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

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

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

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$.
To prove that $I$ satisfies the Palais-Smale condition, we note that
if $|I(u_n)| , =) 0\}.
$$
\proclaim{Lemma 4.1} If $\int b<0$, then for sufficiently small
$\la$, (i) $\La^+_\la$ is closed in $\W$ and open in $\La_\la$
when $\ga>p$, and (ii) $\La^-_\la$ is closed in $\W$ and open in $\La_\la$
when $\ga \int(|\na u_n|^p-\la_n a|u_n|^p)=\int b|u_n|^\ga\to0.\eqno(4.1)
$$
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 $\ga p$ and on $\La_\la^- $ if $\ga p$ 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 $\ga c$
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