© Wuhan University 2024

0 Introduction

Consider the following semi-linear elliptic problem

$\{\begin{array}{c}-\mathrm{\Delta }u=\lambda a(x)f(u),x\in {\mathbb{R}}^N,\\ u=\mathrm{0},\begin{array}{ccc}& & \mathrm{a}\mathrm{s}\left|x\right|\end{array}\to +\mathrm{\infty },\end{array}$(1)

where $\lambda$ is a real parameter, $N\ge \mathrm{3}$, and $a\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\alpha }({\mathbb{R}}^N,\mathbb{R})$ for some $\alpha \in (\mathrm{0,1})$ is a weighted function which can be sign-changing and $f\in C(\mathbb{R},\mathbb{R})$, and $f(s)s>\mathrm{0}$ for any $s\ne \mathrm{0}$. Edelson et al^[1,2] studied the existence of positive solution and the existence of global branches of minimal solutions of the problem (1) by the Schauder-Tychonoff fixed point theorem and Dancer global bifurcation theorems^[3], respectively. By using Rabinowitz global bifurcation method^[4], Rumbos et al^[5] showed the existence of positive minimal solution of the problem (1). In 2017, Dai et al^[6] established a global bifurcation result for problem (1).

On the other hand, Lions^[7] studied the following Kirchhoff type problem

$\{\begin{array}{c}-M({\int }_{\mathrm{\Omega }}{\left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x)\mathrm{\Delta }u=\lambda a(x)u+g(x,u,\lambda ),\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=\mathrm{0},\begin{array}{ccc}& & \begin{array}{cccc}& & \begin{array}{ccc}\begin{array}{ccc}& & \end{array}& & \end{array}& \mathrm{o}\mathrm{n}\partial \mathrm{\Omega }\end{array}\end{array},\end{array}$(2)

where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with a smooth boundary $\partial \mathrm{\Omega }$. The problem (2) is nonlocal as the appearance of the term $\underset{\mathrm{\Omega }}{\int }{\left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x$ which implies that it is not a pointwise identity. By applying the bifurcation techniques, Liang et al^[8] and Figueiredo et al^[9] also studied equation (2). Dai et al^[10] studied the problem (2) by Rabinowitz^[4].

Motivated by the above papers, we shall study the following Kirchhoff type problem

$\{\begin{array}{c}-M({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x)\mathrm{\Delta }u=\lambda a(x)[u+g(u)],x\in {\mathbb{R}}^N,\\ u=\mathrm{0},\begin{array}{ccc}& & \begin{array}{ccc}& \begin{array}{ccc}\begin{array}{ccc}& & \end{array}& & \end{array}& \mathrm{a}\mathrm{s}\left|x\right|\end{array}\end{array}\to +\mathrm{\infty },\end{array}$(3)

where $\lambda$ is a real parameter. By Ref. [6], set $I(\mathrm{\Omega }):=\left\{a\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\alpha }(\mathrm{\Omega },\mathbb{R}):\left\{x\in \mathrm{\Omega }:a(x)>\mathrm{0}\right\}\ne \mathrm{\varnothing }\right\}.$

For any $u\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$, we define ${\Vert u\Vert }_{\mathrm{1}}=({\int }_{\mathrm{\Omega }}{\left|\nabla u\right|}^{\mathrm{2}}{\mathrm{d}x)}^{\mathrm{1}/\mathrm{2}}$. Denote by $D^{\mathrm{1,2}}(\mathrm{\Omega })$ the completion of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with respect to the norm ${\Vert u\Vert }_{\mathrm{1}}$. Denote by $S({\mathbb{R}}^N)$ the set of all measurable real functions defined on ${\mathbb{R}}^N$. Two functions in $S({\mathbb{R}}^N)$ are considered as the same element of $S({\mathbb{R}}^N)$ when they are equal almost everywhere. Let $L^{\mathrm{2}}({\mathbb{R}}^N;\left|a\right|):=\left\{u\in S({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^N}\left|a\right|u^{\mathrm{2}}\mathrm{d}x<+\mathrm{\infty }\right\}.$

We assume that $a$, $g(\cdot )$ and $M(\cdot )$ satisfy the following conditions:

(A1) Let $a\in I({\mathbb{R}}^N)$. If there exist two continuous positive radially symmetric functions $p$ and $P$, where $P\in L^{\mathrm{2}{q}^{\text{'}}r}({\mathbb{R}}^N)$ (where $q$ and $r$ are given in (A3) and Section 1) such that $\mathrm{0}<p\le a(x)\le P(\left|x\right|),\forall x\in {\mathbb{R}}^N$ and ${\int }_{{\mathbb{R}}^N}{\left|x\right|}^{\mathrm{2}-N}P(\left|x\right|)\mathrm{d}x<+\mathrm{\infty }.$

Furthermore, if $P$ satisfies the following stronger condition (with $r=\left|x\right|$)

${\int }_{\mathrm{0}}^{+\mathrm{\infty }}{r}^{N-\mathrm{1}}P(r)\mathrm{d}r<+\mathrm{\infty }.$(4)

(A2) $g\in C(\mathbb{R},\mathbb{R})$ is a Holder continuous function with exponent $\alpha$ and $\underset{s\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}g(s)/s=\mathrm{0}.$

(A3) There exist $c>\mathrm{0}$ and $q\in (\mathrm{1},{\mathrm{2}}^{*}]$ such that $\left|g(s)\right|\le c(\mathrm{1}+{\left|s\right|}^{q-\mathrm{1}}),$ where

${\mathrm{2}}^{*}=\{\begin{array}{c}\frac{\mathrm{2}N}{N-\mathrm{2}},N>\mathrm{2},\\ +\mathrm{\infty },\begin{array}{cc}& N\le \mathrm{2}\end{array}.\end{array}$

(A4) $M(t)\in C({\mathbb{R}}^{+})$ is increasing, $M(\mathrm{0})>\mathrm{0}$, ${\mathbb{R}}^{+}=[\mathrm{0},+\mathrm{\infty })$.

(A5) There exists $m_{\mathrm{1}}>\mathrm{0}$, such that $\underset{t\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}M(t)=m_{\mathrm{1}}$.

Furthermore, we shall investigate the existence of one-sign solutions for the following Kirchhoff type problems

$\{\begin{array}{c}-M({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x)\mathrm{\Delta }u=\lambda a(x)f(u),x\in {\mathbb{R}}^N,\\ u=\mathrm{0},\begin{array}{c}\begin{array}{cccc}& & \begin{array}{ccc}\begin{array}{cc}& \end{array}& & \end{array}& \mathrm{a}\mathrm{s}\left|x\right|\end{array}\end{array}\to +\mathrm{\infty }.\end{array}$(5)

We assume that $a$ satisfies (A1), and $f$ satisfies the following assumptions:

(H1) $f\in C(\mathbb{R},\mathbb{R})$ is a Holder continuous function with exponent $\alpha$ such that $sf(s)>\mathrm{0}$ for any $s\ne \mathrm{0}$.

(H2) $f_{\mathrm{0}},f_{\mathrm{\infty }}\in (\mathrm{0},\mathrm{\infty }).$

(H3) $f_{\mathrm{0}}\in (\mathrm{0},\mathrm{\infty }),f_{\mathrm{\infty }}=\mathrm{\infty }$.

(H4) $f_{\mathrm{0}}\in (\mathrm{0},\mathrm{\infty }),f_{\mathrm{\infty }}=\mathrm{0}.$

(H5) $f_{\mathrm{0}}=\mathrm{\infty },f_{\mathrm{\infty }}\in (\mathrm{0},\mathrm{\infty }).$

(H6) $f_{\mathrm{0}}=\mathrm{0},f_{\mathrm{\infty }}\in (\mathrm{0},\mathrm{\infty }).$

(H7) $f_{\mathrm{0}}=\mathrm{\infty },f_{\mathrm{\infty }}=\mathrm{0}.$

(H8) $f_{\mathrm{0}}=\mathrm{0},f_{\mathrm{\infty }}=\mathrm{\infty }.$

(H9) $f_{\mathrm{0}}=\mathrm{0},f_{\mathrm{\infty }}=\mathrm{0}.$

(H10) $f_{\mathrm{0}}=\mathrm{\infty },f_{\mathrm{\infty }}=\mathrm{\infty }.$

where $f_{\mathrm{0}}=\underset{\left|s\right|\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(s)}{s},f_{\mathrm{\infty }}=\underset{\left|s\right|\to \to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(s)}{s}.$

Finally, we shall study the exact multiplicity of one-sign solutions for (5) by Implicit Function Theorem, the stability properties and condition (A6).

(A6) $f\in C^{\alpha }(\mathbb{R},\mathbb{R})$ such that $f(s)/s$ is decreasing in $(\mathrm{0},+\mathrm{\infty })$ and is increasing in $(-\mathrm{\infty },\mathrm{0})$.

The rest of this paper is arranged as follows. In Section 1, we give some preliminaries and establish the unilateral global bifurcation result for the problem (3). In Section 2, on the above unilateral global bifurcation result, we prove the existence of one-sign solutions for the Kirchhoff type problem (5). In Section 3, we study the exact multiplicity of one-sign solutions for (5).

1 Preliminaries

Let $E:=H^{\mathrm{1}}({\mathbb{R}}^N)$ with the norm $\Vert u\Vert ={({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x)}^{\mathrm{1}/\mathrm{2}}$. Let $P^{+}=\left\{u\in E|u(x)>\mathrm{0},x\in {\mathbb{R}}^N\right\}$ and set $P^{-}=-P^{+}$ and $P=P^{+}\bigcup P^{-}$.

Now, from Theorem 1.1 in Ref. [6], we know that the following eigenvalue problem

$\{\begin{array}{c}-\mathrm{\Delta }u=\lambda a(x)u,\begin{array}{cc}& \mathrm{i}\mathrm{n}\end{array}{\mathbb{R}}^N,\\ u(x)\to \mathrm{0},\begin{array}{c}\mathrm{a}\mathrm{s}\left|x\right|\to +\mathrm{\infty },\end{array}\end{array}$(6)

possesses a unique principal eigenvalue ${\lambda }_{\mathrm{1}}$, and ${\lambda }_{\mathrm{1}}$ is simple and isolated.

To prove Theorem 1, by Section 4 of Ref. [6], we first consider the following problem

$\{\begin{array}{c}-\mathrm{\Delta }u=h(u),\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ u(x)\to \mathrm{0},\begin{array}{c}\end{array}\mathrm{a}\mathrm{s}\left|x\right|\to +\mathrm{\infty }.\end{array}$(7)

Let us define the operator $T:E\to E$ by

$u(x)=T[h](x)={\int }_{{\mathbb{R}}^N}{\Gamma }_N(x-y)h(y)\mathrm{d}y,$(8)

where ${\Gamma }_N(x-y)=\frac{\mathrm{1}}{N(N-\mathrm{2}){\omega }_N}{\left|x-y\right|}^{\mathrm{2}-N},$${\omega }_N$ being the volume of the unit ball in ${\mathbb{R}}^N$.

Then by an argument similar to that of Ref. [1], we can show that $u$ is a one-sign $C^{\mathrm{2}+\alpha }$ solution of problem (7) if and only if $u$ is a solution of the operator equation $u(x)=T(h)$. Similar to proposition 1 in Ref. [5], we also can show that $T:E\to E$ is linear completely continuous and (8) is equivalent to (7).

The first main result for (3) is the following unilateral global bifurcation theorem.

Theorem 1   Assume that (A1)-(A5) hold. The pair $({\lambda }_{\mathrm{1}}M(\mathrm{0}),\mathrm{0})$ is a bifurcation point of the problem (3) and there are two distinct unbounded continua $D^{+}$ and $D^{-}$ in $\mathbb{R}\times H^{\mathrm{1}}({\mathbb{R}}^N)$ of solutions of the problem (3) emanating from $({\lambda }_{\mathrm{1}}M(\mathrm{0}),\mathrm{0})$. Moreover, we have $D^{\nu }\subset ((\mathbb{R}\times P^{\nu })\bigcup \left\{({\lambda }_{\mathrm{1}}M(\mathrm{0}),\mathrm{0})\right\}),$ where $\mu \in \left\{+,-\right\}$.

Proof   By p.5960-5961 in Ref. [6], it is clear that the problem (3) can be equivalently written as $u=G(\lambda ,u)=\lambda \cdot \frac{T(au)}{M(\mathrm{0})}+H(\lambda ,u),$ where

$H(\lambda ,u)=\frac{\lambda (M(\mathrm{0})-M({\Vert u\Vert }^{\mathrm{2}}))}{M(\mathrm{0})M({\Vert u\Vert }^{\mathrm{2}})}T(au)+\frac{T[\lambda a(x)g(u)]}{M({\Vert u\Vert }^{\mathrm{2}})}.$

From conditions (A1)-(A5) and noting $\mathrm{2}<{\mathrm{2}}^{*}$, we can see that $H:\mathbb{R}\times E\to E$ is completely continuous. Furthermore, it follows that $G:\mathbb{R}\times E\to E$ is completely continuous and $G(\lambda ,\mathrm{0})=\mathrm{0},\forall \lambda \in \mathbb{R}.$

Next, we show $\underset{\Vert u\Vert \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}H(\lambda ,u)/\Vert u\Vert =\mathrm{0}$ at $u=\mathrm{0}$ uniformly on bounded $\lambda$ sets. Without loss of generality, we may assume that $q>\mathrm{2}$. Otherwise, we can consider $\tilde{q}=cq,c>\mathrm{1}$ such that $\tilde{q}\in (\mathrm{2},{\mathrm{2}}^{*})$. From $q<{\mathrm{2}}^{*}$, we can see $\frac{q^{\text{'}}(q-\mathrm{2})}{{\mathrm{2}}^{*}}<\frac{\mathrm{2}-q^{\text{'}}}{{\mathrm{2}}^{*}}.$ So we can choose a real number $r>\mathrm{1}$ such that $\frac{q^{\text{'}}(q-\mathrm{2})}{{\mathrm{2}}^{*}}\le \frac{\mathrm{1}}{r}\le \frac{\mathrm{2}-q^{\text{'}}}{{\mathrm{2}}^{*}}.$

It follows

$q^{\text{'}}r(q-\mathrm{2})\le {\mathrm{2}}^{*},q^{\text{'}}{r}^{\text{'}}\le {\mathrm{2}}^{*}.$(9)

By (A2) and (A3), for any $\epsilon >\mathrm{0}$, we can choose positive numbers $\delta =\delta (\epsilon )$ and $M=M(\delta )$ such that the following relations hold: $\left|g(s)/s\right|\le \epsilon ,\mathrm{f}\mathrm{o}\mathrm{r}\left|s\right|\le \delta .$$\left|g(s)/s\right|\le M{\left|s\right|}^{q-\mathrm{2}},\mathrm{f}\mathrm{o}\mathrm{r}\left|s\right|>\delta .$

Then we can obtain

$\begin{array}{l}{\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)g(u)}{u}\right|}^{q^{\text{'}}r}\mathrm{d}x\\ \le \epsilon {\int }_{{\mathbb{R}}^N}(a{(x))}^{q^{\text{'}}r}\mathrm{d}x+M^{q^{\text{'}}r}{\int }_{{\mathbb{R}}^N}(a{(x))}^{q^{\text{'}}r}{\left|u\right|}^{q^{\text{'}}r(q-\mathrm{2})}\mathrm{d}x\\ \le \epsilon {\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{q^{\text{'}}r}\mathrm{d}x\\ +M^{q^{\text{'}}r}{({\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{\mathrm{2}{q}^{\text{'}}r}\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}{({\int }_{{\mathbb{R}}^N}{\left|u\right|}^{\mathrm{2}{q}^{\text{'}}r(q-\mathrm{2})}\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}.\end{array}$

By $P\in L^{\mathrm{2}{q}^{\text{'}}r}({\mathbb{R}}^N)$ and the continuous embedding of $L^{\mathrm{2}{q}^{\text{'}}r}({\mathbb{R}}^N)\hookrightarrow L^{q^{\text{'}}r}({\mathbb{R}}^N)$, we have

${\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{\mathrm{2}{q}^{\text{'}}r}\mathrm{d}x<+\mathrm{\infty },{\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{q^{\text{'}}r}\mathrm{d}x<+\mathrm{\infty }$

Moreover, as $u\to +\mathrm{\infty }$, we obtain that

${\left|\frac{a(x)g(u)}{u}\right|}^{q^{\text{'}}}\to \mathrm{0}\begin{array}{cc}\mathrm{i}\mathrm{n}& L^r({\mathbb{R}}^N)\end{array}$

Let $v=u/\Vert u\Vert ,$ by the boundedness of $v\in E$, $q^{\text{'}}{r}^{\text{'}}\le {\mathrm{2}}^{*}$ and the continuous embedding of $E\hookrightarrow L^{{\mathrm{2}}^{*}}({\mathbb{R}}^N)$, we have ${\int }_{{\mathbb{R}}^N}{\left|v\right|}^{q^{\text{'}}{r}^{\text{'}}}\mathrm{d}x<c,$ furthermore, we can get

$\begin{array}{l}{\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)g(u)}{\Vert u\Vert }\right|}^{q^{\text{'}}}\mathrm{d}x={\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)g(u)}{\left|u\right|}\right|}^{q^{\text{'}}}{\left|v\right|}^{q^{\text{'}}}\mathrm{d}x\\ \le {({\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)g(u)}{\left|u\right|}\right|}^{q^{\text{'}}r}\mathrm{d}x)}^{\frac{\mathrm{1}}{r}}{({\int }_{{\mathbb{R}}^N}{\left|v\right|}^{q^{\text{'}}{r}^{\text{'}}}\mathrm{d}x)}^{\frac{\mathrm{1}}{r^{\text{'}}}}\to \mathrm{0}.\end{array}$

We obtain

$\underset{\Vert u\Vert \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}a(x)g(u)/\Vert u\Vert =\mathrm{0},\mathrm{i}\mathrm{n}{L}^{q^{\text{'}}}({\mathbb{R}}^N)$(10)

uniformly $x\in {\mathbb{R}}^N.$

By (10), we have $\underset{\Vert u\Vert \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}H(\lambda ,u)/\Vert u\Vert =\mathrm{0}$ uniformly for $x\in {\mathbb{R}}^N$ and $\lambda$ on bounded sets, i.e. $H(\lambda ,u)=o(\Vert u\Vert )$ at $u=\mathrm{0}$ uniformly on bounded $\lambda$ sets.

Furthermore, applying the similar proof method of Theorem 1.3 in Ref. [6] and the Rabinowitz global bifurcation theorem^[4], one can obtain that $({\lambda }_{\mathrm{1}}M(\mathrm{0}),\mathrm{0})$ is a bifurcation point of the problem (3) and there exists one unbounded continua $D$ of solutions of the problem (3) emanating from $({\lambda }_{\mathrm{1}}M(\mathrm{0}),\mathrm{0})$.

Moreover, by the Dancer unilateral global bifurcation theorem^[11], we have that there are two distinct unbounded bifurcation continua $D^{+}$ and $D^{-}$ in $\mathbb{R}\times H^{\mathrm{1}}({\mathbb{R}}^N)$ of solutions of the problem (3) emanating from $({\lambda }_{\mathrm{1}}M(\mathrm{0}),\mathrm{0})$. Moreover, we have

$D^{\nu }\subset ((\mathbb{R}\times P^{\nu })\bigcup \left\{({\lambda }_{\mathrm{1}}M(\mathrm{0}),\mathrm{0})\right\}),$ where $\nu \in \left\{+,-\right\}$.

2 One-Sign Solutions for Kirchhoff Type Problem

We first have the following results.

Remark 1   From (H1) and (H2), we can see that there exist two positive constants $\mathrm{0}<\rho <\sigma$ such that $\rho \le \frac{f(s)}{s}\le \sigma$ for all $s\ne \mathrm{0}$.

By an argument similar to that of Lemma 4.1, 4.2 in Ref. [6], we can obtain Lemma 1 and 2.

Lemma 1   Let (H1) and (H2) hold. By Remark 1, the problem (5) has no one-sign solution for any $\lambda \in ({\lambda }_{\mathrm{1}}{m}_{\mathrm{1}}/\rho ,+\mathrm{\infty }).$

Lemma 2   Let (H1) and (H2) hold. By Remark 1, the problem (5) has no positive solution for any $\lambda \in (\mathrm{0},{\lambda }_{\mathrm{1}}M(\mathrm{0})/\sigma ).$ The main results of this section are the following theorem.

Theorem 2   Let (A1), (A4), (A5), (H1) and (H2) hold. For any $\lambda \in (\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{\infty }}}{m}_{\mathrm{1}},\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0})\right\},\mathrm{m}\mathrm{a}\mathrm{x}\{\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{\infty }}}{m}_{\mathrm{1}},$$\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0})\}),$ the problem (5) possesses two solutions $u_{\mathrm{1}}^{+}$, $u_{\mathrm{1}}^{-}$ such that $u_{\mathrm{1}}^{+}>\mathrm{0}$ and $u_{\mathrm{1}}^{-}<\mathrm{0}$ in ${\mathbb{R}}^N$. Therefore, we have

$\underset{\left|x\right|\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left|x\right|}^{N-\mathrm{2}}{u}_{\mathrm{1}}^{+}(x)=c_{\mathrm{1}}>\mathrm{0},\underset{\left|x\right|\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left|x\right|}^{N-\mathrm{2}}{u}_{\mathrm{1}}^{-}(x)=c_{\mathrm{2}}<\mathrm{0}$

for some constants $c_{\mathrm{1}}$ and $c_{\mathrm{2}}$.

Proof   Let $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a continuous function such that $f(u)=f_{\mathrm{0}}u+\zeta (u)$, with $\underset{\left|s\right|\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\zeta (s)}{s}=\mathrm{0},$$\underset{\left|s\right|\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\zeta (s)}{s}=f_{\mathrm{\infty }}-f_{\mathrm{0}}$, uniformly a.e. in ${\mathbb{R}}^N$. Equation (5) can be divided in the form

$\{\begin{array}{c}-\mathrm{\Delta }u=\frac{\lambda f_{\mathrm{0}}a(x)u(x)}{M(\mathrm{0})}+H_{\mathrm{1}}(\lambda ,u),\begin{array}{cc}& \mathrm{i}\mathrm{n}\end{array}{\mathbb{R}}^N,\\ u(x)\to \mathrm{0},\begin{array}{cc}& \begin{array}{ccc}\begin{array}{cc}\begin{array}{c}\end{array}& \end{array}& & \end{array}\mathrm{a}\mathrm{s}\left|x\right|\to +\mathrm{\infty },\end{array}\end{array}$(11)

where

$H_{\mathrm{1}}(\lambda ,u)=\frac{\lambda (M(\mathrm{0})-M({\Vert u\Vert }^{\mathrm{2}}))}{M(\mathrm{0})M({\Vert u\Vert }^{\mathrm{2}})}(f_{\mathrm{0}}a(x)u)+\frac{\lambda a(x)\zeta (u)}{M({\Vert u\Vert }^{\mathrm{2}})}.$

Using the same method to prove (10) with obvious changes, it follows that

$\underset{\Vert u\Vert \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\zeta (u)/\Vert u\Vert =\mathrm{0},\mathrm{i}\mathrm{n}{L}^{q^{\text{'}}}({\mathbb{R}}^N).$

Moreover, we have

$\underset{\Vert u\Vert \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}{H}_{\mathrm{1}}(\lambda ,u)/\Vert u\Vert =\mathrm{0},\mathrm{i}\mathrm{n}{L}^{q^{\text{'}}}({\mathbb{R}}^N)$

uniformly on bounded $\lambda$ sets.

By Theorem 1, there are two distinct unbounded continua $D^{+}$ and $D^{-}$ in $\mathbb{R}\times H^{\mathrm{1}}({\mathbb{R}}^N)$ of solutions of the problem (5) emanating from $(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0}),\mathrm{0})$, such that

$D^{\nu }\subset ((\mathbb{R}\times P^{\nu })\bigcup \left\{(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0}),\mathrm{0})\right\}),$ where $\mu \in \left\{+,-\right\}$.

We will show that $D^{\nu }$ joins $(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0}),\mathrm{0})$ to $(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{\infty }}}{m}_{\mathrm{1}},+\mathrm{\infty })$. Let

$({\mu }_n,u_n)\in D^{\nu }\backslash \left\{(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0}),\mathrm{0})\right\}$ satisfy $\left|{\mu }_n\right|+\left|u_n\right|\to +\mathrm{\infty }.$

By Remark 1 and Lemma 2, one can obtain ${\mu }_n>\mathrm{0}$ for all $n\in \mathbb{N}$. It follows from Lemma 1 that there exists a constant $M$ such that ${\mu }_n\in (\mathrm{0},M]$ for any $n\in \mathbb{N}$. Therefore, we get $\Vert u_n\Vert \to +\mathrm{\infty }.$

One can get that $D^{\nu }$ joins $(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0}),\mathrm{0})$ to $(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{\infty }}}{m}_{\mathrm{1}},+\mathrm{\infty })$. Let $\xi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a continuous function such that $f(u)=f_{\mathrm{\infty }}u+\xi (u)$, with

$\underset{s\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\xi (s)}{s}=\mathrm{0},\underset{s\to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\xi (s)}{s}=f_{\mathrm{0}}-f_{\mathrm{\infty }}$(12)

uniformly a.e. in ${\mathbb{R}}^N$. We divide the equation

$\{\begin{array}{c}-\mathrm{\Delta }u=\frac{\lambda f_{\mathrm{\infty }}a(x)u(x)}{m_{\mathrm{1}}}+H_{\mathrm{2}}(\lambda ,u),\begin{array}{c}\mathrm{i}\mathrm{n}\end{array}{\mathbb{R}}^N,\\ u(x)\to \mathrm{0},\begin{array}{c}\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& & \end{array}\mathrm{a}\mathrm{s}\left|x\right|\to +\mathrm{\infty },\end{array}\end{array}$(13)

where

$H_{\mathrm{2}}(\lambda ,u)=\frac{\lambda (m_{\mathrm{1}}-M({\Vert u\Vert }^{\mathrm{2}}))}{m_{\mathrm{1}}M({\Vert u\Vert }^{\mathrm{2}})}(f_{\mathrm{\infty }}a(x)u)+\frac{\lambda a(x)\xi (u)}{M({\Vert u\Vert }^{\mathrm{2}})}.$

By (12), for any $\epsilon >\mathrm{0}$, we can choose positive numbers $\delta =\delta (\epsilon )$ and $M=M(\epsilon )$ such that for a.e. $x\in {\mathbb{R}}^N$,the following relations hold:

$\begin{array}{l}\left|\xi (s)/s\right|\le \epsilon ,\mathrm{f}\mathrm{o}\mathrm{r}\left|s\right|>\delta ;\\ \left|\xi (s)/s\right|\le M,\mathrm{f}\mathrm{o}\mathrm{r}\left|s\right|\le \delta .\end{array}$

Then we can obtain

$\begin{array}{l}{\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)\xi (u)}{u}\right|}^{q^{\text{'}}r}\mathrm{d}x\le \epsilon {\int }_{{\mathbb{R}}^N}(a{(x))}^{q^{\text{'}}r}\mathrm{d}x+M^{q^{\text{'}}r}{\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)}{u}\right|}^{q^{\text{'}}r}\mathrm{d}x\\ \le \epsilon {\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{q^{\text{'}}r}\mathrm{d}x\\ +M^{q^{\text{'}}r}{({\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{\mathrm{2}{q}^{\text{'}}r}\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}{({\int }_{{\mathbb{R}}^N}{\left|u\right|}^{\mathrm{2}{q}^{\text{'}}r(q-\mathrm{2})}\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}.\end{array}$

By $P\in L^{\mathrm{2}{q}^{\text{'}}r}({\mathbb{R}}^N)$, we have

${\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{\mathrm{2}{q}^{\text{'}}r}\mathrm{d}x<+\mathrm{\infty },{\int }_{{\mathbb{R}}^N}(P{(\left|x\right|))}^{q^{\text{'}}r}\mathrm{d}x<+\mathrm{\infty }.$

Moreover, as $u\to +\mathrm{\infty }$, we obtain

${\left|\frac{a(x)\xi (u)}{u}\right|}^{q^{\text{'}}}\to \mathrm{0}\begin{array}{cc}\mathrm{i}\mathrm{n}& L^r({\mathbb{R}}^N)\end{array}.$

Let $v=u/\Vert u\Vert ,$ by the boundedness of $v\in E$, (9) and the continuous embedding of $E\hookrightarrow L^{{\mathrm{2}}^{*}}({\mathbb{R}}^N)$, we have

${\int }_{{\mathbb{R}}^N}{\left|v\right|}^{q^{\text{'}}{r}^{\text{'}}}\mathrm{d}x<c.$

Furthermore, we can get

$\begin{array}{l}{\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)\xi (u)}{\Vert u\Vert }\right|}^{q^{\text{'}}}\mathrm{d}x={\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)\xi (u)}{\left|u\right|}\right|}^{q^{\text{'}}}{\left|v\right|}^{q^{\text{'}}}\mathrm{d}x\\ \le {({\int }_{{\mathbb{R}}^N}{\left|\frac{a(x)\xi (u)}{\left|u\right|}\right|}^{q^{\text{'}}r}\mathrm{d}x)}^{\frac{\mathrm{1}}{r}}{({\int }_{{\mathbb{R}}^N}{\left|v\right|}^{q^{\text{'}}{r}^{\text{'}}}\mathrm{d}x)}^{\frac{\mathrm{1}}{r^{\text{'}}}}\to \mathrm{0}.\end{array}$

It follows from that

$\underset{\Vert u\Vert \to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}a(x)\xi (u)/\Vert u\Vert =\mathrm{0},\mathrm{i}\mathrm{n}{L}^{q^{\text{'}}}({\mathbb{R}}^N)$(14)

uniformly $x\in {\mathbb{R}}^N.$ Furthermore, one obtain

$\underset{\Vert u\Vert \to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{H}_{\mathrm{2}}(\lambda ,u)/\Vert u\Vert =\mathrm{0},\begin{array}{cc}\mathrm{i}\mathrm{n}& L^{q^{\text{'}}}({\mathbb{R}}^N)\end{array}$

uniformly for $\lambda$ on bounded sets.

By the compactness of $T^{-\mathrm{1}}$, we obtain

$\{\begin{array}{c}-\mathrm{\Delta }u=\frac{\mu f_{\mathrm{\infty }}a(x)u(x)}{m_{\mathrm{1}}},\mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ u(x)\to \mathrm{0},\begin{array}{cc}& \mathrm{a}\mathrm{s}\left|x\right|\to +\mathrm{\infty },\end{array}\end{array}$(15)

where $\mu =\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\mu }_n$, again choosing a subsequence and relabeling if necessary. Thus it is clear that $u\in {\dot{D}}^{\nu }\subseteq D^{\nu }$ since $D^{\nu }$ is closed in $\mathbb{R}\times E$. Moreover, by (15), $\mu f_{\mathrm{\infty }}={\lambda }_{\mathrm{1}}{m}_{\mathrm{1}}$, so that $\mu =\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{\infty }}}{m}_{\mathrm{1}}$. Thus $D^{\nu }$ joins $(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0}),\mathrm{0})$ to $(\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{\infty }}}{m}_{\mathrm{1}},+\mathrm{\infty })$. Now the existence of $u_{\mathrm{1}}^{+}$ and $u_{\mathrm{1}}^{-}$ is clear.

Similar to the proof of the Theorem 1.3 in Ref. [6], we have

$\underset{\left|x\right|\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left|x\right|}^{N-\mathrm{2}}{u}_{\mathrm{1}}^{+}(x)=c_{\mathrm{1}}>\mathrm{0},\underset{\left|x\right|\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left|x\right|}^{N-\mathrm{2}}{u}_{\mathrm{1}}^{-}(x)=c_{\mathrm{2}}<\mathrm{0}$

for some constants $c_{\mathrm{1}}$ and $c_{\mathrm{2}}$.

Theorem 3   Let (A1), (A4), (A5), (H1) and (H3) hold. If ${\lambda }_{\mathrm{1}}\in (\mathrm{0},\frac{{\lambda }_{\mathrm{1}}}{f_{\mathrm{0}}}M(\mathrm{0})),$ the problem (5) possesses two solutions $u_{\mathrm{1}}^{+}$, $u_{\mathrm{1}}^{-}$ such that $u_{\mathrm{1}}^{+}>\mathrm{0}$ and $u_{\mathrm{1}}^{-}<\mathrm{0}$ in ${\mathbb{R}}^N$. Therefore, we have

$\underset{\left|x\right|\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left|x\right|}^{N-\mathrm{2}}{u}_{\mathrm{1}}^{+}(x)=c_{\mathrm{1}}>\mathrm{0},\underset{\left|x\right|\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left|x\right|}^{N-\mathrm{2}}{u}_{\mathrm{1}}^{-}(x)=c_{\mathrm{2}}<\mathrm{0}$

for some constants $c_{\mathrm{1}}$ and $c_{\mathrm{2}}$.

Proof   Inspired by the idea of Ref. [12], we define the cut-off function of $f$ as the following

$f^{[n]}(s)=\{\begin{array}{c}ns,s\in [-\mathrm{\infty },-\mathrm{2}n]\bigcup [\mathrm{2}n,+\mathrm{\infty }],\\ \frac{\mathrm{2}{n}^{\mathrm{2}}+f(-n)}{n}(s+n)+f(-n),s\in (-\mathrm{2}n,-n),\\ \frac{\mathrm{2}{n}^{\mathrm{2}}-f(n)}{n}(s-n)+f(n),s\in (n,\mathrm{2}n),\\ f(s),s\in [-n,n].\end{array}$

We consider the following problem

$\{\begin{array}{c}-M({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^{\mathrm{2}})\mathrm{\Delta }u=\lambda a(x)f^{[n]}(u),x\in {\mathbb{R}}^N,\\ u=\mathrm{0},\begin{array}{c}\begin{array}{cccc}& & & \begin{array}{ccc}\begin{array}{cc}& \end{array}& & \end{array}\mathrm{a}\mathrm{s}\left|x\right|\end{array}\end{array}\to +\mathrm{\infty }.\end{array}$(16)

Clearly, we can see that $\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{f}^{[n]}(s)=f(s)$, and $(f^{[n]}{)}_{\mathrm{\infty }}=n$.

The Proposition 4.1 in Ref.[13] implies that there exist two sequence unbounded continua $D^{\nu [n]}$ of solution set of problem (16) emanating from $(\frac{{\lambda }_{\mathrm{1}}}{n{f}_{\mathrm{\infty }}}{m}_{\mathrm{1}},\mathrm{\infty })$,such that $D^{\nu [n]}\subset ((\mathbb{R}\times P^{\nu })\bigcup \left\{(\frac{{\lambda }_{\mathrm{1}}}{n{f}_{\mathrm{\infty }}}{m}_{\mathrm{1}},\mathrm{\infty })\right\}),$ where $\nu \in \left\{+,-\right\}$.