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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 152, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/152\hfil Bifurcation and multiplicity results]
{Bifurcation and multiplicity results for a
nonhomogeneous semilinear elliptic problem}

\author[K.-J. Chen\hfil EJDE-2008/152\hfilneg]
{Kuan-Ju Chen}

\address{Kuan-Ju Chen \newline
Department of Applied Science, Naval Academy,
P.O. BOX 90175 Zuoying, Taiwan}
\email{kuanju@mail.cna.edu.tw}

\thanks{Submitted May 13, 2008. Published November 4, 2008.}
\subjclass[2000]{35A15, 35J20, 35J25, 35J65}
\keywords{Nonhomogeneous semilinear elliptic problems; bifurcation}

\begin{abstract}
 In this article we consider the  problem
 \begin{gather*}
 -\Delta u(x)+u(x)=\lambda (a(x)u^{p}+h(x))\quad\text{in }\mathbb{R}^N, \\
 u\in H^{1}(\mathbb{R}^N),\quad u>0\quad\text{in }\mathbb{R}^N,
 \end{gather*}
 where $\lambda$ is a positive parameter.
 We assume there exist $\mu >2$ and $C>0$ such that
 $a(x)-1\geq -Ce^{-\mu |x|}$  for all $x\in \mathbb{R}^N$.
 We prove that there exists a positive $\lambda^*$
 such that there are at least two positive solutions for
 $\lambda\in (0,\lambda^*)$ and a unique positive solution
 for $\lambda =\lambda^*$. Also we show that
 $(\lambda ^{*},u(\lambda^*))$ is a bifurcation point in
 $C^{2,\alpha}(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction} \label{secIntroduction}

 We consider the existence and properties of multiple positive solutions
for the nonhomogeneous semilinear elliptic problem
\begin{equation} \label{e1.1}
\begin{gathered}
-\Delta u(x)+u(x)=\lambda (a(x)u^{p}+h(x))\quad\text{in }\mathbb{R}^N, \\
u\in H^{1}(\mathbb{R}^N),\quad u>0\quad\text{in }\mathbb{R}^N,
\end{gathered}
\end{equation}
where $\lambda >0$, $1<p<\frac{N+2}{N-2}$ ($N\geq 3$),
$1<p<\infty $ $(N=1 , 2)$, $h(x)\in H^{-1}(\mathbb{R}^N)$,
$0\not\equiv h(x)\geq 0$ in $\mathbb{R}^N$, $h(x)\to 0$ as
 $|x|\to \infty $, and $a(x)$
satisfies the following conditions:
\begin{itemize}
\item[(A1)] $a(x)\in C(\mathbb{R}^N)$; $a(x)\in (0,1]$ for all
$x\in \mathbb{R}^N$; $a(x)\to 1$ as $|x|\to \infty $; $a(x)\not\equiv 1$;
\item[(A2)] there exist $\mu >2$ and $C>0$ such that
\[
a(x)-1\geq -Ce^{-\mu |x|}\quad\text{for all }x\in \mathbb{R}^N.
\]
\end{itemize}
Here the constant $\mu $  corresponds to a convergent rate
(from below), and the condition $\mu >2$ play an important role in
our existence result.

The main result  is as follows.

\begin{theorem}\label{thm:1.1}
Assume {\rm(A1)--(A2)} hold.
\begin{itemize}
\item[(1)] There exists $\lambda^{\ast}$, $0<\lambda^*< \infty$,
such that \eqref{e1.1} has at least two distinct positive solutions
$u(\lambda )$, $U(\lambda )$ for any
$\lambda \in (0,\lambda^*)$, $u(\lambda )$,
$U(\lambda ) \in C^{2,\alpha}(\mathbb{R}^N)\cap H^{1}(\mathbb{R}^N)$
for any $\lambda \in (0,\lambda^*)$, and a unique positive
solution $u(\lambda^{\ast } )$ for $\lambda =\lambda^*$;
 moreover, $u(\lambda )$ is the minimal positive solution of
\eqref{e1.1}, $u(\lambda )$ is strictly increasing in
$\lambda $ for $\lambda \in (0,\lambda^*)$, and $\lambda
\to u(\lambda )\in C^{2,\alpha }(\mathbb{R}^N)\cap
H^{2}(\mathbb{R}^N)$ is continuous.

\item[(2)] Denote by
$Q=\{(\lambda ,u) : u\text{solves \eqref{e1.1}}\}$,
the set of positive solutions of \eqref{e1.1}, for any
$(\lambda ,u)\in Q$, $u(x)$ and $|\nabla u(x)|$ have uniform limit zero as
$|x|\to \infty $.

\item[(3)] $(\lambda^*,u(\lambda^{\ast }))\in C^{2,\alpha }
(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$
is a bifurcation point for problem \eqref{e1.1}.
Consequently $Q$ is unbounded in
$(0,\lambda^*) \times C^{2,\alpha }(\mathbb{R}^N)\cap H^{1}
(\mathbb{R}^N)$ but is
bounded in $[\epsilon,\lambda^*] \times C^{1,\alpha
}(\mathbb{R}^N)\cap H^{1}(\mathbb{R}^N)$ for any $\epsilon>0$.
\end{itemize}
\end{theorem}

We write $u_{\pm }(x)=\max \{\pm u(x), 0\}$,
$\|u\|^{2}=\int_{\mathbb{R} ^N}(|\nabla u|^{2}+u^{2})dx$, and
we define for given $a(x)$ and $h(x)$,
\[
I(u)=\frac{1}{2}\|u\|^{2}-\frac{\lambda }{p+1}\int_{\mathbb{R}
^N}a(x)u_{+}^{p+1}dx-\lambda \int_{\mathbb{R}^N}h(x)u\,dx.
\]

Many authors have studied the existence of positive solutions of the
semilinear elliptic boundary value problem
\begin{equation}
\begin{gathered}
-\Delta u+u=g(x,u),\quad x\in \Omega \subset \mathbb{R}^N, \\
u|_{\partial \Omega }=0,
\end{gathered}  \label{1.2}
\end{equation}
a problem that occurs in varies branches of geometry and mathematical
physics. There are many results about the existence of the positive
solutions of \eqref{1.2} when $g(x,u)$ is a ``homogeneous'' function (i.e.
$g(x,0)\equiv 0$), see \cite{Be}, \cite{Ber}, \cite{Di},
\cite{Lions}. For the ``nonhomogeneous'' function (i.e.
$g(x,0)\not\equiv 0$), some existence of two solutions have been
obtained in \cite{DF} when $g(x,u)$ is less
than critical growth in the sense that $\lim_{u\to \infty }\frac{
g(x,u)}{u^{q}}=0$ with $q=\frac{N+2}{N-2}$ and $\Omega $ is
bounded. A substantial difference between the problems on bounded
domain and on unbounded domain is the lack of compactness for
Sobolev embedding when we deal with the latter. Thus there seems
to be little progress on the existence theory for the
``nonhomogeneous'' function of \eqref{1.2} when $\Omega $ is
unbounded. Zhu \cite{Zhu}, Zhu-Zhou \cite{ZZ}, Cao-Zhou
\cite{Cao}, Chen \cite{Chen}, and Jeanjean \cite{Je} showed the
existence of
multiple positive solutions in unbounded domains under the assumption:
\begin{equation}
a(x)\geq 1 \quad \text{for all }x\in \mathbb{R}^N \label{1.3}
\end{equation}
or
\begin{equation}
g(x,u)\geq g^{\infty }(u)\quad \Big( =\lim_{|x|\to\infty }g(x,u)\Big)
\quad \text{for all }x\in \mathbb{R}^N\text{ and }u\geq 0.  \label{1.4}
\end{equation}
Now we will show the existence of solutions without assuming (\ref{1.3}),
(\ref{1.4}).

In Section 2, we assert that there exists
$\lambda^*$, $\infty>\lambda^*>0$, such that \eqref{e1.1}
has a minimal positive solution for all
$\lambda \in (0,\lambda^*)$ by the
implicit function theorem and the barrier method.
In Section 3, by the variational method and the concentration-compactness
principle, we establish the existence of second positive solution,
the following energy estimate (\ref{1.5}) plays an important role.
We find $R_{0}\geq 1$ such that
\begin{equation}
\sup_{t\geq 0}J(t\omega (x-Re))<I^{\infty }(\omega ) \quad
\text{for all }R\geq R_{0},  \label{1.5}
\end{equation}
where $J$ is the energy functional of \eqref{3.1}, $e$ is a fixed
unit vector in $\mathbb{R}^N$, $\omega (x)$ is a ground state
solution of the limit problem \eqref{3.2} and $I^{\infty }$ is the
energy functional of \eqref{3.2}. To obtain the energy estimate,
assumption (A2) plays an important role. In Section 4, we discuss
problem \eqref{e1.1} has at least two distinct positive
solutions for any $\lambda \in (0,\lambda^*)$, a unique
positive solution for $\lambda =\lambda^*$, $u(\lambda^*)$ is a
bifurcation point in $C^{2,\alpha}(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$,
and further analyzes of the
set of positive solutions are made.

\section{Existence of minimal positive solution}

 In this section, the implicit function
theorem and the barrier method enable us to deduce the existence
of a minimal positive solution $u(\lambda )$ of \eqref{e1.1}
for all $\lambda $ in some finite interval $(0,\lambda^*)$.

\begin{lemma} \label{lemma:2.1}
For each $h\in C^{\alpha }(\mathbb{R}^N)\cap L^{2}(\mathbb{R}^N)$,
the linear problem
\begin{equation}
-\Delta u+u=h,\quad u\in H^{1}(\mathbb{R}^N)  \label{2.1}
\end{equation}
has a solution $u\in C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$.
\end{lemma}

\begin{proof}
Consider the functional
$\Phi (u)=\frac{1}{2}\|u\|^{2}-\int_{\mathbb{R}^N}hudx$,
$u\in H^{1}(\mathbb{R}^N)$. From the H\"older's and Young's
inequality we have, for any $\epsilon >0$ that
\begin{equation} \label{2.2}
\begin{aligned}
\Phi (u) &\geq \frac{1}{2}\|u\|^{2}-\|h\|_{L^{2}}\|u\|_{L^{2}}\\
&\geq (\frac{1}{2}-\epsilon )\|u\|^{2}-C_{\epsilon }\|h\|_{L^{2}}^{2}\\
&\geq -C_{\epsilon }\|h\|_{L^{2}}^{2}
\end{aligned}
\end{equation}
if we choose $\epsilon$  small.

Let $\{u_{n}\}\subset H^{1}(\mathbb{R}^N)$ be a minimizing
sequence of the variational problem
$ d=\inf \{\Phi (u):u\in H^{1}(\mathbb{R}^N)\}$.
 From (\ref{2.2}),
\[
\big( \frac{1}{2}-\epsilon \big) \|u_{n}\|^{2}
\leq \Phi (u_{n})+C_{\epsilon }\|h\|_{L^{2}}^{2}
=d+C_{\epsilon }\|h\|_{L^{2}}^{2}+o(1) \quad \text{as } n\to \infty ,
\]
we can also deduce that $\{u_{n}\}$ is bounded in
$H^{1}(\mathbb{R}^N)$ if we choose $\epsilon $ small. Thus we may
assume that
\begin{gather*}
u_{n} \to u\quad\text{weakly in }H^{1}(\mathbb{R}^N)\quad\text{as }
n\to \infty , \\
u_{n} \to u\quad\text{a.e. in }\mathbb{R}^N\quad\text{as }n\to
\infty .
\end{gather*}
By Fatou's lemma,
\[
\|u\|^{2}\leq \liminf_{n\to \infty } \|u_{n}\|^{2}.
\]
The weak convergence implies $
\int_{\mathbb{R}^N}hu_{n}dx\to \int_{\mathbb{R}^N}hudx$
as $n\to \infty$, therefore
$ \Phi (u)\leq \lim_{n\to \infty }\Phi (u_{n})=d $ and hence
$\Phi (u)=d$.

By the assumption $h\in C^{\alpha }(\mathbb{R}^N)\cap L^{2}(\mathbb{R}^N)$,
from \cite[Proposition 4.3]{Stu}
that $u\in H^{2}(\mathbb{R}^N)$; the standard elliptic regular
theorem
yield $u\in C^{2,\alpha }(\mathbb{R}^N)$.
\end{proof}

\begin{theorem}\label{thm:2.2}
 Assume {\rm (A1)--(A2)} hold. Then there exists $\lambda^*>0$ such
that for any $\lambda \in (0,\lambda^*)$, \eqref{e1.1} has
a minimal positive solution $u(\lambda )$ ; $u(\lambda )$ is strictly
increasing in $\lambda $ for $\lambda \in (0,\lambda^*)$; $\lambda
\to u(\lambda )\in C^{2,\alpha }(\mathbb{R}^N)\cap
H^{2}(\mathbb{R}^N)$ is continuous.
\end{theorem}

\begin{proof}
Define $F:\mathbb{R}\times C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}
^N)\to C^{\alpha }(\mathbb{R}^N)\cap L^{2}(\mathbb{R}^N)$ by
\[
F(\lambda ,u)=-\Delta u+u-\lambda (a(x)u^{p}+h(x)),
\]
by Lemma \ref{lemma:2.1}, $F_{u}(0,0)u=-\Delta u+u$ is an isomorphism of
$C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$ onto $C^{\alpha }({
\mathbb{R}}^N)\cap L^{2}(\mathbb{R}^N)$. Applying the implicit function
theorem to $F$, we can find that the solutions of $F(\lambda ,u)=0$ near
$(0,0)$ are given by a continuous curve $(\lambda ,u(\lambda ))$
with $u(0)=0$.

Denote $\lambda^*=\sup \{\lambda >0$:there exists $u(\lambda )$
such that $u(\lambda )$ is a unique continuous function of $\lambda $ into
$C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$, $F(\lambda ,u(\lambda
))=0$ and $F_{u}(\lambda ,u(\lambda ))$ is nonsingular$\}$,
then $0<\lambda^*\leq +\infty $; by the maximum principle,
$u(\lambda )>0$ for all $\lambda \in (0,$ $\lambda^*)$.

Let $\lambda $, $\lambda '\in (0,\lambda^*)$,
$\lambda <\lambda '$, $u(\lambda )$, $u(\lambda ')$ are
corresponding positive solutions, then $u(\lambda ')$ is a
supersolution of \eqref{e1.1}. From $h(x)\geq 0$ we deduce that $0$ is
obviously a subsolution of \eqref{e1.1}; the standard barrier method
and the maximun principle enable us to find a solution $v(\lambda )$ such
that $0<v(\lambda )<u(\lambda ')$. Moreover, we may select
$v(\lambda )$ as a minimal positive solution of \eqref{e1.1}. In the
same way one finds a minimal positive solution $v(\lambda )$ of
\eqref{e1.1} for each $\lambda \in (0,\lambda^*)$,
$v(\lambda )$ is strictly increasing in $\lambda $ for $\lambda
\in (0,\lambda^*)$. The uniqueness of $u(\lambda )$ yields
$u(\lambda )=v(\lambda )$ for $\lambda \in (0,\lambda^*)$.\hfill
\end{proof}

We establish the decay estimate for positive solutions of \eqref{e1.1}
which we will use later on.

\begin{theorem}\label{thm:2.3} Assume {\rm(A1)} holds.
If $u\in H^{1}(\mathbb{R}^N)$ is
a positive solution of \eqref{e1.1}, then
\begin{itemize}
\item[(1)] $u\in L^{q}(\mathbb{R}^N)$ for $q\in [ 2,\infty )$;

\item[(2)] $u(x)$ and $|\nabla u(x)|$ have uniform limit zero as
$|x|\to \infty $.
\end{itemize}
\end{theorem}

\begin{proof} (1)  follows by the classical regularity theory based on
a result of Brezis-Kato \cite{Bre}. We will write it in detail for
the reader's convenience.
For $s\geq 0$, $l\geq 1$, let $\varphi =\varphi _{s,l}=u\min \{|u|^{2s}$,
$l^{2}\}\in H^{1}(\mathbb{R}^N)$. Testing \eqref{e1.1} with
$\varphi$, then
\begin{equation}  \label{2.3}
\begin{aligned}
&\int_{\mathbb{R}^N }|\nabla u|^{2}\min
\{|u|^{2s},l^{2}\}dx+2s\int_{\{|u|^{s}\leq l\}}|\nabla u|^{2}|u|^{2s}dx    \\
&\leq C\int_{\mathbb{R}^N}|u|^2dx+C\int_{\mathbb{R}^N
}|u|^{2+2s}dx+C\int_{\mathbb{R}^N }|u|^{p+1}dx \\
&\quad +C\int_{\mathbb{R}^N }|u|^{2s+p+1}dx +C\int_{\mathbb{R}^N
}|h\|u|\min \{|u|^{2s},l^{2}\}dx.
\end{aligned}
\end{equation}
Suppose $u\in L^{2s+p+1}(\mathbb{R}^N)$, then by (\ref{2.3}) and applying the H
\"older's inequality and the Sobolev embedding, we have
\begin{align*}
&\int_{\mathbb{R}^N}|\nabla (u\min
\{|u|^{s},l\})|^{2}dx \\
&\leq \int_{\mathbb{R}^N}|\nabla u|^{2}\min
\{|u|^{2s},l^{2}\}dx+(s^{2}+2s)\int_{\{|u|^{s}\leq l\}}|\nabla
u|^{2}|u|^{2s}dx \\
&\leq C+C\int_{\mathbb{R}^N}|h\|u|dx+C\int_{\mathbb{R}^N}|h\|u|^{2}\min
\{|u|^{2s},l^{2}\}dx \\
&\leq C+CK\int_{\mathbb{R}^N}|u|^{2s+2}dx+C\int_{\{|h|\geq K\}}|h\|u\min
\{|u|^{s},l\}|^{2}dx \\
&\leq C(1+K)+C \Big(\int_{\{|h|\geq K\}}|h|^{\frac{N}{2}}dx \Big)^{2/N}
\Big(\int_{\mathbb{R}^N}|u\min \{|u|^{s},l\}|^{\frac{2N}{N-2}}dx
 \Big)^{\frac{N-2}{N}}
 \\
&\leq C(1+K)+C\varepsilon (K)\int_{\mathbb{R}^N}|\nabla
 (u\min\{|u|^{s},l\})|^{2}dx,
\end{align*}
where
\[
\varepsilon (K)=\Big(\int_{\{|h|\geq K\}}|h|^{\frac{N}{2}}dx
\Big)^{2/N}\to 0\quad\text{as }K\to \infty .
\]
Fix $K$ such that $\varepsilon (K)=1/2C$ and observe that for this choice of
$K$ (and $s$ as above) we now may conclude that
\[
\int_{\{|u|^{s}\leq l\}}|\nabla (|u|^{s+1})|^{2}dx\leq
C\int_{\mathbb{R}^N}|\nabla (u\min \{|u|^{s},l\})|^{2}dx\leq C(1+K)
\]
for any $l\geq 1$. Hence we may let $l\to \infty $ to derive that
\[
|u|^{s+1}\in H^{1}(\mathbb{R}^N)\hookrightarrow L^{2^{\ast }}(\mathbb{R}^N).
\]
It is easy to see that $u\in L^{\frac{(s+1)2N}{N-2}}(\mathbb{R}^N)$.

Now iterate, letting $s_{0}=0$, $2s_{i}+p+1=(s_{i-1}+1)\frac{2N}{N-2}$ for
$i\geq 1$, then $u\in L^{2s_{i-1}+p+1}(\mathbb{R}^N)$ implies $u\in
L^{2s_{i}+p+1}(\mathbb{R}^N)$. It is easily to see $s_{i}\to \infty $
as $i\to \infty $. Therefore $u\in L^{q}(\mathbb{R}^N)$ for $2\leq
q<\infty $.

(2) Obviously $u$ satisfies the linear problem
\[
-\Delta u+u=F(x)=\lambda (a(x)u^{p}+h(x)), x\in \mathbb{R}^N, u\in H^{1}(\mathbb{R}^N).
\]
Choose $q>\max \{\frac{Np}{2},\frac{2N}{N-2}\}$, by the H\"older's
inequality in $B_{2}(x)$ we get
\[
\|u\|_{L^{2}(B_{2}(x))}\leq C\|u\|_{L^{q}(B_{2}(x))},
\]
then
\[
\|F\|_{L^{\frac{q}{p}}(B_{2}(x))}\leq C\left(
\|u\|_{L^{q}(B_{2}(x))}+\|h\|_{L^{q}(B_{2}(x))}\right) .
\]

It is deduced by elliptic regular theory that
$u\in C^{2,\alpha }(\mathbb{R}^N)$. By \cite[Theorem 8.24]{GT},  we have
\begin{equation}
\|u\|_{C^{\alpha }(B_{1}(x))}\leq C\left(
\|u\|_{L^{q}(B_{2}(x))}+\|h\|_{L^{q}(B_{2}(x))}\right) ,
\label{2.4}
\end{equation}
then $u(x)\to 0$ as $|x|\to \infty $ since $u\in L^{q}({\mathbb{R}}^N)$.

By \cite[Theorem 8.32]{GT},
\begin{equation}
\|u\|_{C^{1,\alpha }(B_{1}(x))}\leq C\left( \|u\|_{C^{\alpha
}(B_{2}(x))}+\|h\|_{L^{\infty }(B_{2}(x))}\right).  \label{2.5}
\end{equation}
Then (\ref{2.4}), (\ref{2.5}) give $|\nabla u(x)|\to 0$ as
$|x|\to \infty $.
\end{proof}

We will verify that $\lambda^*$ is finite by considering linear
eigenvalue problems related to decaying positive solutions of
\eqref{e1.1}.

\begin{theorem} \label{thm:2.4}
Assume {\rm (A1)-(A2)} hold. Then
\begin{itemize}
\item[(1)] the minimization problem
\begin{align*}
\mu _{\lambda }&=\mu _{\lambda }(u(\lambda ))\\
&=\inf \big\{ \int_{\mathbb{R}
^N}(|\nabla v|^{2}+v^{2})dx:v\in H^{1}(\mathbb{R}^N),\int_{\mathbb{R}
^N}pa(x)u(\lambda )^{p-1}v^{2}dx=1\big\},
\end{align*}
where $u(\lambda )$ is the minimal positive solution of \eqref{e1.1},
can be achieved by some $\overline{v}_{\lambda }>0$;

\item[(2)] $\mu _{\lambda }>\lambda $, $\mu _{\lambda }$ is strictly
decreasing in $\lambda $ for $\lambda \in (0,\lambda^*)$;

\item[(3)] $\lambda^*$ is finite.
\end{itemize}
\end{theorem}

\begin{proof}
It is easy to see that $0<\mu _{\lambda }<\infty $. Let
$\{v_{n}\}\subset H^{1}(\mathbb{R}^N)$ be a minimizing sequence of
$\mu _{\lambda }$, then $\{v_{n}\}$ is bounded in
$H^{1}(\mathbb{R}^N)$. Without loss of generality (at least by
choosing a subsequence) we can assume that, for some
$\overline{v}_{\lambda }\in H^{1}(\mathbb{R}^N)$,
\begin{gather*}
v_{n} \rightharpoonup \overline{v}_{\lambda }\quad\text{weakly in }H^{1}({
\mathbb{R}}^N), \\
v_{n} \to \overline{v}_{\lambda }\quad\text{a.e. in
}\mathbb{R}^N, \\
v_{n} \to \overline{v}_{\lambda }\quad\text{strongly in
}L_{\rm loc}^{s}(\mathbb{R}^N)\quad\text{for }2\leq s<\frac{2N}{N-2}.
\end{gather*}

By Theorem \ref{thm:2.3} and (A1), we have $pa(x)u(\lambda
)^{p-1}\to 0$ as $|x|\to \infty $, hence for any
$\varepsilon >0$, there exists $R>0$ such that for $x\in \mathbb{R}^N$,
$|x|\geq R$, $|pa(x)u(\lambda )^{p-1}|<\varepsilon $. Consequently, there
exists a constant $C>0$ such that
$
|pa(x)u(\lambda )^{p-1}|\leq C\quad\text{for }x\in \mathbb{R}^N.
$
Then
\begin{align*}
&\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p-1}|v_{n}-\overline{v}_{\lambda
}|^{2}dx \\
&= \int_{B_{R}}pa(x)u(\lambda )^{p-1}|v_{n}-\overline{v}_{\lambda
}|^{2}dx+\int_{\mathbb{R}^N\backslash B_{R}}pa(x)u(\lambda )^{p-1}|v_{n}-
\overline{v}_{\lambda }|^{2}dx \\
&\leq C\int_{B_{R}}|v_{n}-\overline{v}_{\lambda }|^{2}dx+\varepsilon \int_{{
\mathbb{R}}^N\backslash B_{R}}|v_{n}-\overline{v}_{\lambda }|^{2}dx.
\end{align*}
Since $v_{n}\to \overline{v}_{\lambda }$ strongly in $L_{\rm loc}^{s}({
\mathbb{R}}^N)$ for $2\leq s<2N/N-2$, $\{v_{n}\}$ is a bounded sequence in
$H^{1}(\mathbb{R}^N)$, taking $n\to \infty $, then $\varepsilon
\to 0$, we obtain $\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p-1}
\overline{v}_{\lambda }^{2}=1$. Therefore $\overline{v}_{\lambda }$ achieves
$\mu _{\lambda }$. Clearly $|\overline{v}_{\lambda }|$ also achieves $\mu
_{\lambda }$. Hence we may assume $\overline{v}_{\lambda }\geq 0$ in
${\mathbb{R}}^N$ and $\overline{v}_{\lambda }$ satisfies
$-\Delta v+v=\mu _{\lambda }pa(x)u(\lambda )^{p-1}v$.
Once again, by the maximum principle we deduce that
$\overline{v}_{\lambda }>0$ in $\mathbb{R}^N$.

To prove (2). Setting $0<\lambda <\lambda '$ and $\lambda $,
$\lambda '\in (0,\lambda^*)$, by Theorem \ref{thm:2.2},
$u(\lambda ')>u(\lambda )$ as $\lambda '>\lambda $. Noting
that $\lambda '>\lambda $, $h(x)\geq 0$,~$a(x)u(\lambda )^{p}>0$,
we get
\begin{equation} \label{2.15}
\begin{aligned}
&-\Delta (u(\lambda ')-u(\lambda ))+(u(\lambda ')-u(\lambda ))   \\
&= \lambda 'a(x)u(\lambda ')^{p}-\lambda a(x)u(\lambda
)^{p}+(\lambda '-\lambda )h(x)   \\
&= (\lambda '-\lambda )a(x)u(\lambda )^{p}+\lambda '(a(x)u(\lambda ')^{p}-a(x)u(\lambda )^{p})+(\lambda '-\lambda )h(x)   \\
&> \lambda pa(x)u(\lambda )^{p-1}(u(\lambda ')-u(\lambda)).
\end{aligned}
\end{equation}
Multiplying (\ref{2.15}) by $\overline{v}_{\lambda }$ and
integrating it over $\mathbb{R}^N$, we get
\[
\mu _{\lambda }\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p-1}(u(\lambda ')-u(\lambda ))\overline{v}_{\lambda }dx>\lambda \int_{\mathbb{R}
^N}pa(x)u(\lambda )^{p-1}(u(\lambda ')-u(\lambda ))\overline{v}
_{\lambda }dx,
\]
which implies that $\mu _{\lambda }>\lambda $, $\lambda \in (0,\lambda^*)$.

Furthermore, let $\lambda $, $\lambda '\in (0,\lambda^*)$,
$\lambda <\lambda '$; by Theorem \ref{thm:2.2},
\[
\int_{\mathbb{R}^N}pa(x)u(\lambda ')^{p-1}\overline{v}_{\lambda }^{2}dx >
\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p-1}\overline{v}_{\lambda
}^{2}dx=1,
\]
so that there exists $0<t<1$ such that
$\int_{\mathbb{R}^N}pa(x)u(\lambda ')^{p-1}t^{2}\overline{v}
_{\lambda }^{2}dx=1$,
therefore
$\mu _{\lambda '}\leq t^{2}\Vert \overline{v}_{\lambda }\Vert
^{2}<\Vert \overline{v}_{\lambda }\Vert ^{2}=\mu _{\lambda }$;
that is $\mu _{\lambda }$ is strictly decreasing in $\lambda $ for
$\lambda \in (0,\lambda^*)$.

Finally, we show that $\lambda^*$ is finite. Fix any
$\lambda _{0}\in (0,\lambda^*)$, then by (2),
$\lambda <\mu _{\lambda }<\mu _{\lambda_{0}}<\infty $ for all
$\lambda \in (\lambda _{0}$, $\lambda^*)$ and
this implies $\lambda^*<+\infty $.
\end{proof}

\begin{remark} \label{remark:2.5} \rm
Let
\[
\mu _{\lambda }(u)=\inf \big\{ \int_{\mathbb{R}^N}(|\nabla
v|^{2}+v^{2})dx:v\in H^{1}(\mathbb{R}^N), \int_{\mathbb{R}
^N}pa(x)u^{p-1}v^{2}dx=1\big\} ,
\]
where $u$ is any positive solution of \eqref{e1.1}, then
from the proof of Theorem \ref{thm:2.4}(1),
$\mu _{\lambda }(u)$ also can be achieved.
\end{remark}

\section{Existence of second positive solution}

 When $\lambda \in (0,\lambda^*)$, we have
shown that \eqref{e1.1} has the minimal positive solution $u(\lambda )$
by Theorem \ref{thm:2.2}. In this section, we want to
prove that \eqref{e1.1} has another positive solution in the form of
$U(\lambda )=u(\lambda )+v_{\lambda }$, where $v_{\lambda }$ is a
positive solution of the following auxiliary problem:
\begin{equation} \label{3.1}
\begin{gathered}
-\Delta v+v=\lambda a(x)((u(\lambda )+v)^{p}-u(\lambda )^{p})
\quad\text{in } \mathbb{R}^N, \\
v\in H^{1}(\mathbb{R}^N),\quad  v>0\quad\text{in }\mathbb{R}^N.
\end{gathered}
\end{equation}
For this equation, we define the energy functional
$J:H^{1}(\mathbb{R}^N)\to \mathbb{R}$ as follows:
\[
J(v)=\frac{1}{2}\|v\|^{2}-\frac{\lambda }{p+1}\int_{\mathbb{R}^N}a(x)
\left[(u(\lambda )+v_{+})^{p+1}-u(\lambda )^{p+1}-(p+1)u(\lambda )^{p}v_{+}
\right] dx.
\]
We know that critical points of $J(v)$ correspond to positive
solutions of \ref{3.1}. The existence of nontrivial critical
points will be deduced by the mountain pass lemma of Ambrosetti
and Rabinowitz \cite{Amb}.

In this section, firstly several technical results will be
established. Let us
recall that a sequence $\{v_{n}\}\subset H^{1}(\mathbb{R}^N)$ is called a
$(PS)_{c}$-sequence if $J(v_{n})\to c$ and
 $J'(v_{n})\to 0$ as $n\to \infty $.
If any $(PS)_{c}$-sequence possesses a convergent subsequence,
 we say $(PS)_{c}$-condition is satisfied.

 Let us now introduce the problem at infinity associated with \eqref{e1.1}
is
\begin{equation}
\begin{gathered}
-\Delta u+u=\lambda u^{p}\quad\text{in }\mathbb{R}^N, \\
u\in H^{1}(\mathbb{R}^N),\quad u>0\quad\text{in }\mathbb{R}^N.
\end{gathered} \label{3.2}
\end{equation}
We state here some known results for \eqref{3.2}. First of all, we
recall that  Lions \cite{Lions} has studied the following
minimization problem closely related to \eqref{3.2}:
\begin{equation}
S^{\infty }=\inf \left\{I^{\infty }(u):u\in
H^{1}(\mathbb{R}^N), u\neq 0, I^{\infty \prime
}(u)=0 \right\}>0,  \label{3.3}
\end{equation}
where $I^{\infty }(u)=\frac{1}{2}\|u\|^{2}-$ $\frac{\lambda }{p+1}\int_{{
\mathbb{R}}^N}u_{+}^{p+1}dx$. For future reference note also that a
minimum exists
and is realized by a ground state $\omega >0$ in $\mathbb{R}^N$ such that
$S^{\infty }=I^{\infty }(\omega )=\sup_{s\geq 0}I^{\infty }(s\omega)$.
Gidas-Ni-Nirenberg \cite{Gi} showed that there exist $a_{1}$,
$a_{2}>0 $ such that for all $x\in \mathbb{R}^N$,
\begin{equation}
a_{1}(|x|+1)^{\frac{-(N-1)}{2}}e^{-|x|}\leq \omega (x)\leq a_{2}(|x|+1)^{
\frac{-(N-1)}{2}}e^{-|x|}.  \label{3.4}
\end{equation}

Secondly we study the break down of the Palais-Smale condition for $J$. The
ground state solution $\omega $ of \eqref{3.2} play an important role to
describe an asymptotic behavior of $(PS)$-sequence for $J$.

\begin{proposition}\label{proposition:3.1}
 Assume {\rm (A1), (A2)} hold. Let $\{v_{n}\}\subset H^{1}(\mathbb{R}^N)$
 be a $(PS)$-sequence for $J$. Then there exist a subsequence
(still denoted $\{v_{n}\}$) for which the following holds: there exist an
integer $m\geq 0$, sequence of points $\{y_{n}^{i}\}\subset \mathbb{R}^N$
for $1\leq i\leq m$, a solution $v_{\lambda }$ of \eqref{3.1} and
solutions $v^{i}$, for $1\leq i\leq m$, of \eqref{3.2} such that as
$n\to \infty $,
\begin{gather*}
v_{n}\rightharpoonup v_{\lambda }\quad\text{weakly in }H^{1}(\mathbb{R}^N), \\
v_{n}-v_{\lambda }-\sum_{i=1}^{m}v^{i}(x-y_{n}^{i})\to 0
\quad\text{strongly in }H^{1}(\mathbb{R}^N), \\
J(v_{n})\to J(v_{\lambda })+\sum_{i=1}^{m}I^{\infty }(v^{i}), \\
|y_{n}^{i}|\to \infty , |y_{n}^{i}-y_{n}^{j}|\to \infty, \quad
\text{for }1\leq i\neq j\leq m,
\end{gather*}
where we agree that in the case $m=0$ the above holds without $v^{i}$,
$y_{n}^{i}$.
\end{proposition}


\begin{proof}
This is a standard result that we give here without proof (see \cite{Ba},
\cite{Be}, \cite{Lions} for analogous statements).
\end{proof}

Thirdly, we show that $J(u)$ possesses the mountain pass structure.

\begin{lemma}\label{lemma:3.2}
Assume {\rm (A1), (A2)} hold. Then there exist small
$\beta >0$ and $\alpha >0$ such that $J(v)\geq \alpha >0$ for all
$v\in S_{\beta }=\{u\in H^{1}(\mathbb{R}^N):\|u\|=\beta \}$.
\end{lemma}

\begin{proof}
 For each $v\in H^{1}(\mathbb{R}^N)$, using Taylor's formula, the definition
of $\mu _{\lambda }$, $a(x)\in (0,1]$, and the Sobolev embedding, we obtain
\begin{align*}
J(v)
&= \frac{1}{2}\|v\|^{2}-\frac{1}{2}\lambda
p\int_{\mathbb{R}^N}a(x)u(\lambda )^{p-1}v_{+}^{2}dx \\ &-\lambda
\int_{\mathbb{R}^N}\int_{0}^{v_{+}}a(x)\left[ (u(\lambda
)+s)^{p}-u(\lambda )^{p}-pu(\lambda )^{p-1}s\right] dsdx \\ &\geq \frac{1}{2}\Big[ \|v\|^{2}-\lambda
p\int_{\mathbb{R}^N}a(x)u(\lambda )^{p-1}v_{+}^{2}dx\Big]
-\lambda \int_{\mathbb{R}^N}a(x)\Big[ \frac{\epsilon }{2}v_{+}^{2}+
\frac{C_{\varepsilon }}{p+1}v_{+}^{p+1}\Big] dx \\
&\geq \big( \frac{1}{2}-\frac{\lambda}{2\mu _{\lambda }} -\frac{1}{2}
\lambda \epsilon \big) \|v\|^{2}-\lambda C\|v\|^{p+1}
\end{align*}
for all $\epsilon >0$. Since $\mu _{\lambda }>\lambda $ by
Theorem \ref{thm:2.4}, we may choose $\varepsilon $ small enough such
that
$\frac{1}{2}-\frac{\lambda}{2\mu _{\lambda }}-\frac{1}{2}\lambda \epsilon
>0$. If we fix $\varepsilon =(\mu _{\lambda }-\lambda )/(2\lambda
\mu _{\lambda })$, then
\[
J(v)\geq \frac{1}{4\mu _{\lambda }}(\mu _{\lambda }-\lambda
)\|v\|^{2}-\lambda C\|v\|^{p+1}.
\]
Hence, there exist small $\beta >0$, $\alpha >0$ such that
$J(v)\geq \alpha>0$ for all
$v\in S_{\beta }=\{u\in H^{1}(\mathbb{R}^N):\|u\|=\beta \}$.
\end{proof}

Let $e$ be a fixed unit vector in $\mathbb{R}^N$. The following
estimates are important to find a path which lies below the first
level of the break
down of $(PS)_{c}$-condition. Here we use an interaction phenomenon between
$0$ and $\omega (x-R_{0}e)$.

\begin{proposition}\label{proposition:3.3}
Assume {\rm (A1)-(A2)} hold. Then
\begin{itemize}
\item[(1)] there exists $t_{0}>0$ such that $J(t\omega (x-Re))<0$ for all
$t\geq t_{0}$ uniformly in $R\geq 1$;
\item[(2)] there exists $R_{0}>1$ such that $\sup_{t\geq 0}J(t\omega
(x-Re))<S^{\infty }$ for all $R\geq R_{0}$.
\end{itemize}
\end{proposition}

To proof the above proposition, we need the following result.

\begin{lemma}\label{lemma:3.4}
There exist some constants $C_{1}$, $C_{2}$ and
$C_{3}>0$ independent of $R\geq 1$ such that
\begin{itemize}
\item[(I)] $\int_{|x|\leq 1}\omega (x-Re)^{2}dx\geq C_{1}R^{-(N-1)}e^{-2R}$
 for $R\geq 1$,

\item[(II)] $\int_{\mathbb{R}^N}e^{-\mu |x|}\omega
(x-Re)^{2}dx\leq C_{2}R^{-(N-1)}e^{-2R}$  for $R\geq 1$,

\item[(III)] $\int_{\mathbb{R}^N}e^{-\mu |x|}\omega
(x-Re)^{p+1}dx\leq C_{3}e^{-\mu R}$ for $R\geq 1$.
\end{itemize}
\end{lemma}

In what follows, we denote various positive constants independent
of $R\geq 1 $ by $C_{1}$, $C_{2}$ and $C_{3}$.


\begin{proof}
(I) From (\ref{3.4}),  for $R\geq 1$, we have
\begin{align*}
\int_{|x|\leq 1}\omega (x-Re)^{2}dx &\geq \int_{|x|\leq
1}a_{1}^{2}(|x-Re|+1)^{-(N-1)}e^{-2|x-Re|}dx \\
&\geq a_{1}^{2}(R+2)^{-(N-1)}e^{-2R-2}\int_{|x|\leq 1}dx \\
&\geq C_{1}R^{-(N-1)}e^{-2R}.
\end{align*}

(II) From (\ref{3.4}) again,
\begin{align*}
\int_{\mathbb{R}^N}e^{-\mu |x|}\omega (x-Re)^{2}dx
&\leq a_{2}^{2}\int_{\mathbb{R}^N}e^{-\mu
 |x|}(|x-Re|+1)^{-(N-1)}e^{-2|x-Re|}dx \\
&= a_{2}^{2}\int_{\mathbb{R}^N}e^{-(\mu
 -2)|x|}(|x-Re|+1)^{-(N-1)}e^{-2(|x-Re|+|x|)}dx \\
&\leq a_{2}^{2}R^{-(N-1)}e^{-2R}\int_{\mathbb{R}^N}e^{-(\mu
 -2)|x|}(\frac{R}{|x-Re|+1})^{N-1}dx.
\end{align*}
We estimate the last integral. First, we observe that
\[
e^{-(\mu -2)|x|}(\frac{R}{|x-Re|+1})^{N-1}\to e^{-(\mu
-2)|x|}\text{ }\quad\text{as }R\to \infty \quad\text{for
all }x\in \mathbb{R}^N.
\]
For $|x|\leq \frac{R}{2}$,
\[
e^{-(\mu -2)|x|}(\frac{R}{|x-Re|+1})^{N-1}
\leq e^{-(\mu -2)|x|}(\frac{R}{
\frac{R}{2}+1})^{N-1}
\leq 2^{N-1}e^{-(\mu -2)|x|}.
\]
For $|x|\geq \frac{R}{2}$,
\[
e^{-(\mu -2)|x|}(\frac{R}{|x-Re|+1})^{N-1}
\leq e^{-(\mu -2)|x|}R^{N-1}
\leq 2^{N-1}e^{-(\mu -2)|x|}|x|^{N-1}.
\]
Thus,
\[
e^{-(\mu -2)|x|}(\frac{R}{|x-Re|+1})^{N-1}\leq 2^{N-1}e^{-(\mu
-2)|x|}\max \{1,|x|^{N-1}\}\in L^{1}(\mathbb{R}^N).
\]
Therefore, we can apply the Lebesgue dominated convergence theorem
and we obtain
\[
\int_{\mathbb{R}^N}e^{-\mu |x|}\omega (x-Re)^{2}dx\leq
a_{2}^{2}R^{-(N-1)}e^{-2R}\Big(
\int_{\mathbb{R}^N}e^{-(\mu -2)|x|}dx+o(1)\Big)
\]
as $R\to \infty $. Thus we have
\[
\int_{\mathbb{R}^N}e^{-\mu |x|}\omega (x-Re)^{2}dx\leq C_{2}R^{-(N-1)}e^{-2R}
 \quad\text{for }R\geq 1.
\]

(III) Since $|x+Re|\geq R-|x|$, we
have
\begin{align*}
&\int_{\mathbb{R}^N}e^{-\mu |x|}\omega (x-Re)^{p+1}dx \\
&\leq a_{2}^{p+1}\int_{\mathbb{R}^N}e^{-\mu
|x|}(|x-Re|+1)^{-(p+1)(N-1)/2}e^{-(p+1)|x-Re|}dx \\
&\leq a_{2}^{p+1}\int_{\mathbb{R}^N}e^{-\mu |x|}e^{-(p+1)|x-Re|}dx \\
&= a_{2}^{p+1}\int_{\mathbb{R}^N}e^{-\mu |x+Re|}e^{-(p+1)|x|}dx \\
&\leq a_{2}^{p+1}e^{-\mu R}\int_{\mathbb{R}^N}e^{-((p+1)-\mu
)|x|}dx.
\end{align*}
Because of the assumption $\mu \in (2,p+1)$, we have
\[
\int_{\mathbb{R}^N}e^{-\mu |x|}\omega (x-Re)^{p+1}dx\leq C_{3}e^{-\mu R}
 \quad\text{for }R\geq 1.
\]
The proof  is complete.
\end{proof}

\begin{proof}[Proof of Proposition \ref{proposition:3.3}]
 Let $\omega _{\alpha }(x)=\omega (x-\alpha e)$,
$\alpha \in [ 0,+\infty )$,
where $\omega $ is the ground state solution of the limit problem
(\ref{3.2}). We also remark that for all $s>0$, $t>0$,
\begin{equation}
(s+t)^{p+1}-s^{p+1}-t^{p+1}-(p+1)s^{p}t\geq 0; \label{3.5}
\end{equation}
and for any $s_{0}>0$ and $r_{0}>0$ there exists $C_{4}(s_{0}$,
$r_{0})>0$ such that for all $s\in [ s_{0}, r_{0}]$,
$t\in[ 0,r_{0}]$,
\begin{equation}
(s+t)^{p+1}-s^{p+1}-t^{p+1}-(p+1)s^{p}t\geq
C_{4}(s_{0},r_{0})t^{2}. \label{3.6}
\end{equation}

(1) From (\ref {3.5}), (A2), and Lemma \ref{lemma:3.4} (III), we
have
\begin{align*}
J(t\omega _{R})
&\leq \frac{1}{2}t^{2}\|\omega _{R}\|^{2}-\frac{\lambda }{p+1}\int_{\mathbb{R}
^N}a(x)(t\omega _{R})^{p+1}dx \\
&\leq \frac{1}{2}t^{2}\|\omega _{R}\|^{2}-\frac{\lambda }{p+1}t^{p+1}\int_{{
\mathbb{R}}^N}(1-Ce^{-\mu |x|})\omega _{R}^{p+1}dx \\
 &\leq \frac{1}{2}t^{2}\|\omega _{R}\|^{2}-\frac{\lambda }{p+1}
t^{p+1}\Big( \|\omega _{R}\|_{L^{p+1}}^{p+1}-CC_{3}e^{-\mu R}\Big) .
\end{align*}
Choosing $t_{0}>0$ large enough, we have (1).

(2) Since $J$ is continuous in $H^{1}(\mathbb{R}^N)$, there exists
$\underline{t}>0$ such that for $t<\underline{t}$,
$J(t\omega _{R})<S^{\infty } \quad\text{for all }R\geq 0$,
and from (1), $J(t\omega _{R})\to -\infty $ as $t\to \infty $
uniformly in $R\geq 1$, then there exists $\overline{t}>0$ such that
$\sup_{t\geq 0}J(t\omega _{R})=\sup_{0\leq t\leq \overline{t}}J(t\omega _{R})$.
Then we only need to verify the inequality
$\sup_{\underline{t}\leq t\leq \overline{t}}J(t\omega _{R})<S^{\infty }$
for all $R$ large enough.
Straightforward computations give us
\begin{align*}
&J(t\omega _{R})\\
&= I^{\infty }(tw_{R}) \\
&\quad -\frac{\lambda }{p+1}\int_{\mathbb{R}^N}a(x)\left[ (u(\lambda
)+t\omega _{R})^{p+1}-u(\lambda )^{p+1}-(t\omega
_{R})^{p+1}-(p+1)u(\lambda )^{p}t\omega _{R}\right]dx\\
&\quad +\frac{\lambda }{p+1}\int_{\mathbb{R}^N}((t\omega
_{R})^{p+1}-a(x)(t\omega _{R})^{p+1})dx \\
&\leq S^{\infty }-\frac{\lambda }{p+1}\Lambda _{1}
 +\frac{\lambda}{p+1}\Lambda _{2},
\end{align*}
where
\[
\Lambda _{1}=\int_{\mathbb{R}^N}a(x)\big[ (u(\lambda )+t\omega
_{R})^{p+1}-u(\lambda )^{p+1}-(t\omega _{R})^{p+1}-(p+1)u(\lambda
)^{p}t\omega _{R}\big] dx
\]
and
\[
\Lambda _{2}=\int_{\mathbb{R}^N}((t\omega _{R})^{p+1}-a(x)(t\omega
_{R})^{p+1})dx.
\]
We need to show that there exists a constant $R_{0}\geq 1$ such that
\begin{equation}
-\Lambda _{1}+\Lambda _{2}<0 \quad\text{for all }t\in
[ \underline{t},\overline{t}].  \label{3.7}
\end{equation}
Now we estimate $\Lambda _{1}$ and $\Lambda _{2}$.
\[
\Lambda _{1} \geq \int_{|x|\leq 1}a(x)\big[ (u(\lambda
)+t\omega _{R})^{p+1}-u(\lambda )^{p+1}-(t\omega
_{R})^{p+1}-(p+1)u(\lambda )^{p}t\omega _{R}\big] dx.
\]
Setting $s_{0}=\min_{|x|\leq 1}u(\lambda )(x)$,
 $r_{0}=\max \{\max_{x\in {\mathbb{R}}^N}u(\lambda )(x)$,
$\overline{t}\max_{x\in
\mathbb{R}^N}\omega (x)\}>0$,
$\underline{a}=\inf_{|x|\leq 1}a(x)$, by (\ref{3.6}), we obtain
\begin{align*}
\Lambda _{1}
\geq\underline{a}
\int_{|x|\leq 1}C_{4}(s_{0},r_{0})(t\omega _{R})^{2}dx
\geq\underline{a}C_{4}(s_{0},r_{0}) \underline{t}^{2}\int_{|x|\leq
1}\omega _{R}^{2}dx\quad \text{for all }t\in [\underline{t},\overline{t}].
\end{align*}
 From Lemma \ref{lemma:3.4} (I), we have for
 $A= \underline{a}C_{1}C_{4}(s_{0},r_{0})\underline{t}^{2}$,
\begin{equation}
\Lambda _{1}\geq AR^{-(N-1)}e^{-2R}.  \label{3.8}
\end{equation}
Next from (A2), we have for any $R\geq 1$,
\begin{align*}
\Lambda _{2} &= \int_{\mathbb{R}^N}((t\omega
_{R})^{p+1}-a(x)(t\omega _{R})^{p+1})dx \\ &\leq \int_{\mathbb{R}^N}Ce^{-\mu |x|}(t\omega _{R})^{p+1}dx \\ &\leq \overline{t}^{p+1}\int_{\mathbb{R}^N}Ce^{-\mu |x|}\omega
_{R}^{p+1}dx.
\end{align*}
 From (II) and (III) of Lemma \ref{lemma:3.4}, we have for
$B=\overline{t}^{p+1}CC_{3}$,
\begin{equation}
\Lambda _{2}\leq Be^{-\mu R}.  \label{3.9}
\end{equation}
and we choose $R_{0}\geq 1$ so that
\begin{equation}
Be^{-\mu R_{0}}<AR_{0}^{-(N-1)}e^{-2R_{0}}.  \label{3.10}
\end{equation}
Thus from (\ref{3.8})-(\ref{3.10}), we obtain (\ref{3.7}).
\end{proof}

\begin{theorem}\label{thm:3.5}
Assume {\rm (A1)-(A2)} hold. Then for $\lambda \in (0,\lambda^*)$,
there is a second positive solution
$U(\lambda)$ of \eqref{e1.1}.
\end{theorem}


\begin{proof}
We only need to prove that problem \eqref{3.1} has a
positive solution. Set
$ \gamma =\inf_{g\in \Gamma}\max_{t\in [ 0,1]}J(g(t))$, where
$ \Gamma =\{g\in C([0,1],H^{1}(\mathbb{R}^N)):g(0)=0, g(1)=t_{0}\omega
_{R_{0}}\}$. By Lemma \ref{lemma:3.2} and Proposition
\ref{proposition:3.3}, we deduce that $ 0<\alpha \leq \gamma
<S^{\infty }$.
The Mountain Pass Lemma insures the existence of a $(PS)$-sequence
$\{v_{n}\} $ for $J$ at level $\gamma $.
By Proposition \ref{proposition:3.1}, we deduce that
\[
\gamma =\lim_{n\to \infty }J(v_{n})=J(v_{\lambda})
+\sum_{i=1}^{m}I^{\infty }(v^{i}),
\]
for some $v_{\lambda }$, $v^{i}$ satisfying $J'(v_{\lambda })=0$ and $I^{\infty \prime }(v^{i})=0$ for $1\leq
i\leq m$.

By the strong maximum principle, we  need to prove only that
$v_{\lambda}\not\equiv 0$. We have
$\gamma =J(v_{\lambda })\geq \alpha >0$ if $m=0$,
and $S^{\infty}>\gamma \geq J(v_{\lambda })+S^{\infty }$
if $m\geq 1$.
This implies $v_{\lambda }\not\equiv 0$.
\end{proof}

\section{The case $\protect\lambda =\protect\lambda^*$:
properties and bifurcation of positive solutions}

 First, we show that the existence of
positive solutions of \eqref{e1.1} for
$\lambda=\lambda^*$.

\begin{proposition}\label{proposition:4.1}
The set of minimal positive solutions
$\{ u(\lambda ):\lambda \in (0,\lambda^*)\}$
of \eqref{e1.1} is uniformly bounded in $\lambda $
in $L^{\infty }(\mathbb{R}^N)\cap H^{1}({\mathbb{R}}^N)$.
\end{proposition}

\begin{proof}
 From Theorem \ref{thm:2.4},
$$
\int_{\mathbb{R}^N}(|\nabla
u(\lambda )|^{2}+u(\lambda )^{2})dx\geq \mu _{\lambda }(u(\lambda
))\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p+1}dx.
$$
Also we have
$$
 \int_{\mathbb{R}^N}(|\nabla u(\lambda )|^{2}+u(\lambda
)^{2})dx=\lambda \int_{\mathbb{R}^N}\left( a(x)u(\lambda
)^{p+1}+h(x)u(\lambda )\right) dx.
$$
By the H\"older's and Young's inequalities we deduce
\[
\Big( 1-\frac{\lambda }{p\mu _{\lambda }(u(\lambda ))}
-\frac{\epsilon }{2}\lambda\Big) \|u(\lambda )\|^{2}
\leq \frac{\lambda}{2\epsilon}\|h\|_{H^{-1}}
\]
for all $\epsilon >0$. Taking $\epsilon $ small enough so that
\[
\Big( 1-\frac{\lambda }{p\mu _{\lambda }(u(\lambda ))}-\frac{\epsilon }{2}\lambda
\Big) >0
\]
and hence we have $\|u(\lambda )\|\leq C$ which shows that $u(\lambda )$ is
uniformly bounded in $H^{1}(\mathbb{R}^N)$.

By Theorem \ref{thm:2.2} the solution $u(\lambda )$ is strictly
increasing with respect to $\lambda $, we may suppose that
$u(\lambda )\to u(\lambda^*)$ weakly in $H^{1}(\mathbb{R}^N)$
as $\lambda \to \lambda^*$,
and hence $u(\lambda^*)$ is a solution of \eqref{e1.1} with
$\lambda=\lambda^*$.

>From Theorem \ref{thm:2.3} we can deduce that
$\|u(\lambda^*)\|_{L^{\infty }}\leq C$.
Then by Theorem \ref{thm:2.2}, the solution
$u(\lambda )$ is strictly increasing with respect to $\lambda $ and
Theorem \ref{thm:4.6}, the uniqueness of $u(\lambda^*)$
(the proof will be given later), we conclude that
$\|u(\lambda)\|_{L^{\infty }}\leq \|u(\lambda^*)\|_{L^{\infty }}\leq C$.
\end{proof}

Secondly, we show that problem \eqref{e1.1} has exactly two
distinct positive solutions $u(\lambda )$, $U(\lambda )$ for
$\lambda \in (0,\lambda^*)$.

Denote by $Q=\{(\lambda ,u)$ : $u$ solves $(1.1)_{\lambda }\}$,
the set of positive solutions of \eqref{e1.1}. For each
$(\lambda ,u)\in Q$, let's recall that $\mu _{\lambda }(u)$ denote
the number defined by
$$
\mu _{\lambda }(u)=\inf \big\{\int_{\mathbb{R}^N}(|\nabla
v|^{2}+v^{2})dx:v\in H^{1}(\mathbb{R}^N), \int_{\mathbb{R}
^N}pa(x)u^{p-1}v^{2}dx=1\big\},
$$
which is the smallest eigenvalue of the following problem:
\begin{equation}
\begin{gathered}
-\Delta v+v=\mu _{\lambda }(u)pa(x)u^{p-1}v\quad\text{in }
\mathbb{R}^N, \\
v\in H^{1}(\mathbb{R}^N), v>0\quad\text{in }\mathbb{R}^N.
\end{gathered}\label{4.1}
\end{equation}

\begin{proposition}\label{proposition:4.2}
Let $(\lambda ,u)\in Q$, $0<\lambda<\lambda ^{*}$. Then
\begin{itemize}
\item[(i)]  $\mu _{\lambda }(u)>\lambda $ if and only if
$u=u(\lambda )$;

\item[(ii)]  $\mu _{\lambda }(U(\lambda ))<\lambda $, where
$U(\lambda )$ is the second positive solution of \ref{e1.1}.
\end{itemize}
\end{proposition}


\begin{proof}
(i) Let $\psi \geq 0$, $\psi \in H^{1}(\mathbb{R}^N)$. Then from
the convexity of $u^{p}$, we obtain
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^N}\nabla (u-u(\lambda))\cdot \nabla \psi dx
+\int_{\mathbb{R}^N}(u-u(\lambda))\psi dx    \\
&= \lambda \int_{\mathbb{R}^N}a(x)(u^{p}-u(\lambda)^{p})\psi dx \\
&= \lambda \int_{\mathbb{R}^N}a(x)\int_{u(\lambda)}^{u}pt^{p-1}\psi dt\,dx  \\
&\leq \lambda \int_{\mathbb{R}^N}pa(x)u^{p-1}(u-u(\lambda))\psi
dx.
\end{aligned}\label{4.2}
\end{equation}
Set $\psi =(u-u(\lambda))^{+}$. If
$\psi\not\equiv 0$, then by (\ref{4.2}) and the definition of
$\mu_{\lambda }(u)$,
\begin{align*}
\int_{\mathbb{R}^N}(|\nabla \psi |^{2}+|\psi |^{2}) dx
&\leq \lambda \int_{\mathbb{R}^N}pa(x)u^{p-1}\psi ^{2}dx \\
&<&\mu _{\lambda }(u)\int_{\mathbb{R}^N}pa(x)u^{p-1}\psi ^{2}dx  \\
&\leq \int_{\mathbb{R}^N}(|\nabla \psi |^{2}+|\psi |^{2})dx,
\end{align*}
which is impossible. Hence $\psi \equiv 0$, i.e. $u=u(\lambda)$ in
$\mathbb{R}^N$. On the other hand, by Theorem \ref{thm:2.4}, $\mu
_{\lambda }(u(\lambda))>\lambda $. This completes the proof of
(i).

 (ii) By (i), we get that $\mu _{\lambda}(U(\lambda))\leq\lambda $ for
$\lambda\in(0,\lambda ^{*})$. We
claim that $\mu _{\lambda }(U(\lambda))=\lambda $ can not occur.
We proceed by contradiction. Set $w=U(\lambda)-u(\lambda)$; we
have
\begin{equation}  \label{4.3}
-\Delta w+w=\lambda a(x)(U(\lambda)^{p}-(U(\lambda)-w)^{p}),\quad w>0
\quad\text{in }\mathbb{R}^N.
\end{equation}
By $\mu _{\lambda }(U(\lambda))=\lambda $, we have that the
problem
\begin{equation}  \label{4.4}
-\Delta \phi+\phi=\lambda pa(x)U(\lambda)^{p-1}\phi, \quad \phi \in
H^{1}(\mathbb{R}^N)
\end{equation}
possesses a positive solution $\phi_{1}$.

Multiplying (\ref{4.3}) by $\phi_{1}$ and (\ref{4.4}) by $w$,
integrating and subtracting we deduce that
\begin{align*}
0&= \int_{\mathbb{R}^N}\lambda a(x)[U(\lambda)^{p}-(U(\lambda)-w)^{p}-pU(\lambda)^{p-1}w]\phi_{1}dx  \\
&= -\frac{1}{2}\int_{\mathbb{R}^N}\lambda
a(x)p(p-1)\xi_\lambda^{p-2}w^{2}\phi_{1}dx,
\end{align*}
where $\xi_\lambda \in (u(\lambda),U(\lambda))$.
Thus $w \equiv 0$, that is $U(\lambda)=u(\lambda)$ for
$\lambda \in (0,\lambda^{*})$. This is a contradiction.
Hence, we have that $\mu_{\lambda }(U(\lambda))< \lambda $
for $\lambda \in (0,\lambda^{*})$.
\end{proof}

\begin{remark}\label{remark:4.3}\rm
 Since $\mu _{\lambda }(U(\lambda ))<\lambda $, one may
employ a similar argument to the one used for $u(\lambda )$ to show that
$U(\lambda )$ is strictly decreasing in $\lambda $,
$\lambda \in (0,\lambda^*)$.
\end{remark}

\begin{remark}\label{remark:4.4}\rm
To the authors' knowledge, it is still unknown about the existence
of solutions for the case of $\lambda>\lambda ^{*}$.
\end{remark}

When $\lambda =\lambda^*$, we have the existence of positive solution
$u(\lambda^*)$ of $(1.1)_{\lambda^*}$ in Proposition
\ref{proposition:4.1}. Now we show that $u(\lambda^*)$ is the
unique positive solution of $(1.1)_{\lambda^*}$.

\begin{lemma}\label{lemma:4.5}
For any $g(x)\in H^{-1}(\mathbb{R}^N)$, the problem
\begin{equation}
-\Delta w+w=\lambda pa(x)u(\lambda )^{p-1}w+g(x), w\in
H^{1}(\mathbb{R}^N)  \label{4.4.1}
\end{equation}
has a solution $w$, where $u(\lambda )$ is the minimal solution of
\eqref{e1.1} for $\lambda \in (0,\lambda^*)$.
\end{lemma}


\begin{proof}
Consider the functional
\[
\Phi (w)=\frac{1}{2}\int_{\mathbb{R}^N}\big( | \nabla w|
^{2}+w^{2}\big) dx-\frac{1}{2}\int_{\mathbb{R}^N}\lambda pa(x)u(\lambda
)^{p-1}w^{2}dx-\int_{\mathbb{R}^N}g(x)wdx.
\]
 From the definition of $\mu _{\lambda }$, $\mu _{\lambda }>\lambda $,
the H\"older's and Young's inequality we have
\begin{equation} \label{4.5}
\begin{aligned}
\Phi (w)
&= \frac{1}{2}\|w\|^{2}-\frac{1}{2}\int_{\mathbb{R}^N}\lambda
 pa(x)u(\lambda )^{p-1}w^{2}dx-\int_{\mathbb{R}^N}g(x)wdx   \\
&\geq \frac{1}{2}\|w\|^{2}-\frac{\lambda }{2\mu _{\lambda }}
 \|w\|^{2}-\int_{\mathbb{R}^N}g(x)wdx   \\
&\geq \big( \frac{1}{2}-\frac{\lambda }{2\mu _{\lambda }}\big) \|w\|^{2}-
 \frac{\varepsilon }{2}\|w\|^{2}-\frac{C_{\varepsilon }}{2}\|g\|_{H^{-1}}^{2} \\
&= \big( \frac{1}{2}-\frac{\lambda }{2\mu _{\lambda }}-\frac{\varepsilon }{2
 }\big) \|w\|^{2}-\frac{C_{\varepsilon
}}{2}\|g\|_{H^{-1}}^{2}  \\
&\geq -C\|g\|_{H^{-1}}^{2}
\end{aligned}
\end{equation}
if we choose $\epsilon$ small.


Let $\{w_{n}\}\subset H^{1}(\mathbb{R}^N)$ be the minimizing
sequence of the variational problem $ d=\inf \{ \Phi (w):w\in
H^{1}(\mathbb{R}^N)\}$. From (\ref{4.5}),
\[
\big( \frac{1}{2}-\frac{\lambda }{2\mu _{\lambda }}-\frac{\varepsilon }{2}
\big) \|w_{n}\|^{2}
\leq \Phi (w_{n})+\frac{C_{\varepsilon }}{2}\|g\|_{H^{-1}}^{2}
= d+\frac{C_{\varepsilon}}{2}\|g\|_{H^{-1}}^{2}+o(1)
\]
as $n\to \infty$.
Since $\mu _{\lambda }>\lambda $, we deduce that $\{w_{n}\}$ is bounded in
$H^{1}(\mathbb{R}^N)$ if we choose $\varepsilon $ small.
So we may suppose that
\begin{gather*}
w_{n} \rightharpoonup w\quad\text{weakly in }H^{1}(\mathbb{R}^N), \\
w_{n} \to w\quad\text{almost everywhere in }\mathbb{R}^N, \\
w_{n} \to w\quad\text{strongly in }L_{\rm loc}^{s}(\mathbb{R}^N)\quad
 \text{for }2\leq s<\frac{2N}{N-2}.
\end{gather*}
We now prove that
\begin{equation}
\int_{\mathbb{R}^N}\lambda pa(x)u(\lambda )^{p-1}\left(
w_{n}-w\right) ^{2}dx\to 0 \quad\text{as }n\to \infty. \label{4.6}
\end{equation}
By Theorem \ref{thm:2.3} and (A1), we have
$pa(x)u(\lambda )^{p-1}\to 0$ as $|x|\to \infty $, for any
$\varepsilon >0$
there exists $R>0$ such that for $x\in \mathbb{R}^N$, $|x|\geq R$,
$|pa(x)u(\lambda )^{p-1}|<\varepsilon $. Consequently, there exists a
constant $C>0$ such that
$|pa(x)u(\lambda )^{p-1}|\leq C$ for $x\in \mathbb{R}^N$.
Then
\begin{align*}
&\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p-1}\left( w_{n}-w\right) ^{2}dx \\
&= \int_{B_{R}}pa(x)u(\lambda )^{p-1}\left( w_{n}-w\right)
^{2}dx+\int_{\mathbb{R}^N\backslash B_{R}}pa(x)u(\lambda )^{p-1}\left(
w_{n}-w\right) ^{2}dx \\
&\leq C\int_{B_{R}}\left( w_{n}-w\right) ^{2}dx+\varepsilon
\int_{\mathbb{R}^N\backslash B_{R}}\left( w_{n}-w\right) ^{2}dx.
\end{align*}
Since $w_{n}\to w$ strongly in $L_{\rm loc}^{s}(\mathbb{R}^N)$ for
$2\leq s<2N/N-2$, $\{w_{n}\}$ is a bounded sequence in
$H^{1}(\mathbb{R}^N)$, taking $n\to \infty $, then
$R\to \infty $, and finally $\varepsilon \to 0$, we deduce our claim.

 From (\ref{4.6}) we deduce that
$\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p-1}w_{n}^{2}dx\to
\int_{\mathbb{R}^N}pa(x)u(\lambda )^{p-1}w^{2}dx$
and
$\int_{\mathbb{R}^N}g(x)w_{n}dx\to \int_{\mathbb{R}^N}g(x)wdx$
as $n\to \infty $. Thus by Fatou's lemma,
\begin{align*}
\Phi (w)
&= \frac{1}{2}\|w\|^{2}-\frac{1}{2}\int_{\mathbb{R}^N}\lambda
 pa(x)u(\lambda )^{p-1}w^{2}dx-\int_{\mathbb{R}^N}g(x)wdx \\
&\leq \frac{1}{2}\liminf_{n\to \infty}\|w_{n}\|^{2}-\frac{1
}{2}\lim_{n\to \infty }\int_{\mathbb{R}^N}\lambda pa(x)u(\lambda
)^{p-1}w_{n}^{2}dx-\lim_{n\to \infty }\int_{\mathbb{R}^N}g(x)w_{n}dx
 \\
&= \liminf_{n\to \infty}\Phi (w_{n})\\
&=d=\inf_{w\in H^{1}(\mathbb{R}^N)}\Phi (w)
\end{align*}
and hence $\Phi (w)=d$,
which implies that $w$ is a solution of  \eqref{4.4.1}.
\end{proof}

\begin{theorem}\label{thm:4.6}
Let $u(\lambda^*)$ be a solution of \eqref{e1.1} with $\lambda=\lambda^*$.
Then $\mu _{\lambda^*}(u(\lambda^*))=\lambda^*$.
Moreover, $u(\lambda^*)$ is the unique positive
solution of \eqref{e1.1} with $\lambda=\lambda^*$.
\end{theorem}


\begin{proof}
Define $F:\mathbb{R}\times H^{1}(\mathbb{R}^N)\to H^{-1}(\mathbb{R}^N)$ by
\[
F(\lambda ,u)=\Delta u-u+\lambda (a(x)u^{p}+h(x)).
\]
Since $\mu _{\lambda }(u(\lambda ))>\lambda $ for
$\lambda \in (0,\lambda^*)$, it follows that
$\mu _{\lambda^*}(u(\lambda^*))\geq \lambda^*$.
If $\mu _{\lambda^*}(u(\lambda^*))>\lambda^*$, the equation
$F_{u}(\lambda^*,u(\lambda^*))\phi =0$ has no
nontrivial solution. From Lemma \ref{lemma:4.5}, $F_{u}$ maps
$\mathbb{R}\times H^{1}(\mathbb{R}^N)$ onto $H^{-1}(\mathbb{R}^N)$.
Applying the implicit function theorem to $F$ we can find a
neighborhood $(\lambda^*-\delta ,\lambda^*+\delta )$
of $\lambda^*$ such that \eqref{e1.1} possesses a
solution $u(\lambda )$ if $\lambda \in (\lambda^*-\delta
,\lambda^*+\delta )$. This is contradictory to the definition of
$\lambda^*$.

Next, we are going to prove that $u(\lambda^*)$ is unique.
In fact, suppose \eqref{e1.1} with $\lambda=\lambda^*$
 has another solution $U(\lambda^*)$. Set
$w=U(\lambda^*)-u(\lambda^*)$; we have
\begin{equation}  \label{4.6.1}
-\Delta w+w=\lambda ^{*} a(x)((w+u(\lambda^*))^{p}-u(\lambda^*)^{p}),
\; w>0 \quad\text{in }\mathbb{R}^N.
\end{equation}
By $\mu _{\lambda ^{*}}(u(\lambda^*))=\lambda ^{*}$, we
have that the problem
\begin{equation}  \label{4.6.2}
-\Delta \phi+\phi=\lambda ^{*}pa(x)u(\lambda^*)^{p-1}\phi,
\phi \in H^{1}(\mathbb{R}^N)
\end{equation}
possesses a positive solution $\phi_{1}$.

Multiplying (\ref{4.6.1}) by $\phi_{1}$ and (\ref{4.6.2}) by $w$,
integrating and subtracting we deduce that
\begin{align*}
0&= \int_{\mathbb{R}^N}\lambda ^{*}a(x)[(w+u(\lambda^*))^{p}-u(\lambda^*)^{p}-pu(\lambda^*)^{p-1}w]\phi_{1}dx  \\
&= \frac{1}{2}\int_{\mathbb{R}^N}\lambda ^{*}p(p-1)\xi_{\lambda
^{*}}^{p-2}w^{2}\phi_{1}dx,
\end{align*}
where $\xi_{\lambda ^{*}}\in (u(\lambda^*),u(\lambda^*)+w)$.
Thus $w \equiv 0$.
\end{proof}

\begin{theorem}\label{thm:4.7}
$(\lambda^*,u(\lambda^*))$ is a bifurcation point in
$C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$.
\end{theorem}


We prove that $(\lambda^*,u(\lambda^*))$ is a bifurcation point in
$C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$ by using an
idea in \cite{KLO}. To this end, we need the following bifurcation theorem.

\subsection*{Bifurcation Theorem  \cite[Theorem 3.2]{CR}}
Let $X$ and $Y$ be Banach spaces. Let $(\bar{\lambda},\bar{x})\in
\mathbb{R}\times X$
and let $G$ be a twice continuously differentiable mapping of an open
neighborhood of $(\bar{\lambda},\bar{x})$ into $Y$.
Let $N(G_{x}(\bar{\lambda},\bar{x}))=\mathop{\rm span}\{x_{0}\}$ be
one dimensional and $codimR(G_{x}(\bar{\lambda},\bar{x}))=1$. Let
$G_{\lambda }(\bar{\lambda},\bar{x})\notin
R(G_{x}(\bar{\lambda},\bar{x}))$.
If $Z$ is the complement of $span\{x_{0}\}$ in $X$, then the solutions of
$G(\lambda ,x)=G(\bar{\lambda},\bar{x})$ near
$(\bar{\lambda},\bar{x})$ form a curve $(\lambda
(s),x(s))=(\bar{\lambda}+\tau (s),\bar{x}+sx_{0}+z(s))$, where
$s\to (\tau (s),z(s))\in \mathbb{R}\times Z$ is a twice
continuously differentiable function near $s=0$ and $\tau (0)=\tau
'(0)=z(0)=z'(0)=0$.

We define $G:\mathbb{R}\times C^{2,\alpha }(\mathbb{R}^N)\cap
H^{2}(\mathbb{R}^N)\to C^{\alpha }(\mathbb{R}^N)\cap
L^{2}(\mathbb{R}^N)$ by
\[
G(\lambda ,u)=\Delta u-u+\lambda (a(x)u^{p}+h(x)),
\]
where $C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$ and
$C^{\alpha }(\mathbb{R}^N)\cap L^{2}(\mathbb{R}^N)$ are endowed with
the natural norms.
Then they become Banach spaces. We show that the Bifurcation Theorem
\cite{CR} applies at $(\lambda^*,u(\lambda^*))$. Indeed, from Remark
\ref{remark:2.5} and Theorem \ref{thm:4.6}, problem (\ref{4.1}) has
a solution $\phi _{1}>0$ in $\mathbb{R}^N$.
$\phi _{1}>0\in C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$ if
$h\in C^{\alpha }(\mathbb{R}^N)\cap L^{2}(\mathbb{R}^N)$. Thus
$G_{u}(\lambda^*,u(\lambda^*))\phi =0$,
$\phi\in C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N)$, has a
solution $\phi _{1}>0$. This implies that
$\mathop{\rm dim}N(G_{u}(\lambda^*,u(\lambda^*)))
=\mathop{\rm dim} \mathop{\rm span}\{\phi _{1}\}=1$ and
$\mathop{\rm codim}R(G_{u}(\lambda^*,u(\lambda^*)))=1$ by the
Fredholm alternative.

It remains to check that
$G_{\lambda }(\lambda^*,u(\lambda^*))\notin R(G_{u}(\lambda^*,u(\lambda^*)))$.
By contradictory, it would imply the existence of $v(x)\not\equiv
0$ such that
\[
\Delta v-v+\lambda^*pa(x)u(\lambda^*)^{p-1}v=-(a(x)u(\lambda^*)^{p }+h(x)),
v\in C^{2,\alpha }(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N).
\]
 From $G_{u}(\lambda^*,u(\lambda^*))\phi _{1}=0$ we conclude that
$\int_{\mathbb{R}^N}(a(x)u(\lambda^*)^{p }+h(x))\phi _{1}dx=0$.
This is impossible because
$h(x)\geq 0$, $h(x)\not\equiv 0$ and $\phi _{1}(x)>0$ in
$\mathbb{R}^N$. Applying the Bifurcation Theorem \cite{CR} we conclude that
$(\lambda^*,u(\lambda^*))$ is the bifurcation point
near which, the solutions of $(1.1)_{\lambda^*}$ form a
curve $(\lambda^*+\tau (s),u(\lambda^*)+s\phi _{1}+z(s))$ with $s$
near $s=0$ and $\tau (0)=\tau '(0)=0$, $z(0)=z'(0)=0$.

We claim that $\tau ''(0)<0$ which implies that the
bifurcation curve turns strictly to the left in $(\lambda, u)$
plane. Since $\lambda=\lambda^*+\tau (s)$, $u=u(\lambda^*)+s\phi _{1}+z(s)$ in
\begin{equation}
-\Delta u+u-\lambda (a(x)u^{p}+h(x))=0, u>0, u\in C^{2,\alpha
}(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N). \label{4.7}
\end{equation}
Differentiate (\ref{4.7}) in $s$ twice we have
\begin{align*}
&-\Delta u_{ss}+u_{ss}-\lambda_{ss}(a(x)u^{p}+h(x))
-\lambda_{s}(pa(x)u^{p-1}u_{s})-\lambda_{s}(pa(x)u^{p-1}u_{s}) \\
&-\lambda(p(p-1)a(x)u^{p-2}u_{s}^{2}+pa(x)u^{p-1}u_{ss})=0.
\end{align*}
Set here $s=0$ and use that $\tau '(0)=0$,
$u_{s}=\phi_{1}(x)$ and $u=u(\lambda^*)$ as $s=0$ we obtain
\begin{equation} \label{4.8}
\begin{aligned}
&-\Delta u_{ss}+u_{ss}-\tau ''(0)(a(x)u(\lambda^*)^{p}+h(x))  \\
&-\lambda^*(p(p-1)a(x)u(\lambda^*)^{p-2}\phi
_{1}(x)^{2}+pa(x)u(\lambda^*)^{p-1}u_{ss})=0.
\end{aligned}
\end{equation}
Multiplying $ G_{u}(\lambda^*,u(\lambda^*))\phi_{1}=0 $ by $u_{ss}$,
and (\ref{4.8}) by $\phi _{1}$, integrating
and subtracting the results we obtain
\[
\lambda^*p(p-1)\int_{\mathbb{R}^N}a(x)u(\lambda^*)^{p-2}\phi _{1}(x)^{3}dx
+\tau ''(0)\int_{\mathbb{R}^N}(a(x)u(\lambda^*)^{p}+h(x))\phi_{1}(x)dx=0
\]
which immediately gives $\tau ''(0)<0$. Thus
\begin{gather*}
u(\lambda)\to u(\lambda^*) \quad\text{in } C^{2,\alpha
}(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N) \quad\text{as } \lambda
\to \lambda^*,
\\
U(\lambda)\to u(\lambda^*) \quad\text{in } C^{2,\alpha
}(\mathbb{R}^N)\cap H^{2}(\mathbb{R}^N) \quad\text{as } \lambda
\to \lambda^*.
\end{gather*}

\begin{proposition}\label{proposition:4.9}
The solution set $Q$ is unbounded in
$C^{2,\alpha }(\mathbb{R}^N)\cap H^{1}(\mathbb{R}^N)$.
\end{proposition}

\begin{proof}
By Proposition \ref{proposition:4.1}, the set of minimal positive solutions
$\{ u(\lambda )\} $ of \eqref{e1.1} is uniformly
bounded in $H^{1}(\mathbb{R}^N)$. We show that $\{ U(\lambda
):\lambda \in (0,\lambda^*)\} $ is unbounded in
$H^{1}(\mathbb{R}^N)$. First, we show that for any $\delta >0$,
$\{U(\lambda ) :\lambda \in [\delta,\lambda^*)\}$ is bounded
in $H^{1}(\mathbb{R}^N)$. Since $U(\lambda )$ is decreasing in
$\lambda $, therefore, we have
\begin{align*}
\|U(\lambda)\|^{2}
&= \lambda \int_{\mathbb{R}^N}\left(a(x)U(\lambda)^{p+1}+h(x)U(\lambda)\right)
dx \\
&\leq \lambda^*\int_{\mathbb{R}^N}\left( U(\delta)^{p+1}+h(x)U(\delta)\right) dx \\
&\leq C_{1}
\end{align*}
for some constant $C_{1}>0$.

By Adams \cite{Ad}, Gilbarg-Trudinger \cite[Theorem 8.32, 9.16]{GT}, and
the Sobolev embedding theorem, for all $0<\alpha <1$,
\begin{align*}
\|U(\lambda)\|_{C^{1,\alpha }}
&\leq C\|U(\lambda)\|_{W^{2,q\alpha }} \\
&\leq C\left( \|U(\lambda)\|_{L^{q\alpha }}+\|\lambda \left(
  a(x)U(\lambda)^{p}+h(x)\right) \|_{L^{q\alpha }}\right) \\
&\leq C\left(\|U(\delta)\|_{L^{q\alpha }}+\|U(\delta
 )^{p}+h\|_{L^{q\alpha }}\right) \\
&\leq C_{2}
\end{align*}
for some constant $C_{2}>0$.

Then we  need only to show that $\{ U(\lambda ):\lambda \in
(0,\delta)\} $ is unbounded in $H^{1}(\mathbb{R} ^N)$. Since
$U(\lambda )=u(\lambda )+v_{\lambda }$, we claim that
$\{v_{\lambda }:\lambda>0\}$ is unbounded in $H^{1}(\mathbb{R}^N)$.
If not, then
\begin{equation}
\|v_{\lambda }\|\leq M  \label{4.9}
\end{equation}
for all $\lambda \in (0,\delta]$. Choose $\lambda _{n}\downarrow
0$ and let $v_{\lambda _{n}}$ be the corresponding solutions of \
$(3.1)_{\lambda_{n} }$. Then $v_{\lambda _{n}}$
satisfies
\begin{align*}
\int_{\mathbb{R}^N}\left( \left| \nabla v_{\lambda _{n}}\right|
 ^{2}+v_{\lambda _{n}}^{2}\right) dx
&= \lambda _{n}\int_{\mathbb{R}^N}a(x)\left( U(\lambda _{n})^{p}
 -u(\lambda_{n})^{p}\right) v_{\lambda _{n}}dx \\
&\leq C\lambda _{n}\|U(\lambda_{n})\|^{p}\|v_{\lambda _{n}}\| \\
&\leq \overline{C}\lambda _{n}
\end{align*}
for some constant $\overline{C}$, independent of $v_{\lambda _{n}}$, where
we have used (\ref{4.9}) and the boundedness of $\{u(\lambda _{n})\}$ in
$H^{1}(\mathbb{R}^N)$. Hence, we have
$lim_{n\to \infty }\|v_{\lambda _{n}}\|^{2}=0$. It implies that
\begin{equation}
\lim_{n\to \infty }\|v_{\lambda _{n}}\|_{L^{2}}=0.
\label{4.10}
\end{equation}
On the other hand, we notice that
$U(\lambda )=u(\lambda)+v_{\lambda }$ is decreasing and
$u(\lambda )$ is increasing in $\lambda $. Therefore,
$ v_{\lambda _{n}}\geq v_{\delta }$ for all $n$.
Then we obtain
\begin{equation}
\|v_{\lambda _{n}}\|_{L^{2}}\geq \|v_{\delta }\|_{L^{2}}>0\quad
\text{for all }n , \label{4.11}
\end{equation}
which contradicts (\ref{4.10}).

Next we claim that $\{ U(\lambda ):\lambda \in
(0,\delta)\} $ is unbounded in $C^{2,\alpha }(\mathbb{R}^N)$.
Otherwise, we suppose $ \|U(\lambda _{n})\|_{C^{2,\alpha }}\leq M$,
 for all $\lambda _{n} \in (0,\delta)$, $\lambda_{n}\downarrow 0$.
The compact Sobolev embedding theorem and a
diagonal process allow us to extract a subsequence of
$\{U(\lambda_{n})\}$, still denoted by $\{U(\lambda _{n})\}$, such that
$U(\lambda _{n})\to U$ in $C^{2}(\mathbb{R}^N)$.
Then $U$ would satisfy $ -\Delta U+U=0$ in $\mathbb{R}^N$, which
implies $U=0$.

However, by Remark \ref{remark:4.4} that
$m_{n}:=\mathop{\rm meas}\{x\in \mathbb{R}^N:U(\lambda _{n})>U(\delta )\}>0$.
If $m_{n}\to 0$, then by the Lebesgue's dominated convergence theorem it
results that
\begin{align*}
\|U(\lambda _{n})\|^{2} &= \lambda _{n}\int_{\mathbb{R}^N}(a(x)
U(\lambda _{n})^{p}+h(x)) U(\lambda _{n})dx \\ &= \lambda
_{n}\int_{U(\lambda _{n}) \leq U(\delta) }+\lambda
_{n}\int_{U(\lambda _{n}) > U(\delta)}(a(x) U(\lambda
_{n})^{p}+h(x)) U(\lambda _{n})dx \\ &\leq \lambda _{n}
\Big(\int_{U \leq U(\delta) }(a(x) U^{p}+h(x))U dx +Cm_{n}\Big)
\to 0 \quad\text{as }n\to \infty.
\end{align*}
Thus
$\|v_{\lambda _{n}}\|^{2}\to 0$,
which contradicts (\ref{4.11}). Therefore, we may assume that
there exists $\alpha >0$ independent of $\lambda _{n}$ such that
$\mathop{\rm meas}\{x\in \mathbb{R}^N:U(\lambda _{n})>U(\delta) \}
\geq \alpha >0$ for all $\lambda _{n}$,
which would imply
$\mathop{\rm meas}\{x\in \mathbb{R}^N:U>U(\delta) \} \geq \alpha >0$.
But $U\equiv 0$, a contradiction.
This completes the proof.
\end{proof}

The proof of Theorem \ref{thm:1.1} follows from
 Theorems \ref{thm:2.2},  \ref{thm:2.3},
\ref{thm:2.4}(3),  \ref{thm:3.5}, Proposition
\ref{proposition:4.1}, Theorems \ref{thm:4.6},
\ref{thm:4.7}, and Proposition \ref{proposition:4.9}.



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