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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1997/05\hfil Neumann problem for the $p$-Laplacian
\hfil\folio}
\def\leftheadline{\folio\hfil Paul A. Binding, Pavel Dr\'abek \& Yin Xi Huang
 \hfil EJDE--1997/05}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1997}(1997), No.\ 05, pp.\ 1--11.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.swt.edu
\hfill\break 
ftp (login: ftp) ejde.math.swt.edu or 147.26.103.110 or 129.120.3.113
\bigskip} }

\topmatter
\title On Neumann boundary value problems for some quasilinear elliptic 
equations
\endtitle

\thanks \noindent
{\it 1991 Mathematics Subject Classifications:} 35J65, 35J70, 35P30.\hfil\break
{\it Key words and phrases:} $p$-Laplacian, positive solutions, 
Neumann boundary value problems.
\hfil\break
\copyright 1997 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted September 11, 1996. Published January 30, 1997. \hfil\break
The research of the authors was supported by  NSERC of Canada and the 
I.W. Killam Foundation, 
the Grant \# 201/94/0008 of the Grant Agency of the Czech Republic, and
a University of Memphis Faculty Research Grant respectively
\endthanks
\author Paul A. Binding\\ Pavel Dr\'abek \\ Yin Xi Huang
    \endauthor
\address Department of Mathematics \& Statistics, 
University of Calgary, Calgary, Alberta, Canada, T2N 1N4 
\endaddress
\email binding\@acs.ucalgary.ca \endemail

\address Department of Mathematics, University of West Bohemia,
P.O. Box 314, 30614 Pilsen, Czech Republic 
\endaddress
\email pdrabek\@kma.zcu.cz \endemail

\address
Department of Mathematical Sciences, University of Memphis, Memphis, 
TN 38152 USA\endaddress
\email huangy\@mathsci.msci.memphis.edu \endemail

\abstract
We study the role played by the indefinite weight
function $a(x)$ on the existence of positive solutions to the problem
$$\left\{ \eqalign{
  -\text{div}\,(|\nabla u|^{p-2}\nabla u)&=
  \lambda a(x)|u|^{p-2}u+b(x)|u|^{\gamma-2}u,
  \quad  x\in\Omega, \cr \frac{\partial u}{\partial n}&=0,  
  \quad x\in\partial\Omega\,,} \right. 
$$
where $\Omega$ is a smooth bounded domain in $\Bbb R^n$, $b$ changes sign, 
$1<p<N$, $1<\gamma<Np/(N-p)$ and $\gamma\ne p$. 
We prove that 
(i) if $\int_\Omega a(x)\, dx\ne 0$ and $b$ satisfies another 
    integral condition, then there exists some
    $\lambda^*$ such that $\lambda^* \int_\Omega a(x)\, dx<0$ and, 
    for $\lambda$ strictly between 0 and $\lambda^*$, 
    the problem has a positive solution and 
(ii) if $\int_\Omega a(x)\, dx=0$, then the problem has a positive 
    solution for small $\lambda$ provided that $\int_\Omega b(x)\,dx<0$.
\endabstract
\endtopmatter

\document
\def\pa{{\partial}}
\def\ga{{\gamma}}
\def\la{{\lambda}}
\def\laq{{\la(q)}}
\def\La{{\Lambda}}
\def\O{{\Omega}}
\def\De{{\Delta}}
\def\na{{\nabla}}
\def\si{{\sigma}}
\def\vp{{\varphi}}
\def\W {{W^{1, p}(\O)}}
\def\div{\mathop{div}}

\subheading{1. Introduction and results}

In this paper we study the existence of positive solutions of 
the Neumann boundary value problem 
$$\left\{ \eqalign{
-\De_pu+g(x, u)&=0,\quad  x\in\O, \cr
\frac{\pa u}{\pa n}&=0,  \quad x\in\pa\O,} \right. \eqno(1.1)
$$
on a bounded domain $\O\subset{\Bbb R}^N$ with smooth boundary $\pa\O$,
where $\De_pu=\div(|\na u|^{p-2}\na u)$ is the $p$-Laplacian with  
$p>1$, and $g(x, u)$ is a Caratheodory function.

A host of literature exists for this type of problem when $p=2$;
see, e.g., [AV], [GO1], [GO2], [G], [TA]  and 
the references therein. Recently Li and Zhen
[LZ] studied (1.1) with $p\ge2$ and obtained some interesting results. 
In this paper we consider a special type of function $g(x, u)$ which was
excluded in [LZ]. More precisely we investigate problems of the type
$$\left\{ \eqalign{
-\De_pu&=\la a(x)|u|^{p-2}u+b(x)|u|^{\ga-2}u,\quad  x\in\O, \cr
\frac{\pa u}{\pa n}&=0,  \quad x\in\pa\O,} \right. \eqno(1.2)_\la
$$
where $a(x), b(x)\in L^\infty(\O)$, and $a(x)$ and $b(x)$  both may change sign.
Also $1<p$ and $1<\ga<p^*$, where $p^*=\infty$ if $p\ge N$ and 
$p^*=Np/(N-p)$ if $p<N$. Here we say a function $f(x)$ changes sign 
if the measures of the sets
$\{x\in\O: f(x)>0\}$ and $\{x\in\O: f(x)<0\}$ are both positive. 

We study the influence of the indefinite weight function $a(x)$ on the 
existence of positive solutions of $(1.2)_\la$. If $c_1\ge a(x)\ge c_2>0$, then 
$\|u\|_{\la a}:=(\int_\O(|\na u|^p-\la a|u|^p))^{1/p}$ defines an
equivalent norm on $\W$ for $\la<0$. Then a standard variational method can
be used to prove the existence of positive solutions to $(1.2)_\la$ (see the proof
of Theorem 1 (ii) below). The case $-c_1\le a(x)\le -c_2<0$ can be dealt with
in the same way. The situation where $a(x)$ changes sign is more complicated
because the related functional 
$$
I(u)=\frac1p\int(|\na u|^p-\la a|u|^p)-\frac1\ga\int b|u|^\ga
$$
may not be coercive. Our method relies on the eigencurve theory
developed in [BH1, BH2]. It turns out that the sign of the integral
$\int_\O a$ plays an important role for the range of $\la$ for 
which $(1.2)_\la$ has a positive solution. 

To be more specific, we introduce some notations and recall some results.
Consider the eigencurve problem 
$$\left\{ \eqalign{
-\De_pu&=\la a(x)|u|^{p-2}u+\mu |u|^{p-2}u,\quad  x\in\O, \cr
\frac{\pa u}{\pa n}&=0,  \quad x\in\pa\O,} \right. \eqno(1.3)
$$
where we treat the eigenvalue $\mu$
associated with a positive eigenfunction as a function of $\la$. By taking
$$
\mu(\la):=\inf_{u\in\W\setminus\{0\}}
\frac{\int_\O|\na u|^p-\la\int_\O a|u|^p}{\int_\O|u|^p}
$$
we can establish the following (see, e.g., [BH1, BH2, H])

\proclaim{Proposition 1} 
Assume that $a\in L^\infty(\O)$. Then $\mu(\la)$ is continuous and concave
and $\mu(0)=0$. If $a(x)>0$, then $\mu(\la)$ is  decreasing,
and if $a(x)<0$, then $\mu(\la)$ is increasing. Assume, now, that
$a$ changes sign in $\O$. (i) If $\int_\O a<0$,    
there exists a unique $\la^+_1>0$ such that $\mu(\la^+_1)=0$ and
$\mu(\la)>0$ for $\la\in (0, \la^+_1)$. (ii) If $\int_\O a=0$, 
then $\mu(0)=0$ and $\mu(\la)<0$ if $\la\ne0$. 
(iii) If $\int_\O a>0$, then 
there exists a unique $\la^-_1<0$ such that $\mu(\la^-_1)=0$ and
$\mu(\la)>0$ for $\la\in (\la^-_1,0)$.  \endproclaim

\noindent{\smc Remark 1.1.} It follows from this proposition that when 
$a$ changes sign and $\int_\O a<0$, the eigenvalue problem
$$\left\{ \eqalign{
-\De_pu&=\la a(x)|u|^{p-2}u,\quad  x\in\O, \cr
\frac{\pa u}{\pa n}&=0,  \quad x\in\pa\O} \right. 
$$
has a positive eigenvalue $\la^+_1$ associated with a positive eigenfunction. 

\medskip

For a given weight function $a(x)$, we define
$$\la_1(a)=\cases +\infty, &\text{if $a(x)<0$},\\
\la^+_1, &\text{if $a$ changes sign and $\int_\O a<0$},\\
0, &\text{if $\int_\O a=0$},\\ 
\la^-_1, &\text{if $a$ changes sign and $\int_\O a>0$},\\ 
-\infty, &\text{if $a(x)>0$}, \endcases
$$
where $\la^+_1$ and $\la^-_1$ are given in Proposition 1. 
Let $\|\cdot\|$ denote the usual norm in $\W$. 
When $\la_1(a)$ is a finite number, we choose a fixed eigenfunction
$\vp_1>0$ associated with $\la_1(a)$ and satisfying $\|\vp_1\|=1$. 
Note that if $\la_1(a)=0$, then we can take $\vp_1\equiv1$. 

With these constructions, we have

\proclaim{Proposition 2} Assume that $a$ changes sign and 
$\int_\O a\ne0$. Then for any $\la$ strictly between
0 and $\la_1(a)$, 
the relation $\|u\|_{\la a}:=(\int_\O(|\na u|^p-\la a|u|^p))^{1/p}$ 
defines an equivalent norm on $\W$. \endproclaim

\demo{Proof} Suppose the contrary. Then there exist $u_n\in \W$ such 
that $\|u_n\|=1$ and $\int_\O(|\na u_n|^p-\la a|u_n|^p)\to0$.   
The variational characterization of $\mu(\la)$ then gives 
$\|u_n\|_{\la a}^p\ge\mu(\la)\int_\O|u_n|^p$. Since $\la$ is between 0 and 
$\la_1(a)$, it follows that $\mu(\la)>0$ so $u_n\to0$ in $L^p(\O)$. 
This implies $\int_\O a|u_n|^p\to0$ and  hence $\int_\O |\na u_n|^p\to 0$. 
This contradicts the fact that $\|u_n\|=1$. 
This proves the proposition. \qed\enddemo

Now we can state our main results. From now on we assume
$1<\ga<p^*$, $\ga\not=p$ and that $a$  and $b$ both change sign. We first 
consider the situation $\int_\O a\ne0$. 

\proclaim{Theorem 1} Let $\int_\O a\ne0$ and $\int b\vp_1^\ga<0$.  
Then there exists a $\la^*\ne0$ with $(\la_1(a)-\la^*)\cdot\int_\O a>0$, 
such that for $\la$ strictly between 0 and $\la^*$, $(1.2)_\la$ has a positive
solution. \endproclaim

The next result deals with the case where $\int a=0$. 

\proclaim{Theorem 2} Assume $\int_\O a=0$ and $\int b<0$.  
Then for small enough $\la\ne0$, $(1.2)_\la$ has a positive solution. \endproclaim

When $\la=0$, we have 

\proclaim{Corollary 1} Assume $\int_\O b<0$. Then the problem
$$\left\{ \eqalign{
-\De_pu&=b(x)|u|^{\ga-2}u,\quad  x\in\O, \cr
\frac{\pa u}{\pa n}&=0,  \quad x\in\pa\O} \right. 
$$
has a positive solution. \endproclaim

\medskip

Throughout this paper we use $c$ to denote various positive constants,
and the integrals are always taken on $\O$ unless otherwise specified.  
We will use variational methods in a similar way to those in [DH].
The proof of Theorem 1 will be divided into three situations: 
(i) $\la=\la_1(a)$, (ii) $0<|\la|<|\la_1(a)|$ and (iii)
$|\la|>|\la_1(a)|$. The details are presented
in Sections 2 and 3.  We then study the case $\int a=0$ in 
Section 4. We conclude with some remarks in Section 5. 

\bigskip\goodbreak

\subheading{2. The case $|\la|\le|\la_1(a)|$} 

We introduce the functional $I$ on the space $\W$  by 
$$
I(u)=\frac1p\int(|\na u|^p-\la a|u|^p)-\frac1\ga\int b|u|^\ga,\eqno(2.1)
$$
and we set
$$\eqalign{
\La&=\{u\in\W: (I'(u),u)=0\}\cr
&=\{u\in\W: \int(|\na u|^p-\la a|u|^p)=\int b|u|^\ga\}.}
$$
We can see that $\vp_1$ does not belong to $\La$ since $\int b\vp_1^\ga<0$. 
Note also that $\La$ is closed and for $u\in\La$, 
$$
I(u)=(\frac1p-\frac1\ga)\int(|\na u|^p-\la a|u|^p)
=(\frac1p-\frac1\ga)\int b|u|^\ga.\eqno(2.2)
$$

The following lemma is needed. 

\proclaim{Lemma 2.1} Under the conditions of Theorem 1 or Theorem 2, any
minimizer or maximizer $z$ of the functional $I$ 
on $\La$ with $I(z)\ne0$ gives a solution of $(1.2)_\la$.  \endproclaim

\demo{Proof} If $z$ is a nonzero maximizer or a
minimizer of $I$ on $\La$, then there exists $\mu\in {\Bbb R}$ such that
$$\eqalign{
&\int|\na z|^{p-2}\na z\na \vp-\int\la a|z|^{p-2}z\vp-\int b|z|^{\ga-2}z\vp\cr
&=\mu\Bigl(p\int|\na z|^{p-2}\na z\na \vp-p\int\la a|z|^{p-2}z\vp-
\ga\int b|z|^{\ga-2}z\vp\Bigr),}
$$
for any $\vp\in\W$. We claim that $\mu=0$, which proves the lemma. 
If $\mu\not=0$, then taking $\vp=z$ and using the fact that $z\in \La$ we get
$$
(\ga-p)\int(|\na z|^p-a|z|^p)=(\ga-p)\int b|z|^{\ga}=0.
$$
Since $I(z)\ne0$, we obtain a contradiction. \qed\enddemo

\demo{Proof of Theorem 1} (i) Here $\la=\la_1(a)$, and we start with
the case $1<\ga<p$. We show that $I$ satisfies the Palais-Smale condition 
on $\La$, so we assume that $\{u_n\}\subset\La$, $|I(u_n)|\le c$ and 
$I'(u_n)\to0$, and we show that $\{u_n\}$ contains a convergent subsequence. 

We first prove that such $\{u_n\}$ is bounded. Suppose this is not 
true.  Let $v_n=u_n/\|u_n\|$. Without loss of generality we may assume that
$v_n\to v_0$ weakly in $\W$ and strongly in $L^p(\O)$ and  $L^\ga(\O)$. 
We claim that $v_0\ne0$. Indeed, 
dividing $|I(u_n)|\le c$ by $\|u_n\|^p$ yields  
$$
\int(|\na v_n|^p-\la a|v_n|^p)\to0.
$$
If $v_0=0$, similarly to the proof of Proposition 2, we have $v_n\to0$ in
$L^p(\O)$ and $\int|\na v_n|^p\to0$. Since $\|v_n\|=1$ and
$\int|\na v_n|^p\to0$, we must have $v_0\ne0$.  This is a contradiction. 
The claim is proved. Thus by the variational characterization of 
$\la=\la_1(a)$ and the weak convergence of $v_n$ to $v_0\ne0$, we have
$$
0=\mu(\la)\le\int(|\na v_0|^p-\la a|v_0|^p) \le\lim_{n\to\infty}
\int(|\na v_n|^p-\la a|v_n|^p) =0.
$$
We conclude that
$$
v_0=k\vp_1 \quad \text{for some nonzero constant}\ k.\eqno(2.3)
$$ 
On the other hand, dividing
$$
\int(|\na u_n|^p-\la a|u_n|^p)=\int b|u_n|^\ga\eqno(2.4)
$$
by $\|u_n\|^\ga$ we obtain
$$
0\le\|u_n\|^{p-\ga}\int(|\na v_n|^p-\la a|v_n|^p)=\int b|v_n|^\ga.
$$  
The fact that $v_n\to k\vp_1$ strongly in $L^\ga(\O)$ together with 
$\int b\vp_1^\ga<0$  then implies that the right hand side of the above 
equality is negative for large $n$. This contradiction shows that 
$\{u_n\}$ is bounded. 

We now can assume that $u_n\to u_0$ weakly in $\W$ and strongly in 
$L^p(\O)$ and $L^\ga(\O)$. Using $I'(u_n)\to0$ we obtain
$$\eqalign{
(I'(u_n)-I'(u_0), u_n-u_0)=
&\int(|\na u_n|^{p-2}\na u_n-|\na u_0|^{p-2}\na u_0)\na(u_n-u_0)\cr
-&\int\la a(|u_n|^{p-2}u_n-|u_0|^{p-2}u_0)(u_n-u_0)\cr
-&\int b(|u_n|^{\ga-2}u_n-|u_0|^{\ga-2}u_0)(u_n-u_0)\to0.}
$$
Due to the continuity of the Nemytskij operators $u\mapsto |u|^{p-2}u$ and
$u\mapsto|u|^{\ga-2}u$ from $L^p(\O)$ into $L^{p/(p-1)}(\O)$ and
$L^p(\O)$ into $L^{\ga/(\ga-1)}(\O)$, respectively, the last two integrals
approach zero. Hence, for $p'=p/(p-1)$, we have (cf. [DH]) 
$$\eqalign{
&\Bigl\{\Bigl(\int|\na u_n|^p\Bigr)^{1/p'}
-\Bigl(\int|\na u_0|^p\Bigr)^{1/p'}\Bigr\}\cdot
\Bigl\{\Bigl(\int|\na u_n|^p\Bigr)^{1/p}
-\Bigl(\int|\na u_0|^p\Bigr)^{1/p}\Bigr\}\cr
&\le \int(|\na u_n|^{p-2}\na u_n-|\na u_0|^{p-2}\na u_0)\na(u_n-u_0)\to0,}
$$
i.e., $\int|\na u_n|^p$ converges to $\int|\na u_0|^p$. 
This together with the weak convergence of
$u_n$ to $u_0$ in $\W$ implies that $u_n\to u_0$  strongly in $\W$. 
Hence $I$ satisfies the Palais-Smale condition on $\La$.

We see that, for $1<\ga<p$, $I(u)<0$ for  $u\in\La\setminus\{0\}$.
We claim that $I$ is bounded from below on $\La$. If, on the contrary, 
there exists $u_n\in\La$ such that $I(u_n)\to-\infty$, then clearly
$\|u_n\|\to\infty$. Let $v_n=u_n/\|u_n\|$. 
Dividing (2.4) by $\|u_n\|^p$ we obtain
$$
\int(|\na v_n|^p-\la a|v_n|^p)=\int b|v_n|^\ga\cdot\|u_n\|^{\ga-p}.\eqno(2.5)
$$
It then follows from $\ga<p$ and $\|v_n\|=1$ that 
$\int(|\na v_n|^p-a|v_n|^p)\to0$. As in the above proof of (2.3) 
we conclude that $v_n$ converges weakly (and, without loss of generality,
strongly in $L^\ga(\O)$) to $k\vp_1$ for some constant $k$. 
This implies that the right hand side of (2.5) is negative for 
large $n$ since $\int b\vp_1^\ga<0$, contradicting the variational
characterization of $\la_1(a)$. 
Since any minimizer $u$ of $I$ on $\La$ now must satisfy $I(u)<0$, 
we find a solution of $(1.2)_\la$ by Lemma 2.1. 
Observe that $|u|$ is also a minimizer of 
$I$ on $\La$. Then the Harnack inequality (cf. e.g. [TR]) implies 
that $u>0$. Thus we obtain a positive solution.

For the case $\ga>p$, let $\La_0:=\La\setminus\{0\}$.
We note that in this case $I(u)>0$ for $u\in\La_0$. 
We first show that $0$  is an isolated point of $\La$. 
Suppose, for some $u_n\in\La_0$, $u_n\to0$. Let $v_n=u_n/\|u_n\|$. 
>From (2.5) and Sobolev's embedding theorem 
we obtain as for (2.3) that $v_n$ converges in $L^\ga(\O)$ 
to $k\vp_1$ for some nonzero constant $k$. It
then follows that the right hand side of (2.5) is negative when $n$ is large,
a contradiction. Now, since $\La$ is closed and $0$ is isolated, 
$\La_0$ is a closed set. 
Thus any minimizer of $I$ on $\La_0$ gives us a nontrivial 
solution. Its positivity is obtained exactly as for the case $1<\ga<p$. 

\medskip

\noindent (ii) Observe that, for the case $0<|\la|<|\la_1(a)|$, 
$\la$ is between 0 and $\la_1(a)$, so 
it follows from Proposition 2 that, for $u\in\W$,
$$
\int|u|^p\le c\int(|\na u|^p-\la a|u|^p).
$$
Thus (2.2) shows that $I$ is bounded from above on $\La$ if 
$\ga<p$ and is bounded from below if $\ga>p$. 
To prove that $I$ satisfies the Palais-Smale condition, we note that
if $|I(u_n)|<c$ then $\|u_n\|$ is bounded. Indeed, 
if $\|u_n\|\to\infty$, we divide $I(u_n)$ by
$\|u_n\|^p$ and obtain  $\int(|\na v_n|^p-\la a|v_n|^p)\to0$,
where $v_n=u_n/\|u_n\|$ and $\|v_n\|=1$. But this is impossible since 
$\la$ strictly between 0 and $\la_1(a)$ gives $\mu(\la)>0$. 
The rest of the proof can be carried out in a similar manner to that of (i). 
\qed\enddemo



\subheading{3. The case $|\la|>|\la_1(a)|$}

We divide $\La$ into three subsets as follows:
$$
\La^+_\la (\text{resp.}\ \La^-_\la, \La^0_\la)=
\{u\in\La: \int(|\na u|^p-\la a|u|^p)>(\text{resp.}\ <, =)
\frac{\ga-1}{p-1}\int b|u|^\ga\}. 
$$ 
We seek critical points of $I$ on one of these sets. Observe that
$$
\La^+_\la (\text{resp.}\ \La^-_\la, \La^0_\la)=
\{u\in\La:(\ga-p)\int b|u|^\ga<(\text{resp.}\ >, =) 0\}.\eqno(3.1)
$$
First we have

\proclaim{Lemma 3.1} Let $\ga>p$, $\int a\ne0$ 
and $\int b\vp_1^\ga<0$. Then there exists $|\la^*|>|\la_1(a)|$, 
such that for any $\la$ strictly between  $\la_1(a)$ and $\la^*$, 
$\La^-_\la$ is closed in $\W$ and open in $\La_\la$. \endproclaim

\demo{Proof} The proof is similar to that of [DH, Lemma 3.3]. 

Assuming this is not true, there exist $\la_n\to\la_1(a)$ and 
$u_n\in\La^-_{\la_n}$ such  that $\int b|u_n|^\ga\to 0$. Observe that, since 
$u_n\in\La^-_{\la_n}$, we also have 
$$
0<\int b|u_n|^\ga=\int(|\na u_n|^p-\la_n a|u_n|^p)\to0.
$$
Let $v_n=u_n/\|u_n\|$. Then we have
$$
0<\int(|\na v_n|^p-\la_n a|v_n|^p)=\int b|v_n|^\ga\cdot\|u_n\|^{\ga-p}
=\int b|u_n|^\ga\cdot\|u_n\|^{-p},
$$
which approaches zero regardless of whether $\|u_n\|\to\infty$ or not.  
We conclude, similarly to the proof of Theorem 1 (i), that 
$v_n\to k\vp_1$ weakly in $\W$ for some constant $k\not=0$. In particular,
$$
\int b|v_n|^{\ga-2}v_n\vp\to0
$$
for all $\vp\in\W$. Taking $\vp=k\vp_1$ in the above we obtain
$\int b|k\vp_1|^\ga=0$, a contradiction. Thus $\La^-_\la$ is closed. 
\qed\enddemo

\demo{Proof of Theorem 1} (iii) Assume first that $\ga>p$. 
We observe that $0\not\in\La^-_\la$, and for  $u\in\La^-_\la$,
$$
I(u)=\frac{\ga-p}{p\ga}\int b|u|^\ga=\frac{\ga-p}{p\ga}
\int(|\na u|^p-\la a|u|^p)>0. 
$$ 
Thus we look for a minimizer of $I$ on the set $\La^-_\la$ when $\ga>p$. 
We assume $\int a<0$, i.e. $\la^*>\la_1(a)$. The other case can be treated
similarly. 

Next we verify that $I$ satisfies the (P-S) condition on $\La^-_\la$ 
when $\la$ is close enough to $\la_1(a)$.  
Let $\{u_n\}$ satisfy the hypotheses of the Palais-Smale condition, i.e.,  
$\{u_n\}\subset\La^-_\la$, $|I(u_n)|\le c$ and $I'(u_n)\to0$. 

We first show that there exist $\si>0$ and $\la^*>\la_1(a)$ such that for
$\la\in (\la_1(a), \la^*)$ and all $u\in\La_\la^-$ 
$$
\int|\na u|^p-\la\int a|u|^p\ge\si\|u\|^p.\eqno(3.2)
$$

Otherwise there are $\la_n>0$ and $u_n\in\La_{\la_n}^-$ such
that 
$$
\int|\na v_n|^p-\la_n\int a|v_n|^p\to0,\quad \text{and}\quad 
\la_n\to\la_1(a), \eqno(3.3)
$$
where $v_n=u_n/\|u_n\|$. Without loss of generality we can assume that
$v_n\to v_0$  
weakly in $\W$ and strongly in $L^p(\O)$, for some $v_0\in\W$. 
Thus $\int a|v_n|^p\to\int a|v_0|^p$ so (3.3) and the variational
characterization of $\la_1(a)$ yield
$$
0\le\int(|\na v_0|^p-\la_1(a)a|v_0|^p)\le
\liminf_{n\to\infty}\int(|\na v_n|^p-\la_n a|v_n|^p)=0.\eqno(3.4)
$$
It follows from (3.4) that either $v_0=0$ or $\la_0=\la_1(a)$ and 
$v_0=\vp_1$. The former case would imply that $v_n\to 0$ in $L^p(\O)$, a
contradiction. In the latter case, 
$\|v_n\|=\|\vp_1\|=1$, so weak convergence of $v_n$ to $\vp_1$
implies that $v_n\to\vp_1$ strongly in $\W$, and hence strongly in $L^\ga(\O)$. 
Since $u_n\in\La_{\la_n}^-$, we get 
$$
0<\int(|\na u_n|^p-\la_na|u_n|^p)<\frac{\ga-1}{p-1}\int b|u_n|^\ga,
$$
and consequently
$$
0<\int b|v_n|^\ga\to\int b\vp_1^\ga<0,
$$
a contradiction. 

Thus by (3.2) we have proved that $\{u_n\}$ is bounded. Now we can follow 
the proof of Theorem 1 (i, ii) to show that such $\{u_n\}$ contains a
convergent subsequence.
Thus we conclude that the Palais-Smale condition is satisfied. 

The standard procedure then implies that the functional $I$ has a minimizer, 
say $z$, on $\La^-_\la$. Since $0\not\in\La^-_\la$, $z\not=0$. 
The fact that $z$ is a positive solution of $(1.2)_\la$ then follows from 
Lemma 2.1 and the Harnack inequality as in the proof of Theorem 1 (i), so
Theorem 1 (iii) is proved for the case $\ga>p$. 

For the case $\ga<p$, we observe that the same procedure shows that
$\La^+_\la$ is a closed set. It is apparent that for $u\in\La^+_\la$, 
$I(u)<0$. Then we can find a nonzero maximizer $z$ of $I$ on $\La^+_\la$ as
above and it follows that this $z$ is a positive solution of $(1.2)_\la$. 
This concludes the proof of Theorem 1 (iii). \qed
\enddemo


\subheading{4. Proof of Theorem 2}

In this section we assume $\int a=0$. Recall that
$$
\La^+_\la (\text{resp.}\ \La^-_\la, \La^0_\la)=
\{u\in\La:(\ga-p)\int b|u|^\ga<(\text{resp.}\ >, =) 0\}.
$$

\proclaim{Lemma 4.1} If $\int b<0$, then for sufficiently small
$\la$, (i) $\La^+_\la$ is closed in $\W$ and open in $\La_\la$ 
when $\ga>p$, and (ii) $\La^-_\la$ is closed in $\W$ and open in $\La_\la$ 
when $\ga<p$. \endproclaim

\demo{Proof} (i) Suppose the contrary. Then there exist $\la_n\to 0$, 
$u_n\in\La_{\la_n}^+$, such that
$$
0>\int(|\na u_n|^p-\la_n a|u_n|^p)=\int b|u_n|^\ga\to0.\eqno(4.1)
$$
    
Let $v_n=u_n/\|u_n\|$ and assume that $v_n\to v_0$ weakly in $\W$ and strongly $L^p(\O)$ and
$L^\ga(\O)$ for some $v_0\in\W$. Dividing (4.1) by $\|u_n\|^p$ we obtain   
                    
$$
0\le\int|\na v_0|^p\le\liminf_{n\to\infty}\int(|\na v_n|^p-\la_na|v_n|^p)
=\liminf_{n\to\infty}\|u_n\|^{\ga-p}\int b|v_n|^\ga.\eqno(4.2)
$$
We claim that $\liminf\|u_n\|^{\ga-p}\int b|v_n|^\ga=0$. Otherwise we obtain
from (4.1) and (4.2) that 
$$
0\le\int b|v_n|^\ga<0\eqno(4.3)
$$
for certain $n$, which is a contradiction. Now, since the right hand side of 
(4.2) is zero, $v_0$ must be a constant. If $v_0\ne0$, then $\int b<0$ gives
$\int b|v_n|^\ga\to\int b|v_0|^\ga<0$, so again we obtain the contradiction 
(4.3). If $v_0=0$, we have
$\int|v_n|^p\to0$ and $\int|\na v_n|^p\to0$ (for a subsequence) from (4.2), 
contradicting $\|v_n\|=1$. So, for $\la$ sufficiently small, $\La_\la^+$ 
is closed. 

\noindent (ii) The case $\ga<p$ is similar. The proof is complete. 
\qed\enddemo

\proclaim{Lemma 4.2} For sufficiently small $\la$, $I$ satisfies the (P-S) 
condition on $\La_\la^+$ if $\ga>p$ and on $\La_\la^- $ if $\ga<p$.
\endproclaim

\demo{Proof} We consider the case $\ga>p$ only. The other case can be dealt
with in a similar way. Assume that the contention is false. Then there
exist $\la_n\to 0$ with an unbounded Palais-Smale sequence in each 
$\La_{\la_n}^+$. Moreover (2.2) shows that we can scale the sequences so
that $\|I(u)\|$ is bounded independently of n for each sequence.
Thus, by a standard diagonal argument, we can find a sequence
$u_n\in\La_{\la_n}^+$ such that $I(u_n)$ is bounded,
$I'(u_n)\to0$ and $\|u_n\|\to\infty$. Let
$v_n=u_n/\|u_n\|$.  Since $I(u_n)$ is bounded,
it follows that $\int b|u_n|^\ga$ is bounded and $\int b|v_n|^\ga\to0$. 
Thus (4.1) holds with $u_n$ replaced by $v_n$, and we obtain a contradiction 
as in the proof of Lemma 4.1. 
We then conclude that the Palais-Smale sequences are bounded for 
sufficiently small $\la$. The rest of the proof is similar
to that of Theorem 1 (i, ii). This concludes the proof. \qed\enddemo

Now we can find a nonzero maximizer of $I$ on $\La^+_\la$ if $\ga>p$ and
a nonzero minimizer of $I$ on $\La^-_\la$ if $\ga<p$, which gives a
positive solution of (1.2)$_\la$. Theorem~2 is proved. 

\demo{Proof of Corollary 1} Note that in this case we have, for $u\in \La$, 
$$
I(u)=(\frac1p-\frac1\ga)\int|\na u|^p=(\frac1p-\frac1\ga)\int b|u|^\ga.
$$
We show that the functional satisfies the Palais-Smale condition on $\La$.
We first claim that any Palais-Smale sequence is bounded. Indeed, suppose for 
some $u_n\in \La$, $|I(u_n)|\le c$, $I'(u_n)\to0$, and $\|u_n\|\to\infty$. 
Then dividing $I(u_n)$ by $\|u_n\|^p$ we obtain $\int|\na v_n|^p\to0$,
where $v_n=u_n/\|u_n\|$. Let $\bar v_n=\int v_n/|\O|$. We have, 
$$
\int|v_n-\bar v_n|^p\le c\int|\na v_n|^p.
$$
We then conclude that $v_n$ converges strongly to some constant $v_0\ne0$ in 
$L^p(\O)$. Since $\O$ is bounded and has smooth boundary, it satisfies a 
uniform interior cone condition. The embedding theorem given in 
[GT, p. 158] then implies 
that $v_n\to v_0$ in $L^\ga(\O)$ strongly. Now dividing 
$\int|\na u_n|^p=\int b|u_n|^\ga$ by $\|u_n\|^\ga$ we obtain
$$
0\le\|u_n\|^{p-\ga}\int|\na v_n|^p=\int b|v_n|^\ga.
$$
It then follows from the strong convergence of $v_n\to v_0$ in $L^\ga(\O)$ 
that $\int b\ge0$, which contradicts the assumption that $\int b<0$. We 
thus conclude that $u_n$ must be bounded. 

Now we can assume that $u_n$ has a subsequence converging strongly to some
$u_0$ in $L^p(\O)$ and $L^\ga(\O)$ and weakly in $\W$. The conclusion that
$u_n$ converges strongly to $u_0$ in $\W$ then follows from similar arguments
to those in the proof of Lemma 2.1. This shows that the functional $I$
satisfies the Palais-Smale condition. The rest of the proof can be carried 
out as for that of Theorem 1 (iii). \qed\enddemo


\subheading{5. Final Remarks}

\noindent (i) When $N=1$, the existence of solutions for problem (1.1) 
with various boundary conditions can
be found in [HM], where the Fu\v cik spectrum was studied and employed. 

\medskip

\noindent(ii)  Existence of positive solutions for Neumann problems when 
$p=2$ has been studied in [BPT]. Part of our results in Theorem 1 (the case
$\int_\O a(x)\ dx<0$) is similar to those in [BPT].  Here we only 
deal with a power type ``nonlinearity," but we allow a higher growth rate 
on the variable $u$. See [TA] for related results.  
A special form of Theorem 1 (for $p=2$ with $\la=\la_1(a)$) has been given 
in [BCN]. We note that the proofs of our results originate with
eigencurve theory and are different from those of [BPT] and [BCN]. Even for 
the case $p=2$, our result for the case $\int_\O a(x)\ dx=0$ is new. 

\medskip

\noindent (iii) An important sign condition on the nonlinear term $g(x,u)$, 
viz., 
a condition of the type either $g(x, u)\cdot u\ge0$ for $|u|>c$  
or $g(x, u)\cdot u\le0$ for $|u|>c$, has been employed 
extensively in the literature (see [G] and [ZL]). 
One easily sees that this does not hold in our case: when $a(x)\equiv0$, 
which is the case studied in [LZ], $b(x)$ must change sign.

\medskip
\noindent{\smc Acknowledgment}: The authors are grateful for the referee's 
valuable suggestions. 

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\enddocument