\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 316, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/316\hfil Multiple positive solutions]
{Multiple positive solutions for superlinear Kirchhoff type
problems on $\mathbb{R}^{N}$}

\author[Y. Duan, C.-L. Tang \hfil EJDE-2015/316\hfilneg]
{Yu Duan, Chun-Lei Tang}

\address{Yu Duan \newline
School of  Mathematics and  Statistics, Southwest University,
Chongqing 400715, China. \newline
College of Science, Guizhou University of Engineering Science,
Bijie, Guizhou 551700, China}
\email{duanyu3612@163.com}

\address{Chun-Lei Tang (corresponding author) \newline
School of  Mathematics and  Statistics, Southwest University,
 Chongqing 400715, China\newline
Phone: +86 23 68253135; fax: +86 23 68253135}
\email{tangcl@swu.edu.cn}

\thanks{Submitted August 28, 2015. Published December 28, 2015.}
\subjclass[2010]{35R09, 35A15, 35B09}
\keywords{Kirchhoff type problems; Pohozaev identity; variational method;
\hfill\break\indent  iterative technique}

\begin{abstract}
 In this article, we study the multiplicity of positive solutions for
 a class of Kirchhoff type problems depending on two real functions
 and a nonnegative parameter on an unbounded domain.
 Using the variational method and iterative techniques, we show that
 if the nonlinearity is subcritical and superlinear at zero and infinity,
 then the Kirchhoff type problems admits at least two positive solutions
 when the parameter is sufficiently small.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

 The purpose of this article is to sutdy the multiplicity of positive solutions
 to the  nonlinear Kirchhoff type problem
\begin{equation} \label{1.1}
\Big(a+\lambda m\Big(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+bu^2)dx\Big)\Big)
(-\Delta u+bu)=f(u)+h(x)|u|^{q-2}u, \quad\text{in }\mathbb{R}^{N},
\end{equation}
where $N\geq 3$, $1<q<2$, $a$, $b$ are positive constants, $\lambda\geq 0$
is a parameter, and $m, f, h$ are  positive continuous functions.

Problem \eqref{1.1} is related to the stationary analogue of the Kirchhoff equation
\begin{equation}
u_{tt}-(a+b\int_{\Omega}|\nabla u|^{2}dx)\Delta u=f(x,u)  \label{003}
\end{equation}
which was proposed by Kirchhoff in 1883 \cite{K} as a generalization of
the well-known d'Alembert's  equation
$$
    \rho \frac{\partial ^2u}{\partial t^2}-\Big(\frac{P_0}{h}+
\frac{E}{2L}\int_0^L|\frac{\partial u}{\partial
x}|^2dx\Big)\frac{\partial ^2u}{\partial x^2}=f(x,u)
$$
for free vibrations of elastic strings. Kirchhoff's model takes into
account the changes in length of the string produced by
transverse vibrations. Here, $L$ is the length of the string, $h$
is the area of the cross section, $E$ is the Young modulus of the
material, $\rho$ is the mass density and $P_0$ is the initial tension.
The readers can find some early classical research of Kirchhoff's equations
in \cite{B,P}.
However, \eqref{003} received great attention only after Lions \cite{L}
proposed an abstract framework to the problem. Some interesting results for
problem \eqref{003} can be found in \cite{AP,Ca,DS} and the references therein.
More recently some mathematicians study the following Kirchhoff type
problems on bounded domain
\begin{equation}
\begin{gathered}
-\Big(a+b\int_{\Omega}|\nabla u|^{2}dx\Big)\Delta u=f(x,u), \quad\text{in }  \Omega,\\
u=0,   \quad\text{on }   \partial\Omega.
\end{gathered}  \label{02}
\end{equation}
Some interesting studies for problem \eqref{02} by variational methods
can be found in \cite{A,C,ZLLS,MZ,PZ,ST,ZSN,ZP} and the references therein.
Especially, the authors \cite{ZSN} studied the existence of positive solution
for Kirchhoff type problem on bounded domain using iterative techniques
and variational methods.

Recently, authors have studied widely  Kirchhoff type problems under various
conditions on $f$ and $V$ on the whole space $\mathbb{R}^{N}$:
\begin{equation}
\Big(a+\lambda\Big(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)u^2)dx\Big)\Big)
\big(-\Delta u+V(x)u\big)=f(x,u),  \quad\text{in }\mathbb{R}^{N}. \label{004}
\end{equation}
When $f(x,u)=|u|^{p-2}u$, $p\in (2,2^*)$, Huang and Liu \cite {HL2014}
considered \eqref{004} and studied existence and nonexistence of positive solution
by variational methods; they also discussed the energy doubling property
of nodal solutions by Nehari manifold. The results in \cite {HL2014}
complement the corresponding results in \cite{LY2014,LLS}.
Li and Ye \cite{LY2014} showed that  \eqref{004} has no nontrivial
solution provided $f(x,u)=|u|^{p-2}u$, $p\in (2,3)$ when $\lambda>0$
is sufficiently large. If $V(x)=b$ and $f(x,u)=f(u)$ is superlinear at infinity,
Li, Li and Shi \cite{LLS} showed that  \eqref{004} has at least one positive
radial solution for $\lambda>0$ sufficiently small. Wu, Huang and Liu \cite {WHL}
gave a total description on the positive solutions to \eqref{004}, and they
made an observation on the sign-changing solutions.
When $f(x,u)$ is asymptotically linear with respect to $u$ at infinity,
Ye and Yin \cite {YY} studied \eqref{004} and proved the existence of
positive solution for $\lambda$ sufficiently small and the nonexistence
result for $\lambda$ sufficiently large. Very recently, some authors
extend the problem \eqref{004} to the p-Kirchhoff elliptic equations,
 see e.g. \cite{CY,CZ,CD,LC} and the references therein.

In the spirit of \cite{LLS,ZSN}, for any continuous function $m$, we
establish a multiplicity criterion of positive radial solutions to \eqref{1.1}
using a variational method and an iterative technique.
The main result of this article reads as follows.

\begin{theorem} \label{thm1.1}
Assume that $N\geq3$, and $a,b$ are positive constants, $\lambda\geq0$ is a
parameter and the following conditions hold:
\begin{itemize}
\item[(H1)] $f\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ and there are positive
constants $c$ and $p\in (2,2^*)$ such that $f(t)\leq c(1+t^{p-1})$ for
$t\geq0$, where
$2^*=\frac{2N}{N-2}$ for $N\geq 3$;

\item[(H2)] $\lim_{t\to 0}\frac{f(t)}{t}=0$;

\item[(H3)] $\lim_{t\to \infty}\frac{f(t)}{t}=\infty$;

\item[(H4)] $0\leq h(x)=h(|x|)\in L^{q'}({\mathbb{R}}^N)$,
$\langle \nabla h(x),x\rangle \in L^{q'}({\mathbb{R}}^N)$, where
$q'=\frac{2^*}{2^*-q}$, $\langle\cdot, \cdot\rangle $ denotes the usual
inner product in ${\mathbb{R}}^N$ and $1<q<2$.
\end{itemize}
Then for any positive  continuous function $m$, there exist two
constants $\tilde{\lambda}>0$  and $m_0>0$ such that for any
$\lambda\in[0,\tilde{\lambda})$,
problem \eqref{1.1} has at least two positive solutions if $\|h\|_{q'}<m_0$.
\end{theorem}

 Since the result in Theorem \ref{thm1.1} holds for $m(t)=t$, our result generalizes
\cite[Theorem 1.1]{LLS}.
In this paper, we give multiplicity results for the positive solutions
of \eqref{1.1}, while the authors \cite {LLS} only studied the existence
of positive solutions. Furthermore, our method is different from that
used in \cite{LLS}, we combine variational methods and iterative technique.

Our result can be regarded as an extension of the bounded case considered
in \cite{ZSN} to the unbounded case. Also we give two positive solutions,
while the authors \cite{ZSN} only studied the existence of positive solutions.

This article is organized as follows:
In Section 2, we give some preliminaries.
In Section 3 and 4 we present the proofs of the main results.
Through out this paper, $C,C_{i}$ are used in various places to denote distinct
constants.

\section{Preliminaries}

Let $H^1(\mathbb{R}^N)$ be the usual Sobolev space equipped with the inner
product and norm
$$
\langle u,v\rangle=\int_{\mathbb{R}^N}(\nabla u\cdot\nabla v +buv)dx,\quad
\|u\|=\langle u,u\rangle^{\frac{1}{2}}.
$$
We denote by $\|\cdot\|_{p}$ the usual $L^{p}(\mathbb{R}^N)$ norm.
We only consider positive solutions to \eqref {1.1}, and we assume that $f(t)=0$
for $t<0$

To obtain our result, we have to overcome various difficulties.
On one hand, it is well known that Sobolev embedding
$H^1(\mathbb{R}^N)\hookrightarrow L^{p}(\mathbb{R}^N)$ is continuous but
 not compact  for $p\in [2,2^{*}]$, and then it is usually difficult to prove
that a minimizing sequence or a Palais-Smale sequence is strongly convergent
if we seek solutions of \eqref {1.1} by variational methods.
To overcome this difficulty, we usually restrict problem \eqref {1.1}
in the radial function space.
Let $H=H_{r}^{1}(\mathbb{R}^N)$ be the subspace of $H^1(\mathbb{R}^N)$
containing only the radial functions. We recall \cite{W},
$H\hookrightarrow L^{p}(R^N)$ compactly (continuously) for
$p\in (2,2^{*})$($p\in [2,2^{*}]$). That is, there exists a
$\gamma_{p}>0$ such that $\|u\|_{p}\leq \gamma_{p}\|u\|$,
$p\in [2,2^{*}]$. On the other hand, the nonlinearity $f$ may not
satisfy (AR) or 4-superlinearity, it is difficult to get the
boundedness of any (PS) sequence even if a (PS) sequence has been obtained.
To overcome this difficulty, we use a ``freezing" technique whose
formulation appears initially in \cite{FGM}. This technique will help
us to change  problem \eqref {1.1} into semilinear equation.
That is, for each fixed $\omega\in H$, we consider the ``freezing"
problem given by
\[
\Big(a+\lambda m\Big(\int_{\mathbb{R}^{N}}(|\nabla \omega|^{2}
+b\omega^2)dx\Big)\Big)\big(-\Delta u+bu\big)=f(u) +h(x)|u|^{q-2}u,
\quad \text{in }\mathbb{R}^{N},
\]
and the associated function $J_{\omega}:H\to  \mathbb{R}$ is defined by
\begin{equation*}
J_{\omega}(u)=\frac{1}{2}\big(a+\lambda m(\|\omega\|^2)\big)\|u\|^2
-\int_{\mathbb{R}^N}F(u)dx-\frac{1}{q}\int_{\mathbb{R}^N}h(x)|u|^{q}dx,\quad
 u\in H,
\end{equation*}
where $F(t)=\int_{0}^{t}f(s)ds$. Clearly, by the assumptions imposed on
$f$, $h$ and $m$, we know that $J_{\omega}(u)$ is well defined on $H$, it is of
class $C^1$  for all $\lambda \geq 0$, and
\begin{align*}
\langle J_{\omega}'(u), v\rangle
&=\big(a+\lambda m(\|\omega\|^2)\big)\int_{\mathbb{R}^N}(\nabla u\cdot\nabla v+buv )dx
-\int_{\mathbb{R}^N}f(u)v\,dx \\
&\quad -\int_{\mathbb{R}^N}h(x)|u|^{q-2}uv\,dx, \quad  u,v\in H.
\end{align*}

Next we recall a monotonicity method by Jeanjean \cite{LJ} and Struwe \cite{MS},
which will be used in our proof. The version here is from \cite{LJ}.

\begin{theorem} \label{thm2.1}
Let $(X,\|\cdot\|)$ be a Banach space and $I\subset \mathbb{R}_+$ an interval.
Consider the family of $C^1$ functionals on $X$
\begin{equation*}
J_\mu(u)=A(u)-\mu B(u),\quad \mu \in I,
\end{equation*}
with $B$ nonnegative and either $A(u)\to  \infty$ or $B(u)\to  \infty$
as $\|u\|\to  \infty$ and such that $J_\mu(0)=0$.
For any $\mu \in I$, we set
\begin{equation*}
\Gamma_\mu=\big\{\gamma\in C([0,1],X):\gamma(0)=0,J_\mu(\gamma(1))<0\big\}.
\end{equation*}
If for every $\mu\in I$, the set $\Gamma_\mu$ is nonempty and
\begin{equation*}
c_{\mu}=\inf_{\gamma\in\Gamma_\mu}\max_{t\in[0,1]}J_{\mu}(\gamma(t))>0,
\end{equation*}
then for almost every $\mu \in I$, there exists a sequence $\{u_n\}\subset X$
such that
\begin{itemize}
\item[(i)] $\{u_n\}$ is bounded;

\item[(ii)] $J_{\mu}(u_n)\to  c_{\mu}$ as $n\to  \infty$;

\item[(iii)] $J_{\mu}'(u_n)\to  0$ as $n\to  \infty$, in the dual space
$X^{-1}$ of $X$.
\end{itemize}
\end{theorem}

\section{First positive solution of \eqref{1.1}}

In this section, we use Theorem \ref{thm2.1} to obtain the first positive solution for
\eqref{1.1}. In the setting of Theorem \ref{thm2.1}, we have
$X=H$, $I=[1/2,1]$, and for each fixed $\omega\in H$,
\begin{equation*}
A_{\omega}(u)=\frac{1}{2}\left(a+\lambda m(\|\omega\|^2)\right)
 \|u\|^2-\frac{1}{q}\int_{\mathbb{R}^N}h(x)(u^{+})^{q}dx, \quad
 B(u)=\int_{\mathbb{R}^N}F(u)dx,
\end{equation*}
where $u^{+}=\max\{u,0\}$.
So the perturbed functional that we study is
\begin{equation*}
I_{\omega,\tau}(u)=\frac{1}{2}\left(a+\lambda m(\|\omega\|^2)\right)\|u\|^2
-\frac{1}{q}\int_{\mathbb{R}^N}h(x)(u^{+})^{q}dx
-\tau\int_{\mathbb{R}^N}F(u)dx,\quad \tau\in I.
\end{equation*}
It follows from (H4) that
\begin{align*}
\frac{1}{2}\left(a+\lambda m(\|\omega\|^2)\right)\|u\|^2
 -\frac{1}{q}\int_{\mathbb{R}^N}h(x)(u^{+})^{q}dx
&\geq  \frac{a}{2}\|u\|^2-\frac{1}{q}\|h\|_{q'}\|u\|_{2^*}^{q}\\
&\geq  \frac{a}{2}\|u\|^2-\frac{\gamma_{2^*}^q}{q}\|h\|_{q'}\|u\|^{q},
\end{align*}
which implies that $A_{\omega}(u)\to  \infty$ as $\|u\|\to  \infty$ and obviously,
$B(u)\geq0$.
Next, we give some lemmas that are important for proving our main result.

\begin{lemma} \label{lem3.1}
For each  $\omega\in H$ and $\tau\in I$, each bounded (PS) sequence
of the functional $I_{\omega,\tau}$ in $H$ admits a convergent subsequence.
\end{lemma}

\begin{proof}
 For each given $\omega\in H$ and $\tau\in I$, let $\{u_{n}\}$ be a bounded
(PS) sequence of the functional $I_{\omega,\tau}$, namely
$\{u_{n}\}$ and $\{I_{\omega,\tau}(u_{n})\}$ are bounded, and
\[
I_{\omega,\tau}'(u_{n})\to  0 \quad \text{in } H^{-1},
\]
where $H^{-1}$ is the dual space of $H$. Since $\{u_{n}\}$ is bounded,
subject to a subsequence, we can assume that there exists
$u\in H$ such that as $n\to  \infty$,
\begin{equation}\label{3.1}
\begin{gathered}
 u_n\rightharpoonup u, \quad \text{in }H; \\
u_n\to  u,   \quad\text{in } L^s(\mathbb{R}^{N})\; (2<s<\frac{2N}{N-2});\\
 u_n\to  u,\quad \text{a.e } x\in \mathbb{R}^{N}.
\end{gathered}
\end{equation}
By (H1) and (H2), for any $\varepsilon>0$, there exists $C_{\epsilon}>0$ such that
\begin{equation}
|f(t)|\leq b\varepsilon|t|+C_{\varepsilon}|t|^{p-1},\quad  t\in \mathbb{R}.
 \label{13}
\end{equation}
It follows from \eqref{13}, the H\"{o}lder inequality, the Sobolev inequality
and the boundedness of $\{u_{n}\}$ that
\begin{align*}
\big|\int_{\mathbb{R}^{N}}f(u_{n})(u_{n}-u)dx\big|
&\leq \int_{\mathbb{R}^{N}}\left|f(u_{n})(u_{n}-u)\right|dx \\
&\leq b\varepsilon\int_{\mathbb{R}^{N}}|u_{n}||u_{n}-u|dx
 +C_{\varepsilon}\int_{\mathbb{R}^{N}}|u_{n}|^{p-1}|u_{n}-u|dx\\
&\leq b\varepsilon \|u_{n}\|_{2}\|u_{n}-u\|_{2}
 +C_{\varepsilon}\|u_{n}\|_{p}^{p-1}\|u_{n}-u\|_{p}\\
&\leq \varepsilon C \|u_{n}\|\|u_{n}-u\|
 +C_{\varepsilon}C\|u_{n}\|^{p-1}\|u_{n}-u\|_{p}\\
&\leq \varepsilon C+C_{\varepsilon}C\|u_{n}-u\|_{p}.
\end{align*}
Then,  by \eqref{3.1} we can obtain
\begin{equation}
\limsup_{n\to  \infty} \big|\int_{\mathbb{R}^{N}}f(u_{n})(u_{n}-u)dx\big|
\leq \varepsilon C.\label{3.3}
\end{equation}
Therefore,  using the arbitrariness of $\varepsilon$ in \eqref{3.3}, we have
\begin{equation}
\int_{\mathbb{R}^{N}}f(u_{n})(u_{n}-u)dx\to  0,\quad \text{as }
 n\to \infty.\label{3.4}
\end{equation}
Using \eqref{3.1}, we have
$$
(u^{+}_n)^{q-1}(u_{n}-u)\to  0,\quad \text{a.e. } x\in \mathbb{R}^{N}.
$$
Since
\begin{align*}
\int_{\mathbb{R}^{N}}\big(u^{+}_n)^{q-1}(u_{n}-u)\big)^{2^*/q}dx
&\leq  \Big(\int_{\mathbb{R}^{N}}(u^{+}_n)^{2^*}dx\Big)^{\frac{q-1}{q}}
 \Big(\int_{\mathbb{R}^{N}}(u_n-u)^{2^*}dx\Big)^{1/q}\\
&\leq  \|u_n\|_{2^*}^{\frac{(q-1)2^*}{q}}\|u_n-u\|_{2^*}^{2^*/q}\\
&\leq  C\|u_n\|^{\frac{(q-1)2^*}{q}}\|u_n-u\|^{2^*/q}<+\infty.
\end{align*}
So, $(u^{+}_n)^{q-1}(u_{n}-u)$ is bounded in $L^{2^*/q}(\mathbb{R}^{N})$.
Hence, going if necessary to a subsequence, we can assume that
$(u^{+}_n)^{q-1}(u_{n}-u)\rightharpoonup 0$ in $L^{2^*/q}(\mathbb{R}^{N})$
and using  (H4),
\begin{equation}
\int_{\mathbb{R}^{N}}h(x)(u^{+}_n)^{q-1}(u_{n}-u)dx\to  0,\quad
\text{as } n\to \infty.\label{3.40}
\end{equation}
Thus, by using \eqref{3.4}, \eqref{3.40} and $I_{\omega,\tau}'(u_{n})\to  0$,
we have
\begin{align*}
\left(a+\lambda m(\|\omega\|^2)\right)\langle u_{n},u_{n}-u\rangle
&=\langle I_{\omega,\tau}'(u_{n}),u_{n}-u\rangle
 +\tau\int_{\mathbb{R}^{N}}f(u_{n})(u_{n}-u)dx\\
&\quad +\int_{\mathbb{R}^{N}}h(x)(u^{+}_n)^{q-1}(u_{n}-u)dx\to  0;
\end{align*}
that is,
$\|u_{n}\|\to  \|u\|$. This together with $u_n\rightharpoonup u$
shows that $u_n\to  u$ in $H$.
\end{proof}


\begin{lemma} \label{lem3.2} 
For each $R>0$ and $\omega\in H$ with
 $\|\omega\|\leq R$, there exists $\tilde{\lambda}=\tilde{\lambda}(R)>0$,
$m_0>0$ and ${\tau_{k}}\subset [1/2,1]$ satisfying that ${\tau_{k}}\to  1$
as $k\to  \infty$, such that $I_{\omega,\tau_{k}}$ has a nontrivial
critical point $u_{\omega,\tau_{k}}$ if $\lambda\in [0,\tilde{\lambda})$,
$\|h\|_{q'}<m_0$.
\end{lemma}

\begin{proof}
 We choose a function $\phi\in C_{0}^{\infty}(\mathbb{R}^N)$ with
$\phi \geq 0$, $\|\phi\|=1$ and $\operatorname{supp} (\phi)\subset B(0,R_0)$
for some $R_0>0$. For given constant $R>0$, there exists
$\tilde{\lambda}=\tilde{\lambda}(R)>0$, such that if
$\lambda\in [0,\tilde{\lambda})$, we have
$\lambda \max_{\xi\in[0,R^2]}m(\xi)\leq 1$.
By (H3), for $\frac{2(a+1)}{\int_{B(0,R_0)}\phi^2dx}>0$, there exists
$C_{1}>0$ such that
\begin{equation*}
 F(t)\geq \frac{2(a+1)}{\int_{B(0,R_0)}\phi^2dx}t^2-C_{1},\quad t\geq0 .
\end{equation*}
So, for $t\geq 0$ we get
\begin{equation}
\begin{aligned}
 I_{\omega,\tau}(t\phi)
&= \frac{t^2}{2}\left(a+\lambda m(\|\omega\|^2)\right)\|\phi\|^2
 -\tau\int_{\mathbb{R}^N}F(t\phi)dx-\frac{t^q}{q}\int_{\mathbb{R}^N}h(x)\phi^q dx\\
&\leq  \frac{t^2}{2}\left(a+\lambda m(\|\omega\|^2)\right)
 -\frac{t^2}{2}\frac{2(a+1)}{\int_{B(0,R_0)}\phi^2dx}\int_{B(0,R_0)}\phi^{2}dx
 -\frac{t^q}{q}\int_{\mathbb{R}^N}h(x)\phi^q dx\\
&\quad +\frac{C_{1}|B(0,R_0)|}{2}\\
&\leq -\frac{t^2}{2}(a+1)+\frac{C_{1}|B(0,R_0)|}{2}
 -\frac{t^q}{q}\int_{\mathbb{R}^N}h(x)\phi^q dx.
\end{aligned} \label{3.5}
\end{equation}
On one hand, by (H4), we can obtain
\begin{equation*}
 I_{\omega,\tau}(t\phi)\leq -\frac{t^2}{2}(a+1)
+\frac{C_{1}|B(0,R_0)|}{2}-\frac{t^q}{q}\int_{\mathbb{R}^N}h(x)\phi^q dx\to
 -\infty,\quad t\to  +\infty;
\end{equation*}
on the other hand, by \eqref{3.5}, we known that there exists a constant
$C=C(R_0)>0$ (depending on $\omega$ and $\tau$) such that
\begin{equation}
 \max_{t\geq0}I_{\omega,\tau}(t\phi)\leq\frac{C_{1}|B(0,R_0)|}{2}:=C. \label{3.50}
\end{equation}
Hence, we can choose $t>0$ large enough such that $I_{\omega,\tau}(t\phi)<0$;
 that is, $\Gamma_{\omega,\tau}\neq \emptyset$, where,
$\Gamma_{\omega,\tau}=\{\gamma\in C([0,1],H):\gamma(0)=0,
I_{\omega,\tau}(\gamma(1))<0\}$.

Using (H1) and (H2), for $\varepsilon\in (0, \frac{a}{2})$, there exists
$C_{2}(\epsilon)>0$ such that
\begin{equation*}
 F(t)\leq \frac{\varepsilon}{2}bt^2+C_{2}(\varepsilon)t^p,\quad t\geq0 .
\end{equation*}
 By Sobolev's embedding theorem, there exists $C_3(\varepsilon)>0$ such that
\begin{align*}
I_{\omega,\tau}(u)
&= \frac{1}{2}\left(a+\lambda m(\|\omega\|^2)\right)\|u\|^2
 -\tau\int_{\mathbb{R}^N}F(u)dx-\frac{1}{q}\int_{\mathbb{R}^N}h(x)(u^{+})^{q}dx\\
&\geq  \frac{a}{2}\|u\|^2-\frac{\varepsilon}{2} b\int_{\mathbb{R}^{N}} u^2 dx
 -C_{2}(\varepsilon)\int_{\mathbb{R}^{N}}|u|^pdx
 -\frac{1}{q}\|h\|_{q'}\|u\|_{2^*}^{q}\\
&\geq  \frac{a}{4}\|u\|^2-C_{3}(\varepsilon)\|u\|^p
 -\frac{\gamma_{2^*}^q}{q}\|h\|_{q'}\|u\|^{q} \\
&\geq  \|u\|^{q}\Big(\frac{a}{4}\|u\|^{2-q}-C_{3}(\varepsilon)\|u\|^{p-q}
 -\frac{\gamma_{2^*}^q}{q}\|h\|_{q'}\Big).
\end{align*}
Setting
$$
g(t)=\frac{a}{4}t^{2-q}-C_{3}(\varepsilon)t^{p-q}
$$
for $t\geq 0$. Since $1<q<2<p<2^*$,  we can choose a constant $\rho>0$
sufficiently small such that $g(\rho)>0$.
Taking $m_0:=\frac{q}{2\gamma_{2^*}^q}g(\rho)$, it then follows that
there exists a constant $c:=\frac{1}{2}g(\rho)\rho^q>0$ which is independent
of $\tau$, $\lambda$ and $\omega$ such that
\begin{equation*}
 I_{\omega,\tau}(u)\big|_{\|u\|= \rho}\geq c>0,
\end{equation*}
for any $\tau\in I$, $\omega\in H$ and all $h$ satisfying $\|h\|_{q'}<m_0$.
Fix $\tau\in I$ and for any $\gamma\in \Gamma_{\omega,\tau}$,
by the definition of $\Gamma_{\omega,\tau}$, we have $\|\gamma(1)\|>\rho$.
Since $\gamma(0)=0$, then from intermediate value theorem, we deduce
that there exists $t_{\gamma}\in (0,1)$ such that $\|\gamma(t_{\gamma})\|=\rho $.
Therefore, for any fixed $\tau\in I$,
\begin{equation*}
c_{\omega,\tau}=\inf_{\gamma \in\Gamma_{\omega,\tau} }
\max_{t\in[0,1]}I_{\omega,\tau}(\gamma(t))
\geq \inf_{\gamma\in \Gamma_{\omega,\tau}}I_{\omega,\tau}
(\gamma(t_{\gamma}))\geq c>0.
\end{equation*}

Following Theorem \ref{thm2.1}, there are $\{\tau_{k}\}\subset [1/2,1)$, with
$\tau_{k}\to  1$ as $k\to  \infty$, and for every $k$, there exists a sequence
$\{u_{n,\omega,\tau_{k}}\}\subset H$, such that $\{u_{n,\omega,\tau_{k}}\}$
is bounded and $I_{\omega,\tau_{k}}(u_{n,\omega,\tau_{k}})
\to  c_{\omega,\tau_{k}},I'_{\omega,\tau_{k}}(u_{n,\omega,\tau_{k}})\to 0$,
 where
\begin{gather*}
c_{\omega,\tau_{k}}=\inf_{\gamma\in\Gamma_{\omega,\tau_{k}}}
\sup_{u\in\gamma([0,1])}I_{\omega,\tau_{k}}(u),
\\
\Gamma_{\omega,\tau_{k}}
=\big\{\gamma\in C([0,1], H)|\gamma(0)=0,
\; I_{\omega,\tau_{k}}(\gamma(1))<0\big\}.
\end{gather*}
Furthermore, by Lemma \ref{lem3.1}, we can suppose that there exists
$u_{\omega,\tau_{k}}\in H$ such that $u_{n,\omega,\tau_{k}}\to  u_{\omega,\tau_{k}}$,
 and then
$$
I_{\omega,\tau_{k}}(u_{\omega,\tau_{k}}) =c_{\omega,\tau_{k}},  \quad
I'_{\omega,\tau_{k}}(u_{\omega,\tau_{k}})=0.
$$
From the above discussion, we get that for given $R>0$ and $\omega\in H$ with
$\|\omega\|\leq R$, there exists $\tilde{\lambda}=\tilde{\lambda}(R)>0$,
 $m_0>0$ and ${\tau_{k}}\subset [\frac{1}{2},1]$ satisfying that
${\tau_{k}}\to  1$ as $k\to  \infty$, such that $I_{\omega,\tau_{k}}$ has
a nontrivial critical point $u_{\omega,\tau_{k}}$ if
$\lambda\in [0,\tilde{\lambda})$, $\|h\|_{q'}<m_0$ and
\begin{equation}
c_{\omega,\tau_{k}}=I_{\omega,\tau_{k}}(u_{\omega,\tau_{k}})
\leq \max_{t\geq0}I_{\omega,\tau_{k}}(t\phi)\leq C, \label{311}
\end{equation}
where $C$ is given in \eqref{3.50}.
\end{proof}

\begin{lemma} \label{lem3.3}
Let $u_{\omega,\tau_{k}}$ be a critical point of $I_{\omega,\tau_{k}}$ at level
$c_{\omega,\tau_{k}}$. Then $\{u_{\omega,\tau_{k}}\}$ are uniformly bounded.
\end{lemma}

\begin{proof}
 It follows from Lemma \ref{lem3.2} that $u_{\omega,\tau_{k}}$ is a weak solution
of the  problem
$$
\left(a+\lambda m(\|\omega\|^2)\right)(-\Delta u+bu)
=\tau_{k}f(u)+h(x)(u^{+})^{q-1};
$$
therefore,
\begin{equation}
\left(a+\lambda m(\|\omega\|^2)\right)(-\Delta u_{\omega,\tau_{k}}
+bu_{\omega,\tau_{k}})
=\tau_{k}f(u_{\omega,\tau_{k}})+h(x)(u^{+}_{\omega,\tau_{k}})^{q-1}.\label{31}
\end{equation}
Hence, we have the following Pohozaev identity
\begin{equation}
\begin{aligned}
&\Big(\frac{N-2}{2}\int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx
+\frac{Nb}{2}\int_{\mathbb{R}^{N}} u_{\omega,\tau_{k}}^{2}dx\Big)
\left(a+\lambda m(\|\omega\|^2)\right)\\
&= N\tau_{k}\int_{\mathbb{R}^{N}}F(u_{\omega,\tau_{k}})dx
+ \frac{1}{q}\int_{\mathbb{R}^{N}}
\Big(Nh+\langle\nabla h(x),x\rangle\Big)(u^{+}_{\omega,\tau_{k}})^{q}dx\,.
\end{aligned} \label{32}
\end{equation}
 The proof is similar to that of \cite[Proposition 1]{BL}, we omit here.

By letting $c_{\omega,\tau_{k}}=I_{\omega,\tau_{k}}(u_{\omega,\tau_{k}})$, we have
\begin{equation}
\begin{aligned}
 c_{\omega,\tau_{k}}
&=\frac{1}{2}\left(a+\lambda m(\|\omega\|^2)\right)
 \|u_{\omega,\tau_{k}}\|^{2}- \tau_{k}\int_{\mathbb{R}^{N}}F(u_{\omega,\tau_{k}})dx\\
&\quad - \frac{1}{q}\int_{\mathbb{R}^{N}}h(x)(u^{+}_{\omega,\tau_{k}})^{q}dx.
\end{aligned} \label{33}
\end{equation}
By (H4) and the H\"{o}lder inequality, we deduce that
\begin{equation}
\begin{aligned}
 \frac{1}{q}\Big|\int_{\mathbb{R}^{N}}\langle\nabla h(x),x\rangle
(u^{+}_{\omega,\tau_{k}})^{q}dx\Big|
&\leq \frac{1}{q}\int_{\mathbb{R}^{N}}\big|\langle\nabla h(x),x\rangle
 (u^{+}_{\omega,\tau_{k}})^{q}\big|dx\\
&\leq \frac{1}{q}\|\langle\nabla h(x),x\rangle\|_{q'}
 \|(u^{+}_{\omega,\tau_{k}})\|_{2^*}^{q}\\
&\leq C_{4}\Big(\int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx
\Big)^{q/2}.
\end{aligned}\label{344}
\end{equation}
Therefore, by \eqref{311} and \eqref{32}-\eqref{344}, we  obtain
\begin{align*}
 \int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx
&= \frac{Nc_{\omega,\tau_{k}}-\frac{1}{q}\int_{\mathbb{R}^{N}}
 \langle\nabla h(x),x\rangle (u^{+}_{\omega,\tau_{k}})^{q}dx}
 {a+\lambda m(\|\omega\|^2)}\\
&\leq \frac{Nc_{\omega,\tau_{k}}+C_{4}
 \left(\int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx\right)^{q/2}}
 {a+\lambda m(\|\omega\|^2)}\\
&\leq \frac{NC+C_{4}\left(\int_{\mathbb{R}^{N}}|\nabla u_{\omega,
 \tau_{k}}|^{2}dx\right)^{q/2}}{a}.
\end{align*}
Because of $1<q<2$, $\int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx$
is uniformly bounded. That is, there exists a constant $C_5>0$, independent of $\tau$,
 $\lambda$ and $\omega$, such that
\begin{equation}
\int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx\leq C_5.\label{34}
\end{equation}
Furthermore, by (H1) and (H2), there exists a constant $C_{6}>0$ such that 
\begin{equation}
|f(t)|\leq \frac{ab}{2}|t|+C_{6}|t|^{2^{*}-1},\quad t\in \mathbb{R}. \label{113}
\end{equation}
 Hence, by \eqref{31} and \eqref{113}, we have
\begin{align*}
& \left(a+\lambda m(\|\omega\|^2)\right)\|u_{\omega,\tau_{k}}\|^{2}\\
&= \tau_{k}\int_{\mathbb{R}^{N}}f(u_{\omega,\tau_{k}})u_{\omega,\tau_{k}}dx
 +\int_{\mathbb{R}^{N}}h(x)(u^{+}_{\omega,\tau_{k}})^{q}dx\\
&\leq \frac{ab}{2}\int_{\mathbb{R}^{N}}|u_{\omega,\tau_{k}}|^{2}dx
  +C_{6}\int_{\mathbb{R}^{N}}|u_{\omega,\tau_{k}}|^{2^{*}}dx\
  +\|h\|_{q'}\big(\int_{R^{N}}|u_{\omega,\tau_{k}}|^{2^{*}}dx\big)^{q/2^*}\\
&\leq \frac{a}{2}\|u_{\omega,\tau_{k}}\|^{2}
  +C_{6}\int_{R^{N}}|u_{\omega,\tau_{k}}|^{2^{*}}dx
  + \|h\|_{q'}\big(\int_{R^{N}}|u_{\omega,\tau_{k}}|^{2^{*}}dx\big)^{q/2^*}.
\end{align*}
Using \eqref{34}, we conclude that
\begin{align*}
\frac{a}{2}\|u_{\omega,\tau_{k}}\|^{2}
&\leq C_{6}\int_{\mathbb{R}^{N}}|u_{\omega,\tau_{k}}|^{2^{*}}dx
 +\|h\|_{q'}\Big(\int_{R^{N}}|u_{\omega,\tau_{k}}|^{2^{*}}dx\Big)^{q/2^*}\\
&\leq C_{7}\Big(\int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx
 \Big)^{2^*/2}+C_{8}\Big(\int_{\mathbb{R}^{N}}|\nabla u_{\omega,\tau_{k}}|^{2}dx
 \Big)^{q/2}\\
&\leq C_{7}C_5^{2^*/2}+C_{8}C_5^{q/2}.
\end{align*}
Then $\|u_{\omega,\tau_{k}}\|^{2}\leq C_{9}$,
where $ C_{9}=\frac{2}{a}C_{7}C_5^{2^*/2}+\frac{2}{a}C_{8}C_5^{q/2}$
which is independent of $\tau$, $\lambda$ and $\omega$.
If we set $R=\sqrt{C_{9}}$, then for any  $\omega\in H$ with
$\|\omega\|\leq R$, there exist $\tilde{\lambda}>0$  $m_0>0$ which are
independent of $\tau$, $\lambda$ and $\omega$, such that
 $I_{\omega,\tau_{k}}$ has a nontrivial critical point $u_{\omega,\tau_{k}}$
with $\|u_{\omega,\tau_{k}}\|\leq R$ when  $\lambda\in [0,\tilde{\lambda})$,
$\|h\|_{2}<m_0$. And also, $\{u_{\omega,\tau_{k}}\}$ is uniformly bounded.

 Now we choose $R=\sqrt{C_{9}}$ as above and construct a family of sequence
by iterative techniques. For every $k$, if we let $\omega=\omega_{0}\equiv 0$,
by the previous arguments, we know $I_{\omega_{0},\tau_{k}}$ has a nontrivial
critical point and denote it by  $u_{1,k}$ with $\|u_{1,k}\|\leq R$.
Let $\omega=u_{1,k}$, then $I_{u_{1,k},\tau_{k}}$ has a nontrivial critical
point and denote it by  $u_{2,k}$ with $\|u_{2,k}\|\leq R$.
 Hence, by induction, we can get a sequence $\{u_{n,k}\}$ with $\|u_{n,k}\|\leq R$,
$n=1,2,\dots$, such that $I'_{u_{n,k},\tau_{k}}(u_{n+1,k})\cdot v=0$
for all $v\in H$.
\end{proof}


\subsection*{Existence of the first positive solution to \eqref{1.1}}
To complete the proof, we proceed in two steps.
\smallskip

\noindent\textbf{Step 1.} For any fixed $k$, the iterative sequence $\{u_{n,k}\}$
constructed in Lemma \ref{lem3.3} is convergent to a function $u_{k}$, which is a
critical point of  $I_{u_{k},\tau_{k}}$.

 Since for fixed $k$ and for all $n\in \mathbb{N}$,  $\|u_{n,k}\|\leq R$,
 if necessary going to a subsequence, we  suppose that there exists $u_k\in H$
such that as $n\to  \infty$,
\begin{equation}
\begin{gathered}
 u_{n,k}\rightharpoonup u_k, \quad\text{in }H; \\
u_{n,k}\to  u_k,     \quad \text{in } L^p(\mathbb{R}^{N})(2<p<2^*);\\
 u_{n,k}\to  u_k, \quad \text{a.e. } x\in \mathbb{R}^{N}.
\end{gathered}\label{3.01}
\end{equation}
Also we have $\|u_{k}\|\leq R$, for all $k\in \mathbb{N}$.
From the subcritical growth of $f$ and \eqref{3.01}, we see that
\begin{gather}
 \int_{R^{N}}\big(f(u_{n,k})-f(u_{k})\big)(u_{n,k}-u_{k})dx\to  0,
\quad\text{as } n\to  \infty;\label{3.10}\\
 \int_{R^{N}}\big(h(x)(u^{+}_{n,k})^{q-1}-h(x)(u^{+}_{k})^{q-1}\big)(u_{n,k}
-u_{k})dx\to  0, \quad\text{as }n\to  \infty.\label{3.101}
\end{gather}
The proof is similar to that of \eqref{3.4} and \eqref{3.40}, and we omit here.
Then we have
\begin{align*}
&\left(a+\lambda m(\|u_{n-1,k}\|^{2})\right)\|u_{n,k}-u_{k}\|^{2}\\
&= \left\langle I'_{u_{n-1,k},\tau_{k}}(u_{n,k})-I'_{u_{n-1,k},\tau_{k}}(u_{k}),u_{n,k}-u_{k}\right\rangle\\
&\quad + \tau_{k}\int_{R^{N}}\Big(f(u_{n,k})-f(u_{k})\Big)(u_{n,k}-u_{k})dx\\
&\quad + \int_{R^{N}}\Big(h(x)(u^{+}_{n,k})^{q-1}-h(x)(u^{+}_{k})^{q-1}
 \Big)(u_{n,k}-u_{k})dx
\to   0\quad \text{as } n\to  \infty;
\end{align*}
that is, $u_{n,k}\to  u_{k}$  in $H$ as $n\to  \infty$.
Thus, for any $v\in H$, as $n\to  \infty$,  we have
\begin{gather*}
a+\lambda m(\|u_{n-1,k}\|^{2})\to  a+\lambda m(\|u_{k}\|^{2}),\\
\int_{\mathbb{R}^{N}}\left(\nabla u_{n,k}\cdot\nabla v+bu_{n,k}v\right)dx
 \to \int_{\mathbb{R}^{N}}\left(\nabla u_{k}\cdot\nabla v+bu_{k}v\right)dx,\\
\tau_{k}\int_{\mathbb{R}^{N}}f(u_{n,k})v\,dx
 \to \tau_{k}\int_{\mathbb{R}^{N}}f(u_{k})v\,dx,\\
\tau_{k}\int_{\mathbb{R}^{N}}F(u_{n,k})dx
\to \tau_{k}\int_{\mathbb{R}^{N}}F(u_{k})dx.
\end{gather*}
Also, we have
\begin{gather*}
\int_{\mathbb{R}^{N}}h(x)(u^{+}_{n,k})^{q-1}v\,dx
 \to  \int_{\mathbb{R}^{N}}h(x)(u^{+}_{k})^{q-1}v\,dx,\\
\int_{\mathbb{R}^{N}}h(x)(u^{+}_{n,k})^{q}dx
\to  \int_{\mathbb{R}^{N}}h(x)(u^{+}_{k})^{q}dx,\quad\text{as } n\to  \infty,
\end{gather*}
the proof is similar to that of \eqref{3.40}, and we omit here.
So, we obtain
\begin{gather*}
I'_{u_{k},\tau_{k}}(u_{k})\cdot v
=\lim_{n\to  \infty}I'_{u_{n-1,k},\tau_{k}}(u_{n,k})\cdot v=0,\\
I_{u_{k},\tau_{k}}(u_{k})=\lim_{n\to  \infty}I _{u_{n-1,k},
\tau_{k}}(u_{n,k})=\lim_{n\to  \infty}c_{u_{n-1,k},\tau_{k}}\geq c>0;
\end{gather*}
that is, for any $v\in H$,
$$
I'_{u_{k},\tau_{k}}(u_{k})\cdot v=0,\quad  I_{u_{k},\tau_{k}}(u_{k})\geq c>0.
$$
\smallskip

\noindent\textbf{Step 2.}
The sequence $\{u_{k}\}$ obtained in step 1 is convergent to a nontrivial
positive solution of \eqref{1.1}.

Since $\|u_{k}\|\leq R$ for all $k\in \mathbb{N}$, without loss of generality,
we can assume that there exists a function $u\in H$ such that
\begin{equation}
\begin{gathered}
 u_{k}\rightharpoonup u, \quad \text{in }H; \\
u_{k}\to  u,  \quad \text{in } L^p(\mathbb{R}^{N})(2<p<2^*);\\
 u_{k}\to  u, \quad \text{a.e. } x\in \mathbb{R}^{N}.
\end{gathered} \label{3.01b}
\end{equation}
By the similar proof to that of \eqref{3.10} and \eqref{3.101}, we have
\begin{gather*}
\int_{\mathbb{R}^{N}}\big(f(u_{k})-f(u)\big)(u_{k}-u)dx=o(1),\\
\int_{R^{N}}\big(h(x)(u^{+}_{k})^{q-1}-h(x)(u^{+})^{q-1}\big)(u_{k}-u)dx=o(1).
\end{gather*}
Now, taking into account that
\begin{align*}
&\left(a+\lambda m(\|u_{k}\|^{2})\right)\|u_{k}-u\|^{2}\\
&= \langle I'_{u_{k},\tau_{k}}(u_{k})-I'_{u_{k},\tau_{k}}(u),u_{k}-u\rangle
 + \tau_{k}\int_{R^{N}}\Big(f(u_{k})-f(u)\Big)(u_{k}-u)dx\\
&\quad + \int_{R^{N}}\Big(h(x)(u^{+}_{k})^{q-1}-h(x)(u^{+})^{q-1}\Big)(u_{k}-u)dx
\to  0\quad \text{as } n\to  \infty,
\end{align*}
we deduce that $u_{k}\to  u$ as $k\to  \infty$.
So for any $v\in H$, as $k\to  \infty$, we have
\begin{gather*}
a+\lambda m(\|u_{k}\|^{2})\to  a+\lambda m(\|u\|^{2}),\\
\int_{\mathbb{R}^{N}}(\nabla u_{k}\cdot\nabla v+bu_{k}v)dx
 \to \int_{\mathbb{R}^{N}}(\nabla u\cdot\nabla v+buv)dx,\\
\tau_{k}\int_{\mathbb{R}^{N}}f(u_{k})v\,dx\to \int_{\mathbb{R}^{N}}f(u)v\,dx,\\
\tau_{k}\int_{\mathbb{R}^{N}}F(u_{k})dx\to \int_{\mathbb{R}^{N}}F(u)dx,\\
\int_{\mathbb{R}^{N}}h(x)(u^{+}_{k})^{q-1}v\,dx
 \to  \int_{\mathbb{R}^{N}}h(x)(u^{+})^{q-1}v\,dx,\\
\int_{\mathbb{R}^{N}}h(x)(u^{+}_{k})^{q}dx
 \to  \int_{\mathbb{R}^{N}}h(x)(u^{+})^{q}dx.
\end{gather*}
So,  for any $v\in H$, as $k\to  \infty$, we can obtain
\begin{align*}
&\left(a+\lambda m(\|u\|^{2})\right)
\int_{\mathbb{R}^{N}}\big(\nabla u\cdot\nabla v+buv\big)dx
 -\int_{\mathbb{R}^{N}}f(u)v\,dx
 -\int_{\mathbb{R}^{N}}h(x)(u^{+})^{q-1}v\,dx\\
&=\lim_{k\to  \infty}I'_{u_{k},\tau_{k}}(u_{k})\cdot v=0,
\end{align*}
and
\begin{align*}
&\frac{1}{2}\left(a+\lambda m(\|u\|^{2})\right)
\int_{\mathbb{R}^{N}}\big(|\nabla u|^2+bu^2\big)dx
-\int_{\mathbb{R}^{N}}F(u)dx
 -\int_{\mathbb{R}^{N}}h(x)(u^{+})^{q}dx\\
&=\lim_{n\to  \infty}I_{u_{k},\tau_{k}}(u_{k})
=c_{u_{k},\tau_{k}}\geq c>0.
\end{align*}
Therefore, $u$ is a nontrivial solution of \eqref{1.1}.
Setting $u^{-}=\max\{-u,0\}$,
Since
\[
(a+\lambda m(\|u\|^2))\langle u,u^-\rangle
-\int_{\mathbb{R}^{N}}f(u)u^-dx
-\int_{\mathbb{R}^{N}}h(x)(u^{+})^{q-1}u^-dx=0,
\]
by (H1) and (H4) we have  $\|u^-\|=0$;
this implies  $u\geq0$ a.e. in $\mathbb{R}^{N}$. So, by the strong maximum
principle, we get that $u$ is positive on $H$. Thus
$u$ is a positive solution of \eqref{1.1} if
 $\lambda\in[0,\tilde{\lambda})$, $\|h\|_{q'}<m_0$.


\section{Second positive solution of \eqref{1.1}}

  In this section, we prove the existence of local minimum solution for
 problem \eqref{1.1} by Ekeland's variational principle. Define the
functional $I:H\to  \mathbb{R}$ by
\begin{equation*}
I_{\lambda}(u)=\frac{a}{2}\|u\|^2+\frac{\lambda}{2}M(\|u\|^2)
-\int_{\mathbb{R}^N}F(u)dx-\frac{1}{q}\int_{\mathbb{R}^N}h(x)(u^+)^{q}dx,
\end{equation*}
where $M(t)=\int_{0}^{t}m(s)ds$. Then, it follows from (H1)--(H4) and the
continuity of $m$ that $I_{\lambda}$ is well defined on $H$ and is $C^{1}$
for all $\lambda\geq 0$, and
\begin{align*}
\langle I_{\lambda}'(u), v\rangle
&=(a+\lambda m(\|u\|^2))\int_{\mathbb{R}^N}(\nabla u\cdot\nabla v+buv )dx\\
&-\int_{\mathbb{R}^N}f(u)v\,dx
 -\int_{\mathbb{R}^N}h(x)(u^{+})^{q-1}v\,dx, \quad  u,v\in H.
\end{align*}

\begin{lemma} \label{lem4.1}
Assume that {\rm (H1), (H2), (H4)} are satisfied. Then there exist constants
$\rho, m_0, \alpha>0$ such that $I_{\lambda}(u)\big|_{\|u\|= \rho}\geq \alpha>0$
with $\|h\|_{2}<m_0$.
\end{lemma}

\begin{proof}
 Using (H1) and (H2), for $\varepsilon\in (0, \frac{a}{2})$, there exists
$C_{12}(\epsilon)>0$ such that
\begin{equation*}
 F(t)\leq \frac{\varepsilon}{2}bt^2+C_{12}(\varepsilon)t^p,\quad t\geq0 .
\end{equation*}
 By Sobolev's embedding theorem, there exists $C_{13}(\varepsilon)>0$ such that
\begin{align*}
I_{\lambda}(u)
&= \frac{a}{2}\|u\|^2+\frac{\lambda}{2}M(\|u\|^2)
 -\int_{\mathbb{R}^N}F(u)dx-\frac{1}{q}\int_{\mathbb{R}^N}h(x)(u^+)^{q}dx\\
&\geq \frac{a}{2}\|u\|^2-\frac{\varepsilon}{12} b\int_{\mathbb{R}^{N}} u^2 dx
 -C_{12}(\varepsilon)\int_{\mathbb{R}^{N}}|u|^pdx-\frac{1}{q}\|h\|_{q'}\|u\|_{2^*}^{q}\\
&\geq \frac{a}{4}\|u\|^2-C_{13}(\varepsilon)\|u\|^p
 -\frac{\gamma_{2^*}}{q}\|h\|_{q'}\|u\|^{q} \\
&\geq \|u\|^{q}\Big(\frac{a}{4}\|u\|^{2-q}-C_{13}(\varepsilon)\|u\|^{p-q}
 -\frac{\gamma_{2^*}}{q}\|h\|_{q'}\Big)
\end{align*}
So, setting
$$
g(t)=\frac{a}{4}t^{2-q}-C_{13}(\varepsilon)t^{p-q}
$$
for $t\geq 0$. Since $1<q<2<p<2^*$,  we can choose a constant $\rho>0$
sufficiently small such that $g(\rho)>0$.
Taking $m_0:=\frac{q}{2\gamma_{2^*}}g(\rho)$, it then follows that there
 exists a constant $\alpha:=\frac{1}{2}g(\rho)\rho^q>0$ which is independent
of $\tau$, $\lambda$ and $\omega$ such that
\begin{equation*}
 I_{\lambda}(u)\big|_{\|u\|= \rho}\geq \alpha>0,
\end{equation*}
for any $\tau\in I$, $\omega\in H$ and all $h$ satisfying $\|h\|_{q'}<m_0$.
\end{proof}

\begin{lemma} \label{lem4.2}
Assume that {\rm(H1)-(H4)} are satisfied, then there exist a function
$e\in H$ with $\|e\|<\rho$ and a constant $0<\lambda^*<\widetilde{\lambda}$
such that $I_{\lambda}(e)<0$ for any $\lambda \in [0,\lambda^*)$,
where $\rho$ and $\widetilde{\lambda}$ are given by Lemma \ref{lem4.1}
 and Lemma \ref{lem3.2}, respectively.
\end{lemma}

\begin{proof}
 We choose a function $0\leq\phi\in C_{0}^{\infty}(\mathbb{R}^N)$ with 
$\int_{B(0,R_0)}h(x)\phi^q dx \geq 0$ for some $R_0>0$.
By (H1), for $t\geq 0$ we obtain
\begin{equation}
\begin{aligned}
 I_{0}(t\phi)
&=  \frac{at^2}{2}\|\phi\|^2-\int_{\mathbb{R}^N}F(t\phi)dx
 -\frac{t^q}{q}\int_{\mathbb{R}^N}h(x)\phi^q dx\\
&\leq \frac{at^2}{2}\|\phi\|^2-\frac{t^q}{q}\int_{B(0,R_0)}h(x)\phi^q dx.
\end{aligned} \label{4.1}
\end{equation}
Since $1<q<2$, it follows from \eqref{4.1} that
$I_{0}(t\phi)< 0$ for $t>0$ sufficiently small,
which implies that there exist $e\in H$ with $\|e\|<\rho$ such that
 $I_0(e)<0$, where $\rho$ is given by Lemma \ref{lem4.1}.
Since $I_{\lambda}(e)\to  I_0(e)$ as $\lambda\to  0^+$, we see that 
there exists $\widetilde{\lambda}>\lambda^*>0$ such that $I_{\lambda}(e)<0$ 
for all $\lambda \in [0,\lambda^*)$, where $\widetilde{\lambda}$ 
is given by Lemma \ref{lem3.2}. 
\end{proof}


\section*{Second positive solution for \eqref{1.1}}
Setting
$$
c_1:=\inf\{I_{\lambda}(u):u\in \overline{B}_{\rho}\},
$$
where $\rho$ is given by Lemma \ref{lem4.1}, $B_{\rho}=\{u\in H: \|u\|<\rho\}$. 
Using Lemma \ref{lem4.1} and Lemma \ref{lem4.2}, we obtain
$$
\inf_{\overline{B}_{\rho}}I_{\lambda}>-\infty, \quad
\inf_{\partial B_{\rho}}I_{\lambda}>\alpha>0, \quad    c_1<0.
$$
By Ekeland's variational principle, there exists a sequence
 $\{u_n\}\subset \overline{B}_{\rho}$ such that
\begin{gather*}
c_1\leq I_{\lambda}(u_n)<c_1+\frac{1}{n},\\
I_{\lambda}(v)\geq I_{\lambda}(u_n)-\frac{1}{n}\|v-u_n\|
\end{gather*}
for all $v\in \overline{B}_{\rho}$. Then by a standard procedure,
 we can show that $\{u_n\}$ is a bounded Palais-Smale sequence of 
$I_{\lambda}$. Using the similar proof to that of Lemma \ref{lem3.1}, 
we conclude that there exists a function $u_1\in B_{\rho}$ such 
that $I_{\lambda}(u_1)=c_1<0$ and $I'_{\lambda}(u_1)=0$.

Setting $u^{-}=\max\{0,-u\}$. Since 
\[
\big(a+\lambda m(\|u_1\|^2)\big)\langle u_1,u_1^-\rangle
-\int_{\mathbb{R}^{N}}f(u_1)u^{-}_{1}dx
-\int_{\mathbb{R}^{N}}h(x)(u^{+}_{1})^{q-1}u^{-}_{1}dx=0, 
\]
 by (H1) and (H4) we have
 $\|u_1^-\|=0$,
which implies  $u_1\geq 0$ a.e. in $\mathbb{R}^{N}$. 
So, by the strong maximum principle, we obtain that $u_1$ is positive on $H$.


\subsection*{Acknowledgments} 
This research was supported by the National Natural Science Foundation of China
(Nos. 11471267, 11361003), and by the Fundamental Research Funds for
the Central Universities (No. XDJK2013C006).
The authors want to express their gratitude to the reviewers for the careful reading 
and helpful suggestions which led to an improvement of the original manuscript.

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\end{thebibliography}

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