lambda ^1$ and $r_\eta ^1$ be the first zeros of $u_\lambda (r)$ and $u_\eta (r),$ respectively. Then $r_\lambda ^1\ge r_\eta ^1$.

Lemma 2.5

Assume that (H1)–(H2) hold. Then for each $\lambda >0,$

$$\begin{aligned} |u_\lambda (r)|\le \exp (\lambda P/(N-2-\kappa )),\quad r\ge 0, \end{aligned}$$

where $P=\int _0^\infty sK(s)ds$.

Lemma 2.6

Assume that (H1)–(H2) hold. Then for $\lambda >0$ and $\eta >0,$

$$\begin{aligned} |u_\lambda (r)-u_\eta (r)|\le |\lambda -\eta |\frac{P}{N-2-\kappa }\exp (\frac{\lambda +\eta }{N-2-\kappa }P),\quad r\ge 0. \end{aligned}$$

Lemma 2.7

Assume that (H1)-(H2) hold. There is $\lambda '>0$ such that if $\lambda \in (0,\lambda ')$, then $u_\lambda (r)>0$ for $r\ge 0$ and $u_\lambda (r)$ satisfies (2.9).

Proof

Let $\lambda '=\frac{N-2-\kappa }{{\int _0^\infty sK(s)ds}}$, then for any $\lambda \in (0,\lambda ')$, denote

$$\begin{aligned} k(\lambda )=1-\frac{\lambda }{N-2-\kappa }\int _0^\infty sK(s)ds>0. \end{aligned}$$

We will consider an operator equation $u_\lambda =\hat{T}u_\lambda$ with $\hat{T}$ by

$$\begin{aligned}\hat{T}u_\lambda (r)=1-\frac{\lambda }{N-2-\kappa }\int ^r_0\Big [1-\big (\frac{s}{r}\big )^{N-2-\kappa }\Big ]sK(s)u_\lambda (s)ds,\quad r>0.\end{aligned}$$

Let

$$\begin{aligned} Y=\{u_\lambda \in C[0,\infty ): k(\lambda )\le u_\lambda (r)\le 1\, \text {for}\ r\ge 0\}, \end{aligned}$$

which is a closed convex subset of $C[0,\infty )$.

We first prove that the operator $\hat{T}$ maps Y into itself. Indeed, if $u_\lambda \in Y$, then obviously $\hat{T}u_\lambda (r)\le 1$ and

$$\begin{aligned} \begin{aligned} \hat{T}u_\lambda (r)&\ge 1-\frac{\lambda }{N-2-\kappa }\int _0^rsK(s)u_\lambda (s)ds\\&\ge 1-\frac{\lambda }{N-2-\kappa }\int _0^rsK(s)ds\\&\ge 1-\frac{\lambda }{N-2-\kappa }\int _0^\infty sK(s)ds\\&= k(\lambda ) \end{aligned} \end{aligned}$$

for any $r\ge 0$.

Next, we show that $\hat{T}$ is continuous. If $u_{\lambda ,m}\in Y, m=1,2,\ldots$ and $u_{\lambda ,m}\rightarrow u_\lambda$ as $m\rightarrow \infty$ uniformly on every compact subinterval of $[0,\infty )$, then $u_\lambda \in Y$ and we have

$$\begin{aligned} |\hat{T}u_{\lambda ,m}(r)-\hat{T}u_\lambda (r)|\le \frac{\lambda }{N-2-\kappa }\int _0^rsK(s)|u_{\lambda ,m}(s)-u_\lambda (s)|ds \end{aligned}$$

for $r\ge 0$. By Lebesgue’s dominated convergence theorem, $\hat{T}u_{\lambda ,m}\rightarrow \hat{T}u_\lambda$ as $m\rightarrow \infty$. Finally, we prove that $\hat{T}Y$ is relative compact. In fact, $\hat{T}Y$ is clearly uniformly bounded at every point of $[0,\infty )$, and from the relation

$$\begin{aligned} |(\hat{T}u_\lambda )'(r)|=\Big |\int _0^r\big (\frac{s}{r}\big )^{N-1-\kappa }K(s)u_\lambda (s)ds\Big |\le \int _0^rK(s)ds, \end{aligned}$$

it follows that $\hat{T}u_\lambda$ is equicontinuous at every point in $[0,\infty )$.

By Schauder–Tychonoff fixed point theorem, there are positive constant $k(\lambda )$ such that the initial value problem $(2.1)_\lambda$, (2.2) has a solution $u_\lambda (r)$ satisfying $0<k(\lambda )\le u_\lambda (r)\le 1$ for $r\ge 0$. Moreover,

$$\begin{aligned} -\left( r^{N-1-\kappa }u_\lambda '(r)\right) '=\lambda r^{N-1-\kappa }K(r)u_\lambda (r)\ge 0,\quad r\in [0,\infty ), \end{aligned}$$

implies that $r^{N-1-\kappa }u_\lambda '(r)$ is non-increasing in $[0,\infty )$. This fact together with $u_\lambda '(0)=0$ yields

$$\begin{aligned} u_\lambda '(r)\le 0, \quad r\in [0,\infty ), \end{aligned}$$

and subsequently $u_\lambda (r)$ is non-increasing in $[0,\infty )$. Thus there exists $c_\lambda \ge k(\lambda )>0$ such that

$$\begin{aligned} \lim \limits _{r\rightarrow \infty } u_\lambda (r)=c_\lambda . \end{aligned}$$

This complete the proof of Lemma 2.7.$\square$

Proof of Theorem 1.1

Define the subsets $\Lambda _0$ and $\Lambda _1^+$ of $\mathbb {R}_+=(0,+\infty )$ by

$$\begin{aligned} \Lambda _0=\{\lambda \in \mathbb {R}_+:{u_\lambda (r)>0}\ \ \text {for}\ r\ge 0\} \end{aligned}$$

and

$$\begin{aligned} \Lambda _1^+=\{\lambda \in \mathbb {R}_+:u_\lambda (r)\ \text {has at least one zero in}\ [0,\infty )\}. \end{aligned}$$

It is clear that $\mathbb {R}_+=\Lambda _0\cup \Lambda _1^+$, $\Lambda _0\cap \Lambda _1^+=\emptyset$. By Lemma 2.7, $\Lambda _0$ is non-empty. It can be shown that $\Lambda _1^+$ is also non-empty. To see this, assume the contrary. Then $\mathbb {R}_+=\Lambda _0$, i.e., $u_\lambda (r)>0$ on $[0,\infty )$ for every $\lambda >0$. Applying Lemma 2.2 to the case of $q(r)=\lambda K(r)$, we find that

$$\begin{aligned} \lambda (r^{N-2-\kappa }-r_0^{N-2-\kappa })\int _r^\infty s^{-N+3+\kappa }K(s)ds\le N-2-\kappa ,\quad r\ge r_0, \end{aligned}$$

(2.12)

for all $\lambda >0$, where $r_0>0$ is an arbitrarily fixed number. In (2.12), fix r and let $\lambda \rightarrow +\infty$. Then we are led to a contradiction. Thus $\Lambda _1^+$ is non-empty. Besides, $\Lambda _1^+$ is an open subset of $\mathbb {R}_+$ because of the continuous dependence of $u_\lambda (r)$ on $\lambda$. From (i) of Lemma 2.4 we see that if $0<\eta <\lambda$ and $\lambda \in \Lambda _0$, then $\eta \in \Lambda _0$. Therefore we can conclude that $\Lambda _0$ and $\Lambda _1^+$ are of the forms

$$\begin{aligned} \Lambda _0=\left( 0,\lambda _1^{\text {rad}}\right] \quad \text {and}\quad \Lambda _1^+=\left( \lambda _1^{\text {rad}},\infty \right) \end{aligned}$$

for some $\lambda _1^{\text {rad}}>0$.

Consider the subsets $\Lambda _0^n$ and $\Lambda _0^p$ of $\mathbb {R}_+$ defined by

$$\begin{aligned} \Lambda _0^n=\{\lambda \in \mathbb {R}_+:u_\lambda (r)\ \text {is positive on}\ [0,\infty )\ \text {and satisfies}\ (2.9)\} \end{aligned}$$

and

$$\begin{aligned} \Lambda _0^p=\{\lambda \in \mathbb {R}_+:u_\lambda (r)\ \text {is positive on}\ [0,\infty )\ \text {and satisfies}\ (2.10)\}. \end{aligned}$$

We have $\Lambda _0=\Lambda _0^n\cup \Lambda _0^p$ and $\Lambda _0^n\cap \Lambda _0^p=\emptyset$. Lemma 2.7 means that $\Lambda _0^n$ is nonempty. By (i) of Lemma 2.4 we see that if $0<\eta <\lambda$ and $\lambda \in \Lambda _0^n$, then $\eta \in \Lambda _0^n$. It follows from Lemma 2.6 that $\Lambda _0^n$ is an open subset of $\mathbb {R}_+$. Therefore $\Lambda _0^n$ is of the form $\Lambda _0^n=(0,\lambda _0)$ for some $\lambda _0$, $0<\lambda _0\le \lambda _1^{\text {rad}}$. Then $\Lambda _0^p=[\lambda _0,\lambda _1^{\text {rad}}]$. The set $\Lambda _0^p$ may consist of a single point.$\square$

3 Component for Elliptic Problem in $\mathbb {R}^N$

Recall

$$\begin{aligned}E:=\left\{ u\in C[0,\infty ) \mid \sup \limits _{r\in [0,\infty )}\frac{|u(r)|}{\varphi _1(r)}<\infty \right\} \end{aligned}$$

with the norm

$$\begin{aligned}\Vert u\Vert _{\varphi _1}=\sup \limits _{r\in [0,\infty )}\frac{u(r)}{\varphi _1(r)}.\end{aligned}$$

$C_b[0,\infty )$ denotes the set of all continuous functions on $[0,\infty )$ which are bounded.

3.1 A Compact Operator

Note that by the assumption on K and f, we have sK(s)f(v(s)), $s^2K(s)f(v(s))\in L^1(0,\infty )$.

Lemma 3.1

For $v\in E,$ the linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }u'(r))'=\lambda r^{N-1-\kappa } K(r)f(v),& r\in (0,\infty ),\\ u'(0)=0,& u(\infty )=0, \end{array}\right. \end{aligned}$$

(3.1)

has a unique solution $u\in C_b[0,\infty )$.

Proof

It is easy to see that the solution u of (3.1) is given by

$$\begin{aligned} u(r)=\lambda \int _0^\infty G(r,s)s^{N-1-\kappa }K(s)f(v(s))ds, \end{aligned}$$

where $G:[0,\infty )\times [0,\infty )\rightarrow [0,\infty )$ is the Green’s function of problem

$$\begin{aligned} -(r^{N-1-\kappa }u'(r))'=0,\quad \ r\in (0,\infty ),\quad \ u'(0)=0,\quad \ u(\infty )=0 \end{aligned}$$

which is explicitly given by

$$\begin{aligned} G(r,s)=\left\{ \begin{array}{ll} \frac{r^{2+\kappa -N}}{N-2-\kappa },& 0<s\le r<\infty ,\\ \frac{s^{2+\kappa -N}}{N-2-\kappa },& 0<r\le s<\infty .\\ \end{array}\right. \end{aligned}$$

Thus

$$\begin{aligned} u(r)&=\lambda \int _0^r \frac{r^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }K(s)f(v(s))ds \nonumber \\&\quad \ +\lambda \int _r^\infty \frac{s^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }K(s)f(v(s))ds \end{aligned}$$

(3.2)

and

$$\begin{aligned} u'(r)=-\frac{1}{r^{N-1-\kappa }}\int _0^r s^{N-1-\kappa }K(s)f(v(s))ds. \end{aligned}$$

(3.3)

By (H2), there exists $r_0>1,$ such that

$$\begin{aligned} K(r) \le \frac{\bar{c}}{r^{\sigma }},\quad \text {for}\ r\ge r_0. \end{aligned}$$

(3.4)

This together with (H3) imply

$$\begin{aligned} \begin{aligned} |u(r)|&\le \lambda \int _0^r \frac{r^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }K(s)|f(v(s))|ds\\&\quad \ +\lambda \int _r^\infty \frac{s^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }K(s)|f(v(s))|ds\\&\le \lambda r^{N-2-\kappa }\frac{r^{2+\kappa -N}}{N-2-\kappa }\int _0^rsK(s)|f(v(s))|ds\\&\quad +\lambda \int _r^\infty \frac{s}{N-2-\kappa }K(s)|f(v(s))|ds\\&=\frac{\lambda }{N-2-\kappa }\Big (\int _0^rsK(s)|f(v(s))|ds+\int _r^\infty sK(s)|f(v(s))|ds\Big )\\&=\frac{\lambda }{N-2-\kappa }\Big (\int _0^{r_0}sK(s)|f(v(s))|ds+\int _{r_0}^\infty sK(s)|f(v(s))|ds\Big )\\&\le C+\frac{\lambda }{N-2-\kappa }\int _{r_0}^\infty sK(s)|f(v(s))|ds\\&\le C+\frac{\lambda }{N-2-\kappa }\int _{r_0}^\infty s\frac{\bar{c}}{s^{\sigma }}|mv(s)-\frac{c_3}{v(s)^\alpha }+\zeta (s)|ds, \end{aligned} \end{aligned}$$

(3.5)

where C is a positive constant. Using Corollary 1.1 and the fact $v\in E$ in (3.4), it concludes

$$\begin{aligned} |u(r)|\le C+D\int _{r_0}^\infty \frac{s}{s^{\sigma }}\left[ \frac{1}{s^{N-2-\kappa }}+(s^{N-2-\kappa })^\alpha +M\right] ds<\infty , \end{aligned}$$

(3.6)

where D is a positive constant.

Next, we show that $u'(0)=0$.

Let $K_M=\underset{s\in [0,1]}{\max }K(s)$ and $F_M=\underset{s\in [0,1]}{\max }|f(v(s))|$. Then

$$\begin{aligned} \begin{aligned} |u'(r)|&\le \frac{1}{r^{N-1-\kappa }}\int _0^r s^{N-1-\kappa }K(s)|f(v(s))|ds\\&\le \frac{1}{r^{N-1-\kappa }}\int _0^r s^{N-1-\kappa }K_Mf_Mds\rightarrow 0, \end{aligned} \end{aligned}$$

thus $u'(0)=0$.

Finally we prove that $u(\infty )=0$.

Note that Eq. (3.2) can be rewritten to

$$\begin{aligned} u(r)=\lambda \frac{1}{N-2-\kappa }R_1(r)+\lambda \frac{1}{N-2-\kappa }R_2(r), \end{aligned}$$

(3.7)

where

$$\begin{aligned}R_1(r)=r^{2+\kappa -N}\int _0^r s^{N-1-\kappa }K(s)f(v(s))ds,\ \ \ \ R_2(r)=\int _r^\infty sK(s)f(v(s))ds.\end{aligned}$$

Since

$$\begin{aligned} \int _{0}^\infty sK(s)f(v(s))ds&\le C_1+\int _{r_0}^\infty sK(s)|f(v(s))|ds\\&\le C_1+\int _{r_0}^\infty s\frac{\bar{c}}{s^{\sigma }}\left| mv(s)-\frac{c_3}{v(s)^\alpha }+\zeta (s)\right| ds\\&\le C_1+\int _{r_0}^\infty \frac{s}{s^{\sigma }}\left[ \frac{1}{s^{N-2-\kappa }}+(s^{N-2-\kappa })^\alpha +M\right] ds,\\ \end{aligned}$$

where $C_1$ is a positive constant, we have $s K(s)f(v(s))\in L^1(0,\infty )$. Obviously, this fact deduces that

$$\begin{aligned} \lim \limits _{r\rightarrow \infty }R_2(r)=0. \end{aligned}$$

(3.8)

So it suffices to show that

$$\begin{aligned} \lim \limits _{r\rightarrow \infty }R_1(r)=0. \end{aligned}$$

(3.9)

In fact,

$$\begin{aligned} \begin{aligned} R_1(r)&=\frac{r^{2+\kappa -N}}{N-2-\kappa }\Big [\int _0^{r_0} s^{N-1-\kappa }K(s)f(v(s))ds+\int _{r_0}^r s^{N-1-\kappa }K(s)f(v(s))ds\Big ]\\&\le \frac{r^{2+\kappa -N}}{N-2-\kappa }\Big [C_2+\int _{r_0}^r s^{N-1-\kappa }K(s)\big [mv(s)-\frac{c_3}{v(s)^\alpha }+\zeta (s)\big ]ds\Big ]\\&\le \frac{r^{2+\kappa -N}}{N-2-\kappa }\Big [C_2+ D_1\int _{r_0}^r s^{N-1-\kappa }\frac{1}{s^{\sigma }}\big [\frac{1}{s^{N-2-\kappa }}+(s^{N-2-\kappa })^\alpha +M\big ]ds\Big ]\\&= C_3r^{2+\kappa -N}+D_2r^{2+\kappa -N}\int _{r_0}^rs^{1-\sigma }+[s^{N-1-\kappa -\sigma +(N-2-\kappa )\alpha }+Ms^{N-1-\kappa -\sigma }]ds\\&= C_3r^{2+\kappa -N}+D_2r^{2+\kappa -N}\big [\frac{ s^{2-\sigma }}{2-\sigma }+\frac{s^{N-\kappa -\sigma +(N-2-\kappa )\alpha }}{N-\kappa -\sigma +(N-2-\kappa )\alpha }+\frac{Ms^{N-\kappa -\sigma }}{N-\kappa -\sigma }\big ]\Big |_{r_0}^r\\&=C_3r^{2+\kappa -N}+D_2r^{2+\kappa -N}\big [\frac{r^{2-\sigma }}{2-\sigma }+\frac{r^{N-\kappa -\sigma +(N-2-\kappa )\alpha }}{N-\kappa -\sigma +(N-2-\kappa )\alpha }+\frac{Mr^{N-\kappa -\sigma }}{N-\kappa -\sigma }\big ]\\&-D_2r^{2+\kappa -N}\big [\frac{r_0^{2-\sigma }}{2-\sigma }+\frac{r_0^{N-\kappa -\sigma +(N-2-\kappa )\alpha }}{N-\kappa -\sigma +(N-2-\kappa )\alpha }+\frac{Mr_0^{N-\kappa -\sigma }}{N-\kappa -\sigma }\big ]\\&=C_3r^{2+\kappa -N}+D_2\big [\frac{r^{2+\kappa -N+2-\sigma }}{2-\sigma }+\frac{r^{2-\sigma +(N-2-\kappa )\alpha }}{N-\kappa -\sigma +(N-2-\kappa )\alpha }+\frac{Mr^{2-\sigma }}{N-\kappa -\sigma }\big ]\\&-D_2r^{2+\kappa -N}\big [\frac{r_0^{2-\sigma }}{2-\sigma }+\frac{r_0^{N-\kappa -\sigma +(N-2-\kappa )\alpha }}{N-\kappa -\sigma +(N-2-\kappa )\alpha }+\frac{Mr_0^{N-\kappa -\sigma }}{N-\kappa -\sigma }\big ], \end{aligned} \end{aligned}$$

(3.10)

where $C_2, C_3, D_1, D_2$ are positive constants. By (H2), we may deduce that

$$\begin{aligned} \sigma>2,\quad \sigma>\kappa +2-(N-2),\quad \sigma >2+(N-2-\kappa )\alpha . \end{aligned}$$

(3.11)

Thus, combining (3.10) and (3.11) and using $\lim \nolimits _{r\rightarrow \infty }r^{2+\kappa -N}=0$, it may deduce that (3.9) is valid. Furthermore, we get desired result $u(\infty )=0$ by using (3.8) and (3.9).$\square$

For $u\in E$, let Tu be defined by

$$\begin{aligned} Tu=\lambda \int _0^\infty G(r,s)s^{N-1-\kappa }K(s)f(u(s))ds. \end{aligned}$$

Lemma 3.2

$T:E\rightarrow C_b[0,\infty )$ is completely continuous.

Proof

First, we show that T is continuous. Let $u_m$ be a sequence in E such that $u_m\rightarrow u\ (m\rightarrow \infty )$ in E. Then,

$$\begin{aligned} \begin{aligned} |Tu_m-Tu|&=\lambda \Big |\int _0^r \frac{r^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }K(s)(f(u_m(s))-f(u(s)))ds\\ &\quad +\int _r^\infty \frac{s^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }K(s)(f(u_m(s))-f(u(s)))ds\Big |\\&\le \lambda \Big |\int _0^r \frac{r^{2+\kappa -N}s^{N-2-\kappa }}{N-2-\kappa }sK(s)(f(u_m(s))-f(u(s)))ds\\ &+\int _r^\infty \frac{s^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }K(s)(f(u_m(s))-f(u(s)))ds\Big |\\&\le \frac{\lambda }{N-2-\kappa }\int _0^\infty s |K(s)||(f(u_m(s))-f(u(s)))|ds. \end{aligned} \end{aligned}$$

By using the dominated convergence theorem, it follows that $T:E\rightarrow C_b[0,\infty )$ is continuous.

Next, we show that $T:E\rightarrow C_b[0,\infty )$ is compact. Let $\{u_m\}$ be a bounded sequence in E. By (3.3), we have $\{Tu_m\}=\{u_m\}$ is uniformly bounded. Since $s^{N-1-\kappa }G(r,s)$ is Lipschitz continuous, there exists a $\hat{C}$ such that $|s^{N-1-\kappa }(G(x,s)-G(y,s))|\le \hat{C}|x-y|$ for every $x,y\in (0,\infty )$ and $s\in (0,\infty )$. Then,

$$\begin{aligned} |u_m(x)-u_m(y)|\le \lambda \hat{C}|x-y|\int _0^\infty |K(s)f(u_m(s))|ds\le \tilde{C}|x-y|, \end{aligned}$$

where $\tilde{C}$ is positive constant independent of m. Thus $\{Tu_m\}$ is uniformly equi-continuous. Hence by Arzela–Ascoli theorem, $\{Tu_m\}$ has a convergent subsequence in $[0,M_1]$ for every $M_1>0$, which we can again denote by $\{Tu_m\}$. Hence for given $\epsilon >0$, there exists $N\in \mathbb {N}$ such that

$$\begin{aligned} |Tu_{m_1}(r)-Tu_{m_2}(r)|\le \frac{\epsilon }{2}, \quad \text {for all}\ r\in [0,{M_1}],\quad m_1,m_2\ge N. \end{aligned}$$

(3.12)

Also,

$$\begin{aligned} \begin{aligned} |Tu_m(r)|&\le \frac{\lambda }{N-2-\kappa }\Big (\Big |\frac{1}{r^{N-2-\kappa }}\int _0^rs^{N-1-\kappa }K(s)f(u_m(s))ds\Big | +\Big |\int _0^rsK(s)f(u_m(s))ds\Big |\Big )\\&\le \frac{\lambda }{N-2-\kappa }\Big (\Big |\frac{1}{r}\int _0^rs^{2}K(s)f(u_m(s))ds\Big | +\Big |\int _0^rsK(s)f(u_m(s))ds\Big |\Big )\\&\le \frac{\lambda }{N-2-\kappa }\Big (\frac{C_1}{r}+\Big |\int _0^rsK(s)f(u_m(s))ds\Big |\Big ), \end{aligned} \end{aligned}$$

where $C_1=\Big |\int _0^\infty s^{2}K(s)f(u_m(s))ds\Big |$. Using the similar method in the proof of Lemma 3.1, with obvious changes, one has $|Tu_m(r)|\rightarrow 0$ as $r\rightarrow +\infty$. Therefore there exists $\hat{M}\gg 1$ such that for all $r>\hat{M}$,

$$\begin{aligned} |Tu_m(r)|\le \frac{\epsilon }{4}, \end{aligned}$$

from which it follows that

$$\begin{aligned} |Tu_{m_1}(r)-Tu_{m_2}(r)|\le \frac{\epsilon }{2},\quad \text {for all}\ r>\hat{M},\quad m_1,m_2\ge N. \end{aligned}$$

(3.13)

By (3.12) and (3.13), $\{Tu_m\}$ is a Cauchy sequence and hence convergent. Hence, T is compact. $\square$

3.2 Auxiliary Problems

We first give the existence result of a linear problem, which is crucial for us to construct the auxiliary problems.

Lemma 3.3

Assume (H1)–(H2) hold. Then problem

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }w'(r))'=r^{N-1-\kappa }K(r)\Big [\frac{c_3}{\varphi _1^\alpha (r)}+M\Big ],& r\in (0,\infty ),\\ w'(0)=0,& w(\infty )=0, \end{array} \right. \end{aligned}$$

(3.14)

has a unique positive solution $\bar{w}\in C_b[0,\infty ),$ where $c_3>0,$ $\alpha \in (0,1)$ and $M>0$ are defined in (H3).

Proof

By (H1)–(H2) and Corollary 1.1, we can obtain

$$\begin{aligned}p(r):=\max \limits _{r\in [0,\infty )}K(r)\Big [\frac{c_3}{\varphi _1^\alpha (r)}+M\Big ]<\frac{\hat{c}_1}{r^{2N-2-\kappa }}+\frac{\hat{c}_2}{r^{\sigma }}\end{aligned}$$

for r large and some constants $\hat{c}_1,\hat{c}_2>0$. The solution w of (3.14) satisfy

$$\begin{aligned} \begin{aligned} |w(r)|&\le \lambda \int _0^r \frac{r^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }p(s)ds+\lambda \int _r^\infty \frac{s^{2+\kappa -N}}{N-2-\kappa }s^{N-1-\kappa }p(s)ds\\&\le \lambda r^{N-2-\kappa }\frac{r^{2+\kappa -N}}{N-2-\kappa }\int _0^rsp(s)ds+\lambda \int _r^\infty \frac{s}{N-2-\kappa }p(s)ds\\&=\frac{\lambda }{N-2-\kappa }\Big (\int _0^rsp(s)ds+\int _r^\infty sp(s)ds\Big )\\&=\frac{\lambda }{N-2-\kappa }\Big (\int _0^{r_0}sp(s)ds+\int _{r_0}^\infty sp(s)ds\Big )\\&\le C+\frac{\lambda }{N-2-\kappa }\int _{r_0}^\infty sp(s)ds\\&\le C+\frac{\lambda }{N-2-\kappa }\int _{r_0}^\infty s\big [\frac{\hat{c}_1}{s^{2N-2-\kappa }}+\frac{\hat{c}_2}{s^{\sigma }}\big ]ds, \end{aligned} \end{aligned}$$

where $r_0\gg 1$ and C are positive constants. We further obtain

$$\begin{aligned} |w(r)|\le C+D\int _{r_0}^\infty s\big [\frac{1}{s^{2N-2-\kappa }}+\frac{1}{s^{\sigma }}\big ]ds<\infty , \end{aligned}$$

where D is a positive constant. Similar to the argument of Lemma 3.1, with obvious changes, we may obtain that $w'(0)=w(\infty )=0$. Thus there exists a unique positive solution $\bar{w}\in C_b[0,\infty )$.$\square$

Let $\tilde{u}(r):=\max \{u(r),\varphi _1(r):\ r\in [0,\infty )\}$. Consider an auxiliary problem

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }\tilde{u}'(r))'=\lambda r^{N-1-\kappa }K(r) f(\tilde{u}),& r\in (0,\infty ),\\ \tilde{u}'(0)=0,& \tilde{u}(\infty )=0. \end{array} \right. \end{aligned}$$

(3.15)

In order to find a positive solution to problem (1.3), it suffices to show that $\tilde{u}$ is a solution of (3.15) with $\tilde{u}>\varphi _1$ in $[0,\infty )$.

Let

$$\begin{aligned}{\tilde{g}(r,s)}=f(s)+\frac{c_3}{\varphi _1^\alpha (r)}+M,\quad \ r\in [0,\infty ),\quad s\in \mathbb {R}^+.\end{aligned}$$

Let $\hat{w}=\lambda \bar{w}$ and $z(r)=\tilde{u}(r)+\hat{w}(r), \ r\in [0,\infty )$, where $\bar{w}$ is obtained in Lemma 3.3. Obviously, for any $r\in [0,\infty )$,

$$\begin{aligned}\tilde{g}(r,z-\hat{w})\ge 0\end{aligned}$$

since $\tilde{u}(r)\ge \varphi _1(r)$ and $|\zeta (s)|\le M$.

In the following, we will consider the existence of solutions for the auxiliary positone problem

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }z'(r))'=\lambda r^{N-1-\kappa }K(r) \tilde{g}(r,z-\hat{w}),& r\in (0,\infty ),\\ z'(0)=0,& z(\infty )=0. \end{array} \right. \end{aligned}$$

(3.16)

It is easy to verify that, $(\lambda ,\tilde{u})$ is a positive solution of (3.15) if and only if $z=\tilde{u}+\hat{w}$ is a solution of (3.16).

3.3 Bifurcation of Positive Solutions from Infinity

We first consider the bifurcation results of problem (3.16) from infinity. For $z\in E$, define

$$\begin{aligned} \Phi (\lambda , z) :=z-\lambda T[\tilde{g}(\cdot ,z-\hat{w})]. \end{aligned}$$

(3.17)

Plainly, any $z> 0$ such that $\Phi (\lambda , z)=0$ is a positive solution of (3.16).

Lemma 3.4

For every compact interval $\Lambda \subset \mathbb {R}^+\setminus \{\lambda _\infty \},$ there exists $R> 0$ such that

$$\begin{aligned} \Phi (\lambda , z) \ne 0,\quad \forall \lambda \in \Lambda , \quad \forall \ ||z||_{\varphi _1}\ge R. \end{aligned}$$

(3.18)

Proof

Suppose on the contrary that there exists a sequence $(\mu _n,z_n)$ satisfying $\mu _n\rightarrow \mu \ge 0$, $\mu \ne \lambda _\infty$, and $||z_n||_{\varphi _1} \rightarrow \infty$ such that

$$\begin{aligned}z_n= \mu _n T[\tilde{g}(\cdot ,z_n-\hat{w})]. \end{aligned}$$

Setting $y_n=z_n||z_n||_{\varphi _1}^{-1}$. Then $y_n$ satisfies the following

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }y_n'(r))'=\mu _nr^{N-1-\kappa }K(r)\Big [my_n(r)-\frac{m\hat{w}(r)}{\Vert z_n\Vert _{\varphi _1}}\\ \ \quad \ \ \quad \ \ \quad \ \quad \ \ \quad \quad \ \ +\Big (\frac{c_3}{\varphi _1^\alpha (r)}-\frac{c_3}{\tilde{u}_n^\alpha (r)}+\zeta (\tilde{u}_n)+M\Big )\frac{1}{\Vert z_n\Vert _{\varphi _1}}\Big ], & r\in (0,\infty ),\\ y_n'(0)=0,& y_n(\infty )=0. \end{array} \right. \end{aligned}$$

(3.19)

Since the right end of the Eq. (3.19) is bounded uniformly in E, it follows from the argument in Lemma 3.1 that $y_n\rightarrow y$ and $y\in C_b[0,\infty )$. Moreover, as $\Vert z_n\Vert _{\varphi _1}\rightarrow \infty$, then

$$\begin{aligned} 0\le \frac{mK(r)\hat{w}}{\Vert z_n\Vert _{\varphi _1}}\rightarrow 0 \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} 0\le \Big (\frac{c_3}{\varphi _1^\alpha (r)}-\frac{c_3}{\tilde{u}_n^\alpha (r)}+\zeta (\tilde{u}_n)+M\Big )\frac{K(r)}{\Vert z_n\Vert _{\varphi _1}}\le \frac{c_3K(r)}{\phi ^\alpha (r)\Vert z_n\Vert _{\varphi _1}}+\frac{2K(r)M}{\Vert z_n\Vert _{\varphi _1}}\rightarrow 0 \end{aligned} \end{aligned}$$

uniformly for $r\in [0,\infty )$. Consequently, y is such that $||y||_{\varphi _1} = 1$ and satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }y'(r))'=\mu mr^{N-1-\kappa }K(r)y(r),& r\in (0,\infty ),\\ y'(0)=0,& y(\infty )=0. \end{array} \right. \end{aligned}$$

Since $y\ge 0$ and $||y||_{\varphi _1} = 1$, we infer that $\mu = \lambda _\infty$, a contradiction that proves the statement.$\square$

Lemma 3.5

If $\lambda >\lambda _\infty ,$ then there exists $R> 0$ such that

$$\begin{aligned}\Phi (\lambda , z) \ne \tau \phi ,\quad \tau \ge 0,\quad ||z||_{\varphi _1}\ge R. \end{aligned}$$

Proof

Let us assume that for some sequence $(z_n)_{n\in \mathbb {N}}$ in E with $||z_n||_{\varphi _1}\rightarrow \infty$ and numbers $\tau _n\ge 0$, $\Phi (\lambda ,z_n)=\tau _n\varphi _1$. Then

$$\begin{aligned} T^{-1}z_n=\lambda \tilde{g}(\cdot ,z_n-\hat{w})+\tau _n\lambda _\infty \varphi _1, \end{aligned}$$

(3.20)

and since $\tau _n\lambda _\infty \varphi _1\ge 0$ in $[0,\infty )$, it follows by maximum principle that $z_n\ge 0$ in $[0,\infty )$.

Let $z_n=v_n+s_n\varphi _1$. Then $s_n=\left( \int _0^\infty z_n\varphi _1 dr\right) \left( \int _0^\infty \varphi _1^2 dr\right) ^{-1}>0$, $\forall n\in \mathbb {N}$. We first prove that

$$\begin{aligned}s_n\rightarrow \infty , \quad \text {as}\ n \rightarrow \infty . \end{aligned}$$

Suppose that the sequence $(s_n)_{n\in \mathbb {N}}$ is bounded. Then $||v_n||_{\varphi _1}\rightarrow \infty$. Set $y_n:= ||v_n||_{\varphi _1}^{-1}v_n$. Let $\mathcal {P}: E\rightarrow E$ be the (continuous) projection of E onto $\text {ker}(T^{-1})$ parallel to $\mathbb {R}\varphi _1$. Applying $\mathcal {P}$ to

$$\begin{aligned}\Phi (\lambda ,z_n)=\tau _n\varphi _1,\end{aligned}$$

we obtain

$$\begin{aligned} \begin{aligned} -y_n''-\frac{N-1-\kappa }{r} y_n'&= \lambda ||v_n||_{\varphi _1}^{-1}\mathcal {P}K(r)\tilde{g}(r,z_n-\hat{w})\\&=\lambda m K(r)y_n +\lambda ||v_n||_{\varphi _1}^{-1} \mathcal {P}K(r)\\&\quad \times \Big [\left( \frac{c_1}{\varphi _1^\alpha }-\frac{c_1}{\tilde{u}_n^\alpha }\right) +(M+\zeta (\tilde{u}_n))-m\hat{w}(r)\Big ]. \end{aligned} \end{aligned}$$

Since $\Big \{K(\cdot )\Big [\left( \frac{c_1}{\varphi _1^\alpha }-\frac{c_1}{\tilde{u}_n^\alpha }\right) +(M+\zeta (\tilde{u}_n))-m(\cdot )\hat{w}\Big ]\Big \}$ is bounded in E, we infer as in the proof of Lemma 3.4 that (for a subsequence) $y_n\rightarrow \bar{y}$ in $C[0,\infty )$, $||\bar{y}||_{\varphi _1}=1$, and $\int _0^\infty \bar{y}(r)\varphi _1(r) dr=0$. Thus $\bar{y}$ has to change sign in $[0,\infty )$. On the other hand, $z_n\ge 0$ in $[0,\infty )$ implies that $y_n\ge -s_n||v_n||_{\varphi _1}^{-1}\varphi _1$ and in the limit $\bar{y} > 0$, a contradiction.

Integrating (3.20) with $\varphi _1$, we get

$$\begin{aligned} \begin{aligned} s_n\lambda _\infty \int _0^\infty mK(r)\varphi _1^2 dr=&\int _0^\infty \big [\lambda K(r)\tilde{g}(r, z_n-\hat{w})\varphi _1+\tau _n\lambda _\infty \varphi _1^2\big ]dr\\ =&\int _0^\infty \lambda mK(r) z_n\varphi _1 dr+\int _0^\infty \lambda K(r)\big [\left( \frac{c_3}{\varphi _1^\alpha }-\frac{c_3}{\tilde{u}_n^\alpha }\right) \\&\quad +(\zeta (\tilde{u}_n)+M)-m\hat{w}\big ]\varphi _1 dr+\int _0^\infty \tau _n\lambda _\infty \varphi _1^2 dr\\ \ge&\int _0^\infty \lambda mK(r)s_n\varphi _1^2 dr\\&\quad +\int _0^\infty \lambda K(r)\Big [\frac{c_3}{\varphi _1^\alpha }-\frac{c_3}{\tilde{u}_n^\alpha }+\zeta (\tilde{u}_n)+M-m\hat{w}\Big ]\varphi _1 dr. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \lambda _\infty&\ge \lambda +\Big [\int _0^\infty \lambda K(r)\left[ \frac{c_3}{\varphi _1^\alpha }-\frac{c_3}{\tilde{u}_n^\alpha }+\zeta (\tilde{u}_n)+M-m\hat{w}\right] \varphi _1 dr\Big ]\\&\quad \ \Big [s_n\int _0^\infty mK(r)\varphi _1^2 dr\Big ]^{-1}\rightarrow \lambda \end{aligned}$$

as $n\rightarrow \infty$, a contradiction to the assumption $\lambda >\lambda _\infty$.$\square$

In order to investigate the bifurcation from infinity, for $z\ne 0$, we set $y =z ||z||_{\varphi _1}^{-2}$. Letting

$$\begin{aligned}\Psi (\lambda , y)=z||z||_{\varphi _1}^{-2}\Phi (\lambda , z)=y-\lambda ||y||_{\varphi _1}^{2} T\left[ \tilde{g}\left( \cdot ,\frac{y}{||y||^2_{\varphi _1}}-\hat{w}\right) \right] . \end{aligned}$$

One has that $\lambda$ is a bifurcation from infinity for $\Phi (\lambda ,z)=0$ if and only if it is a bifurcation from the trivial solution $y= 0$ for $\Psi (\lambda , y)=0$.

From Lemma 3.4 it follows by homotopy that

$$\begin{aligned} \text {deg}(\Psi (\lambda , \cdot ), B_{1/R}, 0)=\text {deg}(\Psi (0,\cdot ),B_{1/R}, 0)=\text {deg}(I, B_{1/R}, 0)=1, \quad \forall \ \lambda <\lambda _\infty . \end{aligned}$$

(3.21)

Similarly, by Lemma 3.5 one infers, for all $\tau \in [0, 1]$ and for all $\lambda > \lambda _\infty$,

$$\begin{aligned}&\text {deg}(\Psi (\lambda , \cdot ), B_{1/R}, 0)=\text {deg}(\Psi (0, \cdot )-\tau \varphi _1, B_{1/R}, 0)=\text {deg}(\Psi (0, \cdot )-\varphi _1,\nonumber \\&\qquad B_{1/R}, 0)=0, \quad \forall \ \lambda >\lambda _\infty . \end{aligned}$$

(3.22)

Let us set

$$\begin{aligned}\mathscr {D}=\{(\lambda , z)\in \mathbb {R}^+\times E:\; z \ne 0, \Phi (\lambda , z)=0\}. \end{aligned}$$

From (3.21) and (3.22) and the preceding discussion, we deduce that

Proposition 3.1

$\lambda _\infty$ is a bifurcation point from infinity for positive solutions of (3.16). More precisely there exists an unbounded closed connected set $\mathscr {D}_\infty \subset \mathscr {D}$ of positive solutions, which meets $(\lambda _\infty ,\infty )$.

Furthermore, let

$$\begin{aligned} \widetilde{\mathscr {D}}_\infty :=\{(\lambda ,\tilde{u}): \; \tilde{u}=z-\hat{w}\ \ \text {and}\ (\lambda , z)\in \mathscr {D}_\infty \}. \end{aligned}$$

(3.23)

From Proposition 3.1, we can obtain the following

Proposition 3.2

$\lambda _\infty$ is a bifurcation point from infinity for positive solutions of (3.15). More precisely $\widetilde{\mathscr {D}}_\infty$ is an unbounded closed connected set of positive solutions, which meets $(\lambda _\infty ,\infty )$.

3.4 Proof of Theorem 1.2

Through the discussion above, if $(\lambda , \tilde{u})\in \widetilde{\mathscr {D}}_\infty$ satisfy $\tilde{u}-\varphi _1>0$ in $\mathbb {R}^N$, then $\tilde{u}$ is also positive solution of (1.3).

Lemma 3.6

If $\{(\lambda _n,\tilde{u}_n)\}\subset \widetilde{\mathscr {D}}_\infty$ satisfies $||\tilde{u}_n||_{\varphi _1}\rightarrow \infty$ and $\lambda _n\rightarrow \lambda _\infty$. Then

$$\begin{aligned}\underset{n\rightarrow \infty }{\lim }\ \frac{\tilde{u}_n(r)}{||\tilde{u}_n||_{\varphi _1}}=\varphi _1(r). \end{aligned}$$

Moreover

$$\begin{aligned}\frac{\tilde{u}_n}{||\tilde{u}_n||_{\varphi _1}}\rightarrow \varphi _1,\quad n\rightarrow \infty \ \ \text {in}\ C^1(\mathbb {R}^N).\end{aligned}$$

Proof

Let $(\lambda _n,\tilde{u}_n)\in \widetilde{\mathscr {D}}_\infty$ such that $||\tilde{u}_n||_{\varphi _1}\rightarrow \infty$ and $\lambda _n\rightarrow \lambda _\infty$. Then $v_n:=\frac{\tilde{u}_n}{||\tilde{u}_n||_{\varphi _1}}$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }v_n')'=\lambda _nr^{N-1-\kappa }K(r)\Big [mv_n-\frac{c_1}{\tilde{u}_n^\alpha \Vert \tilde{u}_n\Vert _{\varphi _1}}+\frac{\zeta (\tilde{u}_n)}{||\tilde{u}_n||_{\varphi _1}}\Big ],& r\in (0,\infty ),\\ v_n'(0)=0,& v_n(\infty )=0. \end{array}\right. \end{aligned}$$

(3.24)

Since the right side of (3.24) is bounded uniformly in $E_0$, it follows from that $v_n\rightarrow v$, $n\rightarrow \infty$ for some $v\in C_b[0,\infty )$. Using the standard argument similar to the proof of [15, Lemma 3.4], with obvious changes, we may deduce that $v=\varphi _1$.$\square$

Lemma 3.7

For every compact interval $\Lambda \subset \mathbb {R}^+\setminus \{\lambda _\infty \},$ there exists $r >0$ such that

$$\begin{aligned} u-\lambda T[f(u)] \ne 0,\quad \forall \lambda \in \Lambda , \quad \forall \ ||u||_{\varphi _1}\ge r \quad \text {with}\ u\ge \varphi _1. \end{aligned}$$

(3.25)

Moreover,

• (i) if $\liminf \nolimits _{s\rightarrow +\infty }\ \zeta (s)>0$ then we can also take $\Lambda = [\lambda _\infty ,\lambda ],$ $\forall \ \lambda >\lambda _\infty ;$

• (ii) if $\limsup \nolimits _{s\rightarrow +\infty }\ \zeta (s)<0$ then we can also take $\Lambda = [0,\lambda _\infty ]$.

Proof

Let $\mu _n\rightarrow \mu \ge 0$, $\mu \ne \lambda _\infty$, and $||u_n||_{\varphi _1} \rightarrow \infty$ be such that

$$\begin{aligned}u_n= \mu _n T[f(u_n)]. \end{aligned}$$

Similar to the proof of Lemma 3.4, setting $y_n=u_n||u_n||_{\varphi _1}^{-1}\rightarrow y$, we can obtain (3.25).

We will only give a sketch of (i). Taking $\mu _n\rightarrow \lambda _\infty ^+$, it follows that $y\ge 0$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} -(r^{N-1-\kappa }y')'=\lambda _\infty mr^{N-1-\kappa }K(r)y,& r\in (0,\infty ),\\ y'(0)=0,& y(\infty )=0, \end{array} \right. \end{aligned}$$

and hence there exists $\beta >0$ such that $y=\beta \varphi _1$. Then one has $u_n=\Vert u_n\Vert _{\varphi _1} y_n\rightarrow +\infty$ for all $r\in [0,\infty )$ for n large. From $u_n-\lambda T[f(u_n)]=0,$ it follows that

$$\begin{aligned} \int _0^\infty \lambda _\infty mK(r)u_n\varphi _1 dr&=\mu _n\int _0^\infty \Big [-\frac{K(r)}{u_n^\alpha }+K(r)\zeta (u_n)\Big ]\varphi _1 dr \\&\quad \ +\mu _n \int _0^\infty mK(r)u_n\varphi _1 dr.\end{aligned}$$

Since $\mu _n>\lambda _\infty$ and $\int _0^\infty mK(r)u_n\varphi _1 dr>0$ for n large, we infer that

$$\begin{aligned} \int _0^\infty \Big [-\frac{K(r)}{u_n^\alpha }+K(r)\zeta (u_n)\Big ]\varphi _1 dr<0 \end{aligned}$$

for n large. By Lebesgue’s dominated convergence theorem,

$$\begin{aligned} 0<\int _0^\infty \underline{\zeta } K(r)\varphi _1 dx\le \underset{n\rightarrow +\infty }{\liminf }\int _0^\infty \Big [-\frac{K(r)}{u_n^\alpha }+K(r)\zeta (u_n)\Big ]\varphi _1 dr<0, \end{aligned}$$

where $\underline{\zeta }:=\liminf \nolimits _{s\rightarrow +\infty }\zeta (s)>0$. This is a contradiction.$\square$

Proof of Theorem 1.2

Setting $y_n =(\tilde{u}_n-\varphi _1) ||\tilde{u}_n||_{\varphi _1}^{-1}$ and using Lemma 3.6, we find that, up to subsequence, $y_n\rightarrow \left( \rho -\frac{1}{\Vert \tilde{u}_n\Vert }\right) \varphi _1$ in $C^1(\mathbb {R}^N)$, and $y=\rho \varphi _1$ with $|\rho |=1$. Then, it follows that $\tilde{u}_n-\varphi _1 > 0$ in $\mathbb {R}^N$ for n large. This means that $\lambda _\infty$ is a bifurcation point from infinity for positive solutions of (1.3). More precisely, there exists

$$\begin{aligned} \mathscr {C}_\infty :=\{(\lambda ,u): \; (\lambda , \tilde{u})\in \widetilde{\mathscr {D}}_\infty \ \ \text {and}\ \tilde{u}>\varphi _1\} \end{aligned}$$

which is an unbounded closed connected set of positive solutions, which meets $(\lambda _\infty ,\infty )$. By Lemma 3.7, $\mathscr {C}_\infty$ bifurcates to the left of $\lambda _\infty$ if $\liminf \nolimits _{s\rightarrow +\infty }\ \zeta (s)>0$ and to the right of $\lambda _\infty$ if $\limsup \nolimits _{s\rightarrow +\infty }\ \zeta (s)<0$.$\square$

Proof of corollary 1.2

It is an immediate consequence of Theorem 1.2. $\square$