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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 69, pp. 1--9.\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 2007 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2007/69\hfil A fibering map approach]
{A fibering map approach to a semilinear elliptic boundary value problem}

\author[K. J. Brown, T.-F. Wu\hfil EJDE-2007/69\hfilneg]
{Kenneth J. Brown, Tsung-Fang Wu}  % in alphabetical order

\address{Kenneth J. Brown  \newline
 School of Mathematical and Computer
Sciences and the Maxwell Institute, Heriot-Watt University,
Riccarton, Edinburgh EH14 4AS, UK}
\email{K.J.Brown@hw.ac.uk}

\address{Tsung-Fang Wu \newline
 Department of Applied Mathematics,
 National University of Kaohsiung, Kaohsiung 811, Taiwan}
\email{tfwu@nuk.edu.tw}


\thanks{Submitted February 27, 2007. Published May 10, 2007.}
\subjclass[2000]{35J20,  36J65}
\keywords{Semilinear elliptic boundary value problem;
 variational methods; \hfill\break\indent Nehari manifold; fibering map}

\begin{abstract}
 We prove the existence of at least two positive solutions for
 the semilinear elliptic boundary-value problem
 $$
 -\Delta u(x)  =  \lambda  a(x) u^q + b(x) u^p \quad\mbox{for } x \in \Omega;
 \quad u(x) =  0  \quad \mbox{for }   x \in \partial  \Omega
 $$
 on a bounded region $\Omega$ by using the Nehari manifold and
 the fibering maps associated with the Euler functional for the problem.
 We show how knowledge of the fibering maps for the problem leads to
 very easy existence proofs.
\end{abstract}

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

\section{Introduction}

We shall discuss the existence of positive solutions of the
semilinear elliptic boundary-value problem
\begin{gather} \label{e1.1}
-\Delta u(x)  =  \lambda  a(x) u^q + b(x) u^p \quad \mbox{for } x \in \Omega; \\
u(x)  =  0  \quad \mbox{for }   x \in \partial  \Omega, \label{e1.2}
\end{gather}
where  $\Omega$ is a bounded region with smooth boundary in $\mathbb{R}^N$,
$0<q<1<p<\frac{N+2}{N-2}$,   $\lambda > 0$ and $a,b: \Omega \to \mathbb{R}$
 are smooth functions which are somewhere positive but which may change
sign on $\Omega$.  Equation \eqref{e1.1},  \eqref{e1.2} has been recently studied
in \cite {dgu2003} by using the Mountain Pass Lemma and in \cite{i2005}
and \cite{w2006} using the Nehari manifold.

In \cite{dp1997} and \cite{bz2003} it was shown that the Nehari manifold
for an equation such as \eqref{e1.1} is closely related to the fibering
maps for the problem. In this paper we show how a fairly complete
knowledge of all possible forms of the fibering maps provides a very
simple and comparatively elementary means of establishing results similar
to those proved in \cite{i2005} and \cite{w2006} on the existence of
multiple solutions of \eqref{e1.1},  \eqref{e1.2}. In section 2 we recall
the properties which we shall require of fibering maps and of the Nehari
manifold. In section 3 we give a fairly complete description of the
fibering maps associated with \eqref{e1.1} and in section 4 we
use this information to give a very simple variational proof of the
existence of at least two positive solutions of \eqref{e1.1}, \eqref{e1.2}
for sufficiently small $\lambda$.

We shall throughout  use the function space $W_0^{1,2}( \Omega )$ with norm
\[
\|u \| =  \Big( \int_{\Omega} | \nabla u | ^2 \, dx \Big) ^{1/2}
\]
and the standard $L^p( \Omega )$ spaces whose norms we denote by $\| u \|_p$.

\section{Fibering Maps and the Nehari manifold}

The Euler functional associated with \eqref{e1.1}, \eqref{e1.2}  is
\[
J_{\lambda } (u) = \frac{1}{2} \int_{\Omega} | \nabla u | ^2 \, dx
- \frac{\lambda }{q+1} \int_{\Omega } a(x) | u |^{q+1} \, dx
- \frac{1}{p+1 } \int_{\Omega} b (x) | u | ^{p+1}  \, dx
\]
for all $u \in W_0^{1,2}(\Omega )$.

As $J_{\lambda}$ is not bounded below on $W_0^{1,2}(\Omega)$, it is useful
to consider the functional on the Nehari manifold
\[
M_{\lambda }(\Omega) = \{ u \in W_0^{1,2} ( \Omega )
: \langle J'_{\lambda} (u), u \rangle = 0 \}
\]
where $\langle , \rangle$ denotes the usual duality. Thus
$u \in M_{\lambda}( \Omega )$ if and only if
\begin{equation}
 \int_{\Omega} | \nabla u | ^2 \, dx    - \lambda \int_{\Omega }
a(x) |u |^{q+1}  \, dx  - \int_{\Omega } b(x) | u | ^ {p +1}\, dx = 0
\end{equation}
Clearly $M_{\lambda}(\Omega)$ is a much smaller set than
$W_0^{1,2} ( \Omega )$ and, as we shall show, $J_{\lambda}$ is
much better behaved on $M_{\lambda}(\Omega)$. In particular,
on $M_{\lambda}(\Omega)$ we have that
\begin{equation} \label{J_on_M} % \label{e2.3}
\begin{aligned}
J_{\lambda } (u) &=  ( \frac{1}{2} - \frac{1}{q+1}) \int_{\Omega}
 | \nabla u | ^2   +  ( \frac{1}{q+1} -
\frac{1}{p+1}) \int_{\Omega } b(x) | u | ^ {p+1}  \\
&=  ( \frac{1}{2} - \frac{1}{p+1}) \int_{\Omega}
 | \nabla u | ^2   - \lambda  ( \frac{1}{q+1} -
\frac{1}{p+1}) \int_{\Omega } a(x) | u | ^ {q+1}
\end{aligned}
\end{equation}

\begin{theorem}
$J_{\lambda}$ is coercive and bounded below on $M_{\lambda}(\Omega)$.
\end{theorem}

\begin{proof} It follows from \eqref{J_on_M} and the Sobolev embedding
theorems that there exist positive constants $c_1, c_2$ and $c_3$ such that
\[
J_{\lambda}(u) \geq c_1 \|u \|^2 - c_2 \int_{\Omega} | u |^{q+1} \, dx
\geq c_1\|u \|^2 - c_3 \| u \|^{q+1}
\]
and so $J_{\lambda}$ is coercive and bounded below on $M_{\lambda}(\Omega)$.
\end{proof}

The Nehari manifold is closely linked to the behaviour of the functions of
 the form $\phi_u : t \to J_{\lambda}(tu)$  $(t > 0)$. Such maps are known
as fibering maps and were introduced by Drabek and Pohozaev in \cite{dp1997}
and are also discussed in Brown and Zhang \cite{bz2003}.
If $u \in W_0^{1,2} ( \Omega )$, we have
\begin{gather}
\phi_u(t) = \frac{1}{2} t^2 \int_{\Omega}  | \nabla u | ^2
 -   \lambda \,  \frac{t^{q+1}}{q+1} \int_{\Omega} a | u | ^{q+1}
 - \frac{t^{p+1}}{p+1} \int_{\Omega}b| u |^{p+1}  \label{e2.4} \\
\phi_u'(t) = t \int_{\Omega}  | \nabla u | ^2   - \lambda \,
 t^q \int_{\Omega} a | u | ^{q+1}  - t^p \int_{\Omega}b| u |^{p+1} \label{e2.5} \\
\phi_u''(t) =  \int_{\Omega}  | \nabla u | ^2     -
\lambda  q t^{q-1} \int_{\Omega} a | u | ^{q+1}  -
pt^{p-1} \int_{\Omega}b| u |^{p+1} \label{e2.6}
\end{gather}
It is easy to see that $u \in M_{\lambda}(\Omega) $ if and only if $\phi_u'(1) = 0$ and,
more generally, that $\phi_u'(t) = 0$ if and only if $tu \in M_\lambda (\Omega)$, i.e.,
elements in $M_\lambda(\Omega)$ correspond to stationary points of fibering maps.
Thus it is natural to subdivide $M_\lambda (\Omega)$ into sets corresponding to local
minima, local maxima and points of inflection and so we define
\begin{gather*}
M^{+}_{\lambda}(\Omega)  = \{ u \in M_\lambda (\Omega): \phi_u''(1) > 0 \},
\\
M^{-}_{\lambda}(\Omega)  = \{ u \in M_\lambda (\Omega): \phi_u''(1) < 0 \},
\\
M^{0}_{\lambda}(\Omega)  = \{ u \in M_\lambda (\Omega): \phi_u''(1) = 0 \},
\end{gather*}
and note that if $u \in  M_{\lambda}(\Omega ) $, i.e., $\phi_u'(1) = 0$, then
\begin{equation} \label{second_der}
\begin{aligned}
\phi_u''(1) &=  (1-q) \int_{\Omega}  | \nabla u | ^2 \, dx - (p-q)
\int b(x) | u | ^{p+1} \, dx  \\
& =  (1-p) \int_{\Omega}  | \nabla u | ^2 \, dx - \lambda(q-p)
\int a(x) | u | ^{q+1} \, dx\,.
\end{aligned}
\end{equation}
Also, as proved in Binding, Drabek and Huang \cite{bdh1997} or in
Brown and Zhang \cite{bz2003}, we have the following lemma.

\begin{lemma} \label{lem2.2}
Suppose that $u_0$ is a local maximum or minimum for $J_{\lambda}$
on $M_{\lambda}(\Omega)$. Then, if $u_0 \not \in M_{\lambda}^0(\Omega)$,
$u_0$ is a critical point of $J_{\lambda}$.
\end{lemma}

\section{Analysis of the Fibering Maps}

In this section we give a fairly complete description of the fibering maps
associated with the problem. As we shall see the essential nature of
the maps is determined by the signs of $\int a(x) | u |^{q+1}\, dx$
and $\int_{\Omega} b(x) | u |^{p+1} \, dx$.
We will find it useful to consider the function
\[
m_u(t) = t^{1-q} \int_{\Omega}  | \nabla u | ^2 \, dx  - t^{p-q} \int_{\Omega} b(x) | u |^{p+1} \, dx.
\]
Clearly, for $t>0$, $tu \in M_{\lambda}(\Omega)$ if and only if $t$ is
a solution of
\begin{equation} \label{m_equn}
m_u(t) = \lambda  \int_{\Omega} a(x) | u |^{q+1} \, dx.
\end{equation}
Morever,
\begin{equation} \label{m_u_der_equn}
m_u'(t) = (1-q) t^{-q}  \int_{\Omega}  | \nabla u | ^2 \, dx
 - (p-q) t^{p-q-1} \int_{\Omega} b(x) | u |^{p+1} \, dx.
\end{equation}
It is easy to see that $m_u$ is a strictly increasing function
for $t\geq 0$ whenever $\int_{\Omega} b(x) | u |^{p+1} \, dx \leq 0$
and $m_u$ is initially increasing and eventually decreasing with a
single turning point as in Figure 1(b)  when
$\int_{\Omega} b(x) | u |^{p+1} \, dx > 0$.

\begin{figure}[ht]
\begin{center}
\scalebox{.5}{\includegraphics{fig1}}
\caption{Possible forms of $m(u)$}
\end{center}
\end{figure}

Suppose $tu \in  M_{\lambda}(\Omega)$. It follows from
(\ref {second_der}) and (\ref{m_u_der_equn}) that
$\phi_{tu}''(1) = t^{q+2} m_u'(t)$ and so
$tu \in   M^+_{\lambda}(\Omega)  (  M^{-}_{\lambda}(\Omega)  )$
provided $m_u'(t) >0$  ($<0$).

We shall now describe the nature of the fibering maps for all possible signs
of $\int_{\Omega} b(x) | u |^{p+1} \, dx$ and
$\int_{\Omega} a(x) | u |^{q+1} \, dx$.
If $\int_{\Omega} b(x) | u |^{p+1} \, dx \leq 0$ and
$\int_{\Omega} a(x) | u |^{q+1} \, dx \leq 0$, clearly
 $\phi_u$ is an increasing function of $t$ and so has graph as shown
in Figure 2(a); thus in this case no multiple of $u$ lies in
$M_{\lambda}(\Omega)$.
If  $\int_{\Omega} b(x) | u |^{p+1} \, dx \leq 0$ and
$\int_{\Omega} a(x) | u |^{q+1} \, dx > 0$, then $m_u$ has graph as
in Figure 1(a), and it is clear that there is exactly one solution
of (\ref{m_equn}). Thus there is a unique value  $t(u) > 0$ such
that $t(u)u \in M_{\lambda}(\Omega)$. Clearly $m_u'(t(u)) > 0$
and so $t(u) u \in M^+_{\lambda}(\Omega)$.  Thus the fibering map $\phi_u$
has a unique critical point  at $t = t(u)$ which is a local minimum.
Since $\lim_{t \to \infty } \phi_u(t) = \infty$, it follows that
$\phi_u$ has graph as shown in Figure 2(c).

 Suppose now  $\int_{\Omega} b(x) | u |^{p+1} \, dx > 0$ and
$\int_{\Omega} a(x) | u |^{q+1} \, dx \leq 0$. Then $m_u$ has graph as
shown in Figure 1(b) and it is clear that there is exactly one positive
solution of (\ref{m_equn}). Thus there is again a unique value  $t(u) > 0$
such that $t(u)u \in M_{\lambda}(\Omega)$ and since $m_u'(t(u)) < 0$ in
this case $t(u)u \in   M^{-}_{\lambda}(\Omega)$. Hence  the fibering
map $\phi_u$ has a unique critical point which is a local maximum.
Since $\lim_{t \to \infty } \phi_u(t) =  -\infty$, it follows that
$\phi_u$ has graph as shown in Figure 2(b).

 Finally we consider the case   $\int_{\Omega} b(x) | u |^{p+1} \, dx > 0$
and $\int_{\Omega} a(x) | u |^{q+1} \, dx >  0$ where  the situation is
 more complicated. As in the previous case $m_u$ has a graph as shown in
Figure 1(b). If $\lambda > 0$ is sufficiently large, (\ref{m_equn})
has no solution and so $\phi_u$ has no critical points - in this case
$\phi_u$ is a decreasing function. Hence no multiple of $u$ lies in
$M_{\lambda}(\Omega)$.  If, on the other hand,  $\lambda > 0$ is sufficiently
 small, there are exactly two solutions $t_1(u) < t_2(u)$ of  (\ref{m_equn})
with $m_u'(t_1(u)) >0$ and $m_u'(t_2(u)) < 0$. Thus there are exactly
two multiples of $u \in M_{\lambda}(\Omega)$, namely
$t_1(u)u \in  M^{+}_{\lambda}(\Omega)$ and
$t_2 (u)u\in  M^{-}_{\lambda}(\Omega)$.  It follows that $\phi_u$ has
exactly two critical  points  - a local minimum at $t=t_1(u)$  and a
local maximum at $t = t_2(u)$; moreover $\phi_u$ is decreasing in $(0,t_1)$,
increasing in $(t_1,t_2)$ and decreasing in $(t_2,\infty)$ as in Figure 2(d).

\begin{figure}[ht]
\begin{center}
\scalebox{.5}{\includegraphics{fig2}}
\caption{Possible forms of fibering maps}
\end{center}
\end{figure}

The following result ensures that when $\lambda$ is sufficiently
small the graph of $\phi_u$ must be as shown in Figure 2(d) for all
non-zero $u$.


\begin{lemma} \label{lem3.1}
There exists $\lambda_1 > 0$ such that, when $\lambda < \lambda_1$,
$\phi_u$ takes on positive values for all non-zero $u \in W_0^{1,2}(\Omega)$.
\end{lemma}

\begin{proof}
If $\int_{\Omega} b(x) | u |^{p+1} \, dx \leq 0$, then $\phi_u(t) > 0$
for $t$ sufficiently large.
Suppose $u \in W_0^{1,2}(\Omega)$ and
$\int_{\Omega} b(x) | u |^{p+1} \, dx > 0$. Let
\[
h_u(t) = \frac{t^2}{2} \int_{\Omega} |  \nabla u |^2 \, dx
- \frac{t^{p+1}}{p+1} \int_{\Omega} b(x) | u |^{p+1} \, dx.
\]
Then elementary calculus shows that $h_u$ takes on a maximum value of
$$
 \frac{p-1}{2(p+1)}\Big\{ \frac{( \int_{\Omega} | \nabla u |^2 \, dx)
^{p+1}}{(\int_{\Omega} b(x) | u |^{p+1} \, dx)^{2}} \Big\}^{\frac{1}{p-1}}
\quad \text{when } t = t_{\rm max}
= \Big( \frac{ \int_{\Omega} | \nabla u |^2 \, dx}{\int_{\Omega} b(x)
| u |^{p+1} \, dx} \Big)^{\frac{1}{p-1}}.
$$
However
\[ \frac{ (\int_{\Omega} | \nabla u |^2 \, dx)^{p+1}}{(\int_{\Omega}
 | u |^{p+1} \, dx)^{2}} \geq \frac{1}{S_{p+1}^{2(p+1)}}
\]
where $S_{p+1}$ denotes the Sobolev constant of the embedding of  
$W_0^{1,2}(\Omega)$ into $L^{p+1} ( \Omega)$. Hence
\[
 h_u( t_{\rm max} ) \geq  \frac{p-1}{2(p+1)} \,
\Big( \frac{1}{\Vert b^+ \Vert_{\infty}^2 \, S_{p+1}^{2(p+1)}}
\Big)^{\frac{1}{p-1}} =\delta
\]
where $\delta$ is independent of $u$.

We shall now show that there exists $\lambda_1 > 0$ such that
$\phi_u(t_{\rm max}) > 0$, i.e.,
\[
h_u(t_{\rm max}) - \frac{\lambda ~ (t_{\rm max})^{q+1}}{q+1}
\int_{\Omega} a(x) | u |^{q+1} \, dx > 0
\]
for all $u \in W_0^{1,2}(\Omega) - \{0 \}$ provided $\lambda < \lambda_1$.
We have
\begin{align*}
&\frac{(t_{\rm max})^{q+1}}{q+1} \int_{\Omega} a(x) | u |^{q+1} \, dx\\
&\leq  \frac{1}{q+1}\Vert a \Vert_{\infty} \, S_{q+1}^{q+1}
\Big( \frac{\int_{\Omega} | \nabla u |^2 \, dx}{\int_{\Omega} b(x)
| u |^{p+1} \, dx} \Big) ^{\frac{q+1}{p-1}}
\Big( \int_{\Omega} | \nabla u |^2 \, dx \Big)^{\frac{q+1}{2}} \\
& = \frac{1}{q+1} \, \Vert a \Vert_{\infty} \,  S_{q+1}^{q+1}
 \Big\{ \frac{\left( \int_{\Omega} | \nabla u |^2 \, dx \right)^{p+1}}
{\left( \int_{\Omega} b(x) | u |^{p+1} \, dx \right)^2}
\Big\}^{\frac{q+1}{2(p-1)}} \\
& = \frac{1}{q+1} \, \Vert a \Vert_{\infty} \, S_{q+1}^{q+1} \,
\big[ \frac{2(p+1)}{p-1} \big]^{\frac{q+1}{2}}
 h_u(t_{\rm max})^{\frac{q+1}{2}} = c \, h_u(t_{\rm max})^{\frac{q+1}{2}}
\end{align*}
where $c$ is independent of $u$.
Hence
\[
\phi_u(t_{\rm max}) \geq  h_u(t_{\rm max})
- \lambda c h_u(t_{\rm max})^{\frac{q+1}{2}}
 = h_u(t_{\rm max})^{\frac{q+1}{2}}
\big( h_u(t_{\rm max})^{\frac{1-q}{2}} - \lambda c \big)
\]
and so, since $h_u(t_{\rm max}) \geq \delta$ for all
$u \in W_{0}^{1,2}(\Omega) - \{ 0 \}$, it follows that
$\phi_u (t_{\rm max}) >0$ for all non-zero $u$ provided
$  \lambda < \delta^{\frac{1-q}{2}}/2c = \lambda_1$. This completes the proof.
\end{proof}

It follows from the above lemma  that  when  $\lambda < \lambda_1$,
$\int_{\Omega} a(x) |u|^{q+1} \, dx >0$ and
$\int_{\Omega} b(x) |u|^{p+1} \, dx > 0$  then $\phi_u$ must have exactly
two critical points as discussed in the remarks preceding the lemma.

  Thus  when $\lambda < \lambda_1$ we have obtained a complete knowledge
of the number of critical points of $\phi_u$, of the intervals on which
 $\phi_u$ is increasing and decreasing and of the multiples of $u$ which
lie in $M_{\lambda}(\Omega)$ for every possible choice of signs of
$\int_{\Omega} b(x) |u|^{p+1} \, dx$ and $\int_{\Omega} a(x) |u|^{q+1} \, dx$.
In particular we have the following result.

\begin{corollary} \label{coro3.2}
$M_{\lambda}^0(\Omega) = \emptyset$ when $0 < \lambda < \lambda_1$.
\end{corollary}

 \begin{corollary} \label{coro3.3}
If $\lambda < \lambda_1 $, then there exists $\delta_1 > 0$ such that
$J_{\lambda}(u) \geq \delta_1$ for all $u \in M_{\lambda}^{-}(\Omega)$.
\end{corollary}

\begin{proof} Consider $u \in M_{\lambda}^{-}(\Omega)$.
Then $\phi_u$ has a positive global  maximum at $t = 1$ and
$\int b(x) | u |^{p+1} \, dx > 0$.
Thus
\begin{align*}
J_{\lambda}(u) &= \phi_u(1) \geq \phi_u(t_{\rm max}) \\
&\geq  h_u(t_{\rm max})^{\frac{q+1}{2}}
\left( h_u(t_{\rm max})^{\frac{1-q}{2}} - \lambda c \right) \\
&\geq \delta^{\frac{q+1}{2}}(\delta^{\frac{1-q}{2}}- \lambda c)
\end{align*}
and the left hand side  is uniformly bounded away from $0$ provided
that $\lambda < \lambda_1$.
\end{proof}

\section{Existence of Positive Solutions}

In this section using the properties of fibering maps we shall give
simple proofs of the existence of two positive solutions, one in
$M_{\lambda}^+(\Omega)$ and one in $M_{\lambda}^{-}( \Omega)$.

\begin{theorem} \label{minimizer+}
If $\lambda <  \lambda_1 $, there exists a minimizer of $J_{\lambda}$ on
$M_{\lambda}^{+} (\Omega)$.
\end{theorem}

\begin{proof} Since $J_{\lambda}$ is bounded below on
$M_\lambda(\Omega)$ and so on $M_{\lambda}^{+}(\Omega)$,
there exists a minimizing sequence
$\{ u_n \} \subseteq  M_{\lambda}^{+}(\Omega)$ such that
\[
\lim_{n \to \infty} J_{\lambda}(u_n)
 = \inf_{u \in M_{\lambda}^+(\Omega)} J_{\lambda}(u).
\]
Since $J_{\lambda}$ is coercive, $\{ u_n \}$ is bounded in
$W_0^{1,2} (\Omega )$. Thus we may assume, without loss of generality,
 that $u_n \rightharpoonup u_0$ in $W_0^{1,2} (\Omega )$ and
$u_n \to u_0$ in $L^r(\Omega)$ for $1< r < \frac{2N}{N-2}$.

If we choose $u \in W_0^{1,2} (\Omega )$ such that
$\int_{\Omega} a(x) | u |^{q+1} \, dx > 0$, then the graph of the
fibering map $\phi_u$ must be of one of the forms shown in Figure 2(c)
or (d) and so  there exists $t_1(u)$ such that
$t_1(u) u \in  M_{\lambda}^{+}( \Omega)$ and $J_{\lambda} ( t_1(u) u) < 0$.
Hence, $\inf_{u \in M_{\lambda}^+(\Omega)} J_{\lambda}(u) < 0$.
By \eqref{J_on_M},
\[
J_{\lambda}(u_n) =  ( \frac{1}{2} - \frac{1}{p+1}) \int_{\Omega}
 | \nabla u_n  | ^2  \, dx - \lambda  \, ( \frac{1}{q+1} -
\frac{1}{p+1}) \int_{\Omega } a(x) | u_n  | ^ {q+1} \, dx
\]
and so
\[
  \lambda  \, ( \frac{1}{q+1} -
\frac{1}{p+1}) \int_{\Omega } a(x) | u_n  | ^ {q+1} \, dx
=  ( \frac{1}{2} - \frac{1}{p+1}) \int_{\Omega} | \nabla u_n  | ^2  \, dx
- J_{\lambda}(u_n).
\]
Letting $n \to \infty$, we see that
$\int_{\Omega} a(x) | u_0 |^{q+1} \, dx > 0$. 

Suppose $u_n \not \to  u_0$ in  $W_0^{1,2} (\Omega )$. We shall obtain a contradiction by discussing the fibering map
$\phi_{u_0}$. Since   $\int_{\Omega} a(x) | u_0 |^{q+1} \, dx > 0$,
 the graph of $\phi_{u_0} $ must be either of the form shown in
Figure 2(c) or (d).  Hence  there exists $t_0>0$ such that
$t_0 u_0 \in M_{\lambda}^{+} ( \Omega )$ and $\phi_{u_0}$ is
decreasing on $(0,t_0)$ with $\phi_{u_0}'(t_0) = 0$.

Since  $u_n \not \to  u_0$ in  $W_0^{1,2} (\Omega )$,
 $\int_{\Omega} | \nabla u_0 |^2 \, dx
< \lim \inf _{n\to \infty} \int_{\Omega} | \nabla u_n |^2 \, dx$. Thus, as
\[
\phi_{u_n}'(t) = t \int_{\Omega} | \nabla u_n |^2 \, dx - \lambda  t^q \int_{\Omega} a(x)  | u_n | ^{q+1} \, dx - t^{p} \int_{\Omega} b(x) | u_n | ^{p+1} \, dx
\]
and
 \[
\phi_{u_0}'(t) = t \int_{\Omega} | \nabla u_0 |^2 \, dx - \lambda
t^q \int_{\Omega} a(x)  | u_0 | ^{q+1} \, dx - t^p \int_{\Omega} b(x)
| u_0 | ^{p+1} \, dx,
\]
it follows that $\phi_{u_n}'(t_0) > 0$ for $n$ sufficiently large.
Since $\{ u_n \} \subseteq M_{\lambda}^+(\Omega)$, by considering the
possible fibering maps it is easy to see that $\phi_{u_n}'(t) <0$ for
$0<t<1$ and $\phi_{u_n} '(1) = 0$ for all $n$. Hence we must have  $t_0 > 1$.
But $t_0u_0 \in M_{\lambda}^{+} (\Omega)$ and so
\[
J_{\lambda}(t_0 u_0 ) < J_{\lambda}(u_0)  < \lim_{n \to \infty} J_{\lambda}(u_n) =
\inf_{u \in M_{\lambda}^+(\Omega)} J_{\lambda}(u)
\]
and this is a contradiction.
Hence $u_n  \to  u_0$ in  $W_0^{1,2} (\Omega )$ and so
\[
J_{\lambda}(u_0) = \lim_{n \to \infty} J_{\lambda}(u_n)
=  \inf_{u \in M_{\lambda}^+(\Omega)} J_{\lambda}(u).
\]
Thus $u_0$ is a minimizer for $J_{\lambda}$ on $M_{\lambda}^+(\Omega)$.
\end{proof}

\begin{theorem} \label{minimizer-}
If $\lambda < \lambda_1 $, there exists a  minimizer of $J_{\lambda}$ on
$M_{\lambda}^{-} (\Omega)$.
\end{theorem}

\begin{proof} By Corollary \ref{coro3.3} we have $J_{\lambda}(u) \geq \delta_1 > 0$
for all $u \in M_{\lambda}^{-} (\Omega)$ and so
$\inf_{u \in M_{\lambda}^{-} (\Omega)} J_{\lambda}(u) \geq \delta_1 $.
Hence there exists a minimizing sequence
$\{ u_n \} \subseteq  M_{\lambda}^{-}(\Omega)$ such that
\[
\lim_{n \to \infty} J_{\lambda}(u_n)
= \inf_{u \in M_{\lambda}^-(\Omega)} J_{\lambda}(u) > 0.
\]
As in the previous proof, since $J_{\lambda}$ is coercive, $\{ u_n \}$
is bounded in $W_0^{1,2} (\Omega )$ and we may assume,
without loss of generality, that $u_n \rightharpoonup u_0$ in
 $W_0^{1,2} (\Omega )$ and $u_n \to u_0$ in $L^r(\Omega)$ for
$1 < r < \frac{2N}{N-2}$.
By (2.2)
\[
J_{\lambda}(u_n) =  ( \frac{1}{2} - \frac{1}{q+1}) \int_{\Omega}
 | \nabla u_n  | ^2  \, dx +  ( \frac{1}{q+1} -
\frac{1}{p+1}) \int_{\Omega } b(x) | u_n  | ^ {p+1} \, dx
\]
and, since $\lim_{n \to \infty} J_{\lambda}(u_n) > 0$  and
$$
\lim_{n \to \infty} \int_{\Omega}b(x) | u_n | ^{p+1} \, dx
 = \int_{\Omega} b(x) | u_0(x) |^{p+1} \, dx,
$$
we must have that $\int_{\Omega} b(x) | u_0(x) |^{p+1} \, dx > 0$. Hence
the fibering map $\phi_{u_0}$ must have graph as shown in Figure 2(b) or (d)
 and so  there exists $\hat t >0$ such that
$\hat t u_0 \in M_{\lambda}^{-}(\Omega)$.

Suppose $u_n \not \to  u_0$ in  $W_0^{1,2} (\Omega )$. Using the facts that
$$
\int_{\Omega} | \nabla u_0 |^2 \, dx
< \lim \inf _{n\to \infty} \int_{\Omega} | \nabla u_n |^2 \, dx
$$
 and that, since $u_n \in M_{\lambda}^{-} ( \Omega)$,
$J(u_n)  \geq J(su_n)$ for all $s \geq 0$, we have
\begin{align*}
J(\hat t u_0)
&= \frac{1}{2}  \hat t^2 \int_{\Omega} | \nabla u_0 |^2 \, dx
- \frac{\lambda \hat t^{q+1}}{q+1} \int_{\Omega } a(x) | u_0 | ^ {q+1} \, dx
-  \frac{ \hat t^{p+1}}{p+1} \int_{\Omega } b(x) | u_0 | ^ {p+1} \, dx\\
&< \lim_{n \to \infty} \Big[  \frac{1}{2} ~ \hat t^2 \int _{\Omega}
| \nabla u_n |^2 \, dx
- \frac{\lambda \hat t^{q+1}}{q+1} \int_{\Omega } a(x) | u_n | ^ {q+1} \, dx \\
&\quad -  \frac{ \hat t^{p+1}}{p+1} \int_{\Omega } b(x) | u_n | ^ {p+1} \, dx
\Big] \\
&= \lim_{n \to \infty}J(\hat t u_n) \\
&\leq \lim_{n \to \infty} J(u_n)
= \inf_{u \in M_{\lambda}^-(\Omega)} J_{\lambda}(u)
\end{align*}
which is a contradiction.
Hence $u_n  \to  u_0$ in  $W_0^{1,2} (\Omega )$  and the proof can
 be completed as in the previous theorem.
\end{proof}

\begin{corollary} \label{coro4.3}
Equation \eqref{e1.1},  \eqref{e1.2} has at least two positive solutions
whenever $0 < \lambda < \lambda_1$.
\end{corollary}

\begin{proof} By Theorems \ref{minimizer+} and \ref{minimizer-} there
exist $u^+ \in M_{\lambda}(\Omega)$ and $u^- \in M_{\lambda}^-(\Omega)$
such that $J(u^+) = \inf_{u \in M_{\lambda}^+(\Omega)} J(u) $ and
$J(u^-) = \inf_{u \in M_{\lambda}^-(\Omega)} J(u)$. Moreover
$J(u^{\pm}) = J(|u^{\pm}|)$ and $|u^{\pm}| \in M_{\lambda}^{\pm}(\Omega)$
 and so we may assume $u^{\pm} \geq 0$. By Lemma \ref{lem2.2} $ u^{\pm}$ are critical
points of $J$ on $W_0^{1,2}(\Omega)$ and hence are weak solutions
(and so by standard regularity results classical solutions) of
 \eqref{e1.1},  \eqref{e1.2}. Finally, by the Harnack inequality due
to Trudinger \cite{t1967}, we obtain that $u^{\pm}$ are positive
solutions of  \eqref{e1.1},  \eqref{e1.2}.
\end{proof}

\subsection*{Acknowledgement}
 We would like to thank the referee for bringing \cite{i2005} to our
attention and for making the important observation that the fact
that $M_{\lambda}^0(\Omega) = \emptyset $ follows from sufficient
knowledge of the fibering maps.

 \begin{thebibliography}{00}


\bibitem{bdh1997}  P. A. Binding, P. Drabek and Y. X. Huang;
 {\em On Neumann boundary value problems for some quasilinear
elliptic equations}, Electron.
J. Differential Equations, {\bf 1997}, 1997, no. 5, 1-11.

\bibitem{bz2003} K. J. Brown and Y. Zhang;
{\em The Nehari manifold for a semilinear elliptic problem with a sign
changing weight function }, J. Differential
Equations, {\bf 193 }, 2003, 481-499.

\bibitem{dgu2003} D. G. de Figueiredo, J. P. Gossez and P. Ubilla;
 {\em Local superlinearity and sublinearity for indefinite semilinear
elliptic problems}, J. Funct. Anal., {\bf 199}, 2003, 452-467.

\bibitem{dp1997} P. Drabek and S. I. Pohozaev;
 {\em Positive solutions for the p-Laplacian: application of the
fibering method}, Proc. Royal Soc. Edinburgh Sect A, {\bf 127},
(1997), 703-726.

\bibitem{i2005} Y. Il'yasov;
{\em On non-local existence results for elliptic operators with
convex-concave nonlinearities}, Nonlinear Analysis, {\bf 61}, 2005 211-236.

\bibitem{t1967} N. S. Trudinger, {\em On Harnack type inequalities and
their application to quasilinear elliptic equations},
Comm. Pure Applied Math. 20 (1967) 721-747.

\bibitem{w2006} T.-F. Wu, {\em Multiplicity results for a semilinear
 elliptic equation involving sign-changing weight function}, to appear
in Rocky Mountain J. Math.

\end{thebibliography}

\end{document}