\documentclass[twoside]{article}
\usepackage{amssymb} % font used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil Antimaximum principle for elliptic problems\hfil EJDE--1999/22}
{EJDE--1999/22\hfil T. Godoy, J.-P. Gossez, \& S. Paczka \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 1999}(1999), No.~22, pp. 1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
Antimaximum principle for elliptic problems with weight
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35J20, 35P05.
\hfil\break\indent
{\em Key words and phrases:} Antimaximum principle, indefinite weight,
Fu\v cik spectrum.
\hfil\break\indent
\copyright 1999 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted February 2, 1999. Published June 17, 1999.} }
\date{}
%
\author{T. Godoy, J.-P. Gossez, \& S. Paczka}
\maketitle
\begin{abstract}
This paper is concerned with the antimaximum principle for the linear
problem with weight $-\Delta u = \lambda m(x) u +h(x)$,
under Dirichlet or Neumann boundary conditions.
We investigate the following three questions:
Where exactly can this principle hold?
If it holds, does it hold uniformly or not?
If it holds uniformly, what is the exact interval of uniformity?
We will in particular obtain a variational characterization of
this interval of uniformity.
\end{abstract}
\newtheorem{theo}{Theorem}[section]
\newtheorem{prop}[theo]{Proposition}
\newtheorem{lem}[theo]{Lemma}
\newtheorem{rem}[theo]{Remark}
\newtheorem{claim}[theo]{Claim}
\def\theequation{\thesection.\arabic{equation}}
\section{Introduction}
This paper is concerned with the antimaximum principle (in brief AMP) for
the problem
\begin{equation}
\label{1.1}
-\Delta u = \lambda m(x) u +h(x) \quad\mbox{in } \Omega, \quad
Bu=0\quad\mbox{on }\partial\Omega\,.
\end{equation}
Here $\Omega$ is a smooth bounded domain in ${\mathbb R}^N$ and $Bu=0$ represents
either the Dirichlet or the Neumann homogeneous boundary conditions.
Let us first consider the case where there is no weight, i.e. $m(x) \equiv 1$
in $\Omega$. It is then a standard consequence of the maximum principle that if
$\lambda < \lambda_1$ (where $\lambda_1$ denotes the principal eigenvalue of
$-\Delta$ under the corresponding boundary conditions) and if $h$ is a
nonnegative function that is not identically zero, then the solution $u$ of
(\ref{1.1}) is strictly positive in $\Omega$. Cl\'ement and Peletier
\cite{Cl-Pe} investigated the situation where $\lambda > \lambda_1$ and proved
the following AMP: given a nonnegative function $h$, not identically zero,
there exists $\delta = \delta(h) > 0$ such that if $\lambda_1 < \lambda <
\lambda_1+\delta$, then any solution $u$ of (\ref{1.1}) is strictly negative in
$\Omega$. To describe this situation, we will say that the AMP holds at the
right of the eigenvalue $\lambda_1$. It is also shown in \cite{Cl-Pe} that
$\delta$ can be taken independent of $h$ for the Neumann problem in dimension
$N=1$. We will say in this latter case that the AMP holds {\it uniformly} and
denote by $\delta_1$ the largest $\delta$ admissible.
Recent works dealing with this AMP (without weight) include \cite{Bi} (irregular
domains), \cite{Sw1} (exact $L^p$ space where $h$ should be taken), \cite{Sw2}
(extension to operators of higher order), \cite{DF-Go} (connection with the Fu\v
cik spectrum), \cite{Fl-Go-Ta-DT} (extension to the $p$-Laplacian),
\cite{Ar-Ca-Go} (variational characterization of $\delta_1$).
When a weight $m(x)$ is introduced in (\ref{1.1}), the situation gets more
involved. Indeed, generally, {\it two} principal eigenvalues (i.e. eigenvalues
associated to nonnegative eigenfunctions) are present. Moreover, as we will see
in Remark 5.4 below, the connection between AMP and Fu\v cik spectrum does not
hold anymore (even when the weight does not change sign). The only work we know
which deals with the AMP in the presence of weight is that of Hess \cite{He1}.
It is proved there, in the Dirichlet case, with a weight $m$ in
$C(\overline{\Omega})$ which changes sign in $\Omega$, that the AMP holds at the
right of the positive principal eigenvalue and at the left of the negative
principal eigenvalue.
Our purpose in this paper is to answer rather completely for (\ref{1.1}) the
following three questions: (i) Where exactly can the AMP hold?
(ii) If it holds, does it hold uniformly or not?
(iii) If it holds uniformly, what is the exact interval of uniformity
(i.e. the value of $\delta_1$ above)?
To give an idea of our results, let us consider the Neumann problem, with a
weight $m
$ in $L^\infty(\Omega)$ which changes sign in $\Omega$. Suppose first
${\int_\Omega m \neq 0}$, say ${\int_\Omega m <0}$.
It is then known that there are two
principal eigenvalues: 0 and a positive one which we denote by $\lambda^*$
(cf.
\cite{Br-Li}, \cite{Se-He} as well as Section 2 below). We show that the
AMP holds at
the right of $\lambda^*$ and at the left of $0$. Moreover it is nonuniform
when $N
\geq 2$ and uniform when $N=1$. In the latter case the intervals of
uniformity are
exactly $\lambda^* < \lambda \leq \bar{\lambda}(m)$ and $-\bar{\lambda}(-m)
\leq
\lambda < 0$, where
\begin{equation} \label{1.2}
\bar{\lambda}(m) := \inf \left\{
\begin{array}{l} \int_\Omega (u')^2\,,\ u \in H^1(\Omega)\,,\
\int_\Omega mu^2 = 1\,,\\
\mbox{ and $u$ vanishes somewhere in } \overline{\Omega} \end{array} \right\}.
\end{equation}
We also show in this latter case that the AMP still holds at the right of
$\bar{\lambda}(m)$ and at the left of $-\bar{\lambda}(-m)$, of course now non
uniformly. Suppose now $\int_\Omega m=0$. In this singular case, 0 is the
unique
principal eigenvalue. We show that the AMP holds at the right {\it and} at
the left
of $0$. Moreover it is nonuniform when $N \geq 2$ and uniform when $N =1$.
In the
latter case the intervals of uniformity are exactly $0 < \lambda \leq
\bar{\lambda}(m)$ and $-\bar{\lambda}(-m) \leq \lambda < 0$, with
$\bar{\lambda}(m)$
as in (\ref{1.2}). In this latter case also the AMP still holds (non
uniformly) at
the right of $\bar{\lambda}(m)$ and at the left of $-\bar{\lambda}(-m)$.
We will also see, as a final answer to question (i) above, that the AMP can
not hold
far away to the right of $\bar{\lambda}(m)$ or to the left of
$-\bar{\lambda}(-m)$
(cf. Theorem 3.6). This is true for all $N$, with a suitable extension of
definition
(\ref{1.2}) to higher dimension (cf. formula (\ref{3.1})).
Our methods of proof are rather different from those in the linear papers
\cite{Cl-Pe}, \cite{He1} (which however deal with more general non-selfadjoint operators).
Our present approach is more in the line of the nonlinear works
\cite{Fl-Go-Ta-DT}, \cite{Ar-Ca-Go}. We observe in particular that an
expression
analogous to (\ref{1.2}) was introduced in \cite{Ar-Ca-Go} in the context
of the
$p$-Laplacian. However in \cite{Ar-Ca-Go} (and also in \cite{DF-Go}), the
answers to
questions (ii), (iii) above were derived from information on the Fu\v cik
spectrum.
As already mentioned, the connection between AMP and Fu\v cik spectrum does
not hold
anymore in the presence of weight, and a different approach has to be
introduced. In
this respect the comparison between $\lambda^*$ and $\bar{\lambda}$ in
Lemma 3.1 as
well as the argument of completing a square in the proof of Theorem 3.6 are
the crucial
steps which lead to sharp answers to questions (i), (ii), (iii) above.
Our results relative to the Neumann problem, as briefly described above,
are stated
in detail in Section 3 and proved in Section 4 (for a general operator in
divergence
form). The somewhat simpler case of the Dirichlet problem is briefly
considered in
Section 5. The main differences in that case concern the spectrum itself
(since there
is no singular case of the type ${\int_\Omega m = 0}$) and the
fact that
the AMP is nonuniform for all dimensions. In Section 2 we collect some
preliminary
results on the principal eigenvalues of a selfadjoint Neumann problem with
weight.
\section{Principal eigenvalues in the Neumann case}
\setcounter{equation}{0}
A large part of this paper is concerned with the Neumann problem
\begin{equation}
\label{2.1}
Lu=\lambda m(x) u + h(x) \quad\mbox{in } \Omega, \quad \partial u/\partial
\nu_L = 0
\quad\mbox{on } \partial \Omega.
\end{equation}
Here $\Omega$ is a $C^{1,1}$ bounded domain in ${\mathbb R}^N$, $L$ is an uniformly
elliptic
symmetric expression of the form
$$
Lu :=- \sum_{i,j=1}^N \frac{\partial}{\partial x_i} \left( a_{ij} (x)
\frac{\partial
u}{\partial x_j}\right)
$$
with real-valued coefficients $a_{ij} \in C^{0,1}(\overline{\Omega})$, and
$\partial/\partial \nu_L$ represents the conormal derivation on $\partial
\Omega$
associated to $L$. The real-valued functions $m$ and $h$ belong respectively to
$L^\infty(\Omega)$ and $L^p(\Omega)$, where $p=1$ if $N=1$ and $p > N$ if
$N \geq 2$.
Unless otherwise stated, we will always assume in addition to the above
that $m$
changes sign in $\Omega$, i.e.
\begin{equation}
\label{2.2}
\mbox{meas}\left\{ x \in \Omega; m(x) > 0 \right\} > 0 \mbox{ and meas}
\left\{ x \in
\Omega; \, m(x) < 0\right\} > 0\,.
\end{equation}
Also, without loss of generality, changing $\lambda$ in (\ref{2.1}) if
necessary, we
can assume
\begin{equation}
\label{2.3}
|m(x)| < 1 \mbox{ a.e. in } \Omega.
\end{equation}
Solutions of (\ref{2.1}) are understood in the weak sense: $u \in
H^1(\Omega)$ with
\begin{equation}
\label{2.4}
a(u,v)=\lambda \int_\Omega m u v + \int_\Omega h v \qquad \forall v \in
H^1(\Omega),
\end{equation}
where $a(u,v)$ denotes the Dirichlet form associated to $L$:
$$
a(u,v) := \sum_{i,j = 1}^N \int_\Omega a_{ij} \frac{\partial u}{\partial x_j}
\frac{\partial v}{\partial x_i}.
$$
By the $L^p$ regularity theory, such a solution $u$ belongs to $W^{2,p}
(\Omega)
\subset C^1(\overline{\Omega})$.
Our purpose in this preliminary section is to collect some results relative
to the
principal eigenvalues of the associated problem
\begin{equation}
\label{2.5}
Lu = \lambda m(x) u \quad\mbox{in } \Omega, \quad \partial u/\partial \nu_L = 0
\quad \mbox{on } \partial \Omega.
\end{equation}
Most of those results can be found in \cite{Br-Li}, \cite{Se-He}, although
not in the
same form nor with the same degree of generality. For the sake of
completeness and
for later reference, some proofs will be sketched.
The fundamental tool is the following form of the strong maximum principle.
\begin{prop}
Let $u$ be a solution of the problem
\begin{equation}
\label{2.6}
Lu+a_0(x)u=h \quad\mbox{in } \Omega, \quad \partial u/\partial \nu_L = 0
\quad\mbox{on }\partial \Omega,
\end{equation}
with $a_0 \in L^\infty(\Omega)$, $a_0 \geq 0$, $h$ as above and $h\gneqq 0$.
Then $u$ satisfies
\begin{equation}
\label{2.7}
u > 0 \quad\mbox{in } \overline{\Omega}.
\end{equation}
\end{prop}
\paragraph{Proof.} As already observed, $u \in C^1(\overline{\Omega})$ so that
(\ref{2.7}) makes
sense. Taking $-{u}^-$ as testing function in (\ref{2.6}), one deduces $u
\geq 0$
in $\Omega$. Theorem 8.19 in \cite{Gi-Tr} then implies $u > 0$ in $\Omega$. The
conclusion (\ref{2.7}) can then be derived by using the Hopf boundary lemma
as given
e.g. in Proposition 1.16 from \cite{DF}. One should observe here that this
last step
involves the verification of the fact that the weak solution $u$ satisfies the
pointwise equality $\partial u/\partial \nu_L=0$ on $\partial \Omega$. This
can be
achieved through a standard argument based on integration by parts. Q.E.D.
We are thus interested in the principal eigenvalues of (\ref{2.5}). Clearly
0 is a
principal eigenvalue, with the nonzero constants as eigenfunctions. We also
observe
that if $\lambda \in {\mathbb R}$ is a principal eigenvalue with eigenfunction
$u \gneqq 0$,
then $u > 0$ in $\overline{\Omega}$. (This follows from Proposition
2.1, by
writing equation (\ref{2.5}) as $Lu \pm \lambda u=\lambda (m \pm 1)u$ and using
(\ref{2.3})).
The following expression will play a central role in our approach:
\begin{equation}
\label{2.8}
\lambda^*(m) := \inf \left\{ a(u,u); \, u \in H^1(\Omega) \mbox{ and }
\int_\Omega m
u^2 = 1 \right\}.
\end{equation}
\begin{prop}
(i) Suppose ${\int_\Omega m < 0}$. Then $\lambda^*(m) > 0$ and
$\lambda^*(m)$ is the unique nonzero principal eigenvalue; moreover the
interval
$]0,\lambda^*(m)[$ does not contain any eigenvalue. (ii) Suppose
${\int_\Omega m \geq 0}$. Then $\lambda^*(m) = 0$; moreover, if
${\int_\Omega m=0}$, then $0$ is the unique principal eigenvalue.
\end{prop}
Proposition 2.2 of course also applies to the weight $-m$. In particular, if
${\int_\Omega m > 0}$, then $-\lambda^*(-m)$ is the unique non
zero
principal eigenvalue of (\ref{2.5}).
\paragraph{Proof of Proposition 2.2} In case (i), using Lemma 2.3 below, one sees
that the
infimum (\ref{2.8}) is achieved, so that $\lambda^*(m) > 0$. Replacing $u$
by $|u|$
if necessary, one observes that this infimum is achieved at a function
$u \gneqq 0$. By Lagrange multipliers, $u$ solves an equation of the form
(\ref{2.5})
for some $\lambda \in {\mathbb R}$. Taking $u$ as testing function in this
equation, one
concludes that $\lambda=\lambda^*(m)$, which shows that $\lambda^*(m)$ is a
principal
eigenvalue. In case (ii), if ${\int_\Omega m > 0}$, then the
infimum
(\ref{2.8}) is achieved at a suitable nonzero constant, so that
$\lambda^*(m)=0$. The
case ${\int_\Omega m=0}$ requires a little more care. (Note
that in this
case, the infimum (\ref{2.8}) is {\it not} achieved). We pick $\psi \in
H^1(\Omega)$
with ${\int_\Omega m \psi > 0}$ and put
${u=u_\epsilon:=(1+\epsilon
\psi)/[\int_\Omega m(1+\epsilon \psi)^2]^{1/2}}$ in (\ref{2.8}), where
$\epsilon > 0$
is chosen sufficiently small so that the denominator in the definition of
$u_\epsilon$ does not
vanish. An easy calculation shows that
$a(u_\epsilon,u_\epsilon)
\to 0$ as $\epsilon \to 0$, which yields $\lambda^*(m)=0$.
The fact that if ${\int_\Omega m < 0}$, then
$]0,\lambda^*(m)[$ does not
contain any eigenvalue easily follows from the definition (\ref{2.8}) of
$\lambda^*(m)$. Finally the two statements relative to the uniqueness of
the principal
eigenvalues follow from the more general Proposition 2.4 below. Q.E.D.
\begin{lem}
Assume ${\int_\Omega m < 0}$. Then there exists a constant $c
> 0$ such
that ${a(u,u) \geq c \int_\Omega u^2}$ for all $u \in
H^1(\Omega)$ with
${\int_\Omega mu^2=1}$
\end{lem}
\paragraph{Proof.} Assume by contradiction that for each $k=1,2,\ldots,$ there
exists $u_k
\in H^1(\Omega)$ with ${\int_\Omega m u_k^2=1}$ and
${a(u_k,u_k) \leq (1/k) \int_\Omega u_k^2}$. One distinguishes two
cases: either $||u_k||_{L^2}$ remains bounded (for a subsequence), or
$||u_k||_{L^2}
\to + \infty$ (for a subsequence). In this latter case one considers the
normalization $v_k=u_k/||u_k||_{L^2}$. It is then easily verified that each
of the
two cases leads to a contradiction with $\int_\Omega m < 0$. Q. E. D.
In the following, we will generally only deal with the case
${\int_\Omega m \leq 0}$. The case ${\int_\Omega
m > 0}$
can be reduced to this one by considering the weight $-m$. The two
propositions below
concern problem (\ref{2.1}), with $\lambda \not \in [0,\lambda^*(m)]$ in
the first
one, and $\lambda \in [0,\lambda^*(m)]$ in the second one.
\begin{prop}
Suppose ${\int_\Omega m \leq 0}$. If $\lambda \not \in
[0,\lambda^*(m)]$, then problem (\ref{2.1}) with $h \geq 0$ has no solution
$u \gneqq 0$.
\end{prop}
\paragraph{Proof.} Assume that there exists a solution $u \gneqq 0$ of
(\ref{2.1}) for some $\lambda \in \mathbb R$ and some $h \geq 0$. Applying
Proposition 2.1, we get $u > 0$ in $\overline{\Omega}$, and so $u$ can be written as
$u=e^{-v}$ with say $v \in C^1(\overline{\Omega})$. We pick $w \in H^1(\Omega) \cap
L^\infty(\Omega)$ and take $e^vw^2$ as testing function in (\ref{2.1}). A
simple
calculation using the idea of ``completing a square'' yields
\begin{equation}
\label{2.9}
\lambda \int_\Omega mw^2 = a(w,w)-\int_\Omega he^v w^2-\int_\Omega
,
\end{equation}
where $A$ denotes the matrix $(a_{ij}(x))$ of the coefficients of $L$ and
$<,>$ the
scalar product in ${\mathbb R}^N$. Consequently
\begin{equation}
\label{2.10}
\lambda \int_\Omega m w^2 \leq a(w,w)
\end{equation}
for all $w \in H^1(\Omega)\cap L^\infty (\Omega)$. Since one can clearly
restrict
oneself to this class of functions in the definition (\ref{2.8}) of
$\lambda^*(m)$,
one deduces from (\ref{2.10}) that $\lambda \leq \lambda^*(m)$, and also that
$-\lambda \leq \lambda^*(-m)$. Since $\lambda^*(-m)=0$ by that part of
Proposition
2.2 which is already proved, we can conclude $\lambda \in
[0,\lambda^*(m)]$. Q. E. D.
\begin{rem}
{\rm The calculation in the above proof will be used again in Section 4. It is
inspired from \cite{He2}.
}
\end{rem}
\begin{prop}
Suppose ${\int_\Omega m \leq 0}$. Then the problem (\ref{2.1})
with $h \gneqq 0$ does not admit any solution if $\lambda =0$ or
$\lambda^*(m)$.
It admits a unique solution, which is $>0$ in $\overline{\Omega}$, if $0 <
\lambda <
\lambda^*(m)$.
\end{prop}
\paragraph{Proof.} The nonexistence results follow easily by taking for testing
function in
(\ref{2.1}) the corresponding eigenfunctions. Suppose now $0 < \lambda <
\lambda^*(m)$, and let $u$ be the unique solution of (\ref{2.1}). Clearly
$u \not
\equiv 0$. We claim that $u \geq 0$. Indeed, if this is not so, then $u^-
\not \equiv
0$, and by taking $-u^-$ as testing function in (\ref{2.1}), one gets
\begin{equation}
\label{2.11}
a(u^-,u^-)=\lambda \int_\Omega m(u^-)^2-\int_\Omega h u^-.
\end{equation}
If $a(u^-,u^-)=0$, then $u$ is a constant $<0$, which is easily seen to be
impossible. If $a(u^-,u^-) > 0$, then (\ref{2.11}) implies
${\int_\Omega
m(u^-)^2 > 0}$ and consequently, by the definition of
${\lambda^*(m),
a(u^-,u^-) \geq \lambda^*(m) \int_\Omega m(u^-)^2}$; combining with
(\ref{2.11}) then
yields again a contradiction. So $u \gneqq 0$, and by
Proposition 2.1, we
conclude $u > 0$ in $\overline{\Omega}$. Q. E. D.
Finally, for later reference, we mention the following result whose proof
can be
carried out exactly as that of Theorem 1.13 in \cite{DF}.
\begin{prop}
Suppose ${\int_\Omega m \leq 0}$. The principal eigenvalues 0 and
$\lambda^*(m)$ are simple.
\end{prop}
\begin{rem}
{\rm
The above results can easily be adapted to the simpler case where $m$ does
not change
sign in $\Omega$, say $m \gneqq 0$. In this case
$\lambda^*(m)=0$ and 0 is the unique
principal eigenvalue. Problem (\ref{2.1}) with $h \gneqq 0$ has
no solution $u \geq
0$ if $\lambda > 0$, and no solution at all if $\lambda=0$; its (unique)
solution is
$>0$ in $\overline{\Omega}$ if $\lambda < 0$.
}
\end{rem}
\section{Antimaximum principle in the Neumann case}
\setcounter{equation}{0}
We consider in this section problem (\ref{2.1}) with the same assumptions
on $\Omega,
L, m$ and $h$ as in Section 2. The following expression will play an
important role
in our study of the AMP:
\begin{equation} \label{3.1}
\bar{\lambda}(m) := \inf \left\{
\begin{array}{l} a(u,u) : \ u \in H^1(\Omega)\,, \ \int_\Omega mu^2=1\, \\
\mbox{and $u$ vanishes on some ball in } \overline{\Omega}
\end{array} \right\}.
\end{equation}
It is easily seen that when $N=1$, this definition coincides with that given in
(\ref{1.2}). (This follows from the fact that if $u \in H^1(]a,b[)$ and
vanishes at
$x_0$ with, say, $x_0 < b$, then $u_\epsilon$ defined for $\epsilon > 0$ by
$u_\epsilon(x)=u(x)$ if $x < x_0$, $u_\epsilon(x)=0$ if $x_0 \leq x \leq x_0 +
\epsilon$, $u_\epsilon(x)=u(x-\epsilon)$ if $x > x_0+\epsilon$, converges
to $u$ in
$H^1(]a,b[)$ as $\epsilon \to 0$). Clearly $\lambda^*(m) \leq
\bar{\lambda}(m)$. The
following lemma makes more precise the relation between these two members.
\begin{lem}
If $N \geq 2$, then $\lambda^*(m)=\bar{\lambda}(m)$. If $N=1$, then
$\lambda^*(m) <
\bar{\lambda}(m)$. Moreover, in the latter case, there is no eigenvalue in
$]\lambda^*(m),\bar{\lambda}(m)]$.
\end{lem}
As in Section 2, we can limit ourselves without loss of generality in the
study of
(\ref{2.1}) to the case
\begin{equation}
\label{3.2}
\int_\Omega m \leq 0\,.
\end{equation}
We recall that if ${\int_\Omega m < 0}$ and $0 < \lambda <
\lambda^*(m)$, then the solution $u$ of (\ref{2.1}) with $h \gneqq 0$ is
$ > 0$ in $\overline{\Omega}$. If ${\int_\Omega m = 0}$, then no
result of
the type ``$h \gneqq 0$ implies $u \geq 0$'' holds. With
respect to the
AMP, we have the following three results.
Theorem 3.2 concerns the AMP in general, and its nonuniformity when $N
\geq 2$. Theorem 3.4
characterizes the interval of uniformity when $N=1$. Theorem 3.5 shows that
some form of the
AMP still holds outside this interval of uniformity.
\begin{theo}
Assume (\ref{3.2}). (i) Given $h \gneqq 0$, there exists $\delta =
\delta(h) > 0$ such that if $\lambda^*(m) < \lambda < \lambda^*(m) + \delta$ or
$-\delta < \lambda < 0$, then any solution $u$ of (\ref{2.1}) satisfies $u
< 0$ in
$\overline{\Omega}$. (ii) If $N \geq 2$, then no such $\delta$ independent of
$h$ exists
(either at the right of $\lambda^*(m)$ or at the left of 0).
\end{theo}
\begin{rem}
{\rm In the case where there is no weight and $L=-\Delta$, the fact that
the AMP is nonuniform
for
$N
\geq 2$ (and for all $N$ under the Dirichlet boundary conditions) was already
observed in \cite{Cl-Pe} by using Green function.
A similar observation was also derived in \cite{Ar-Ca-Go} (see also
\cite{DF-Go}) in the case
of the $p$-Laplacian (without weight) by using the Fu\v cik spectrum. It is
not clear whether
the approach based on Green function can be adapted to the context of
Theorem 3.2. On the
other hand the approach based on the Fu\v cik spectrum can not be adapted,
as we will see in
Remark 5.4.}
\end{rem}
\begin{theo}
Assume (\ref{3.2}) and $N=1$. (i) If $\lambda^*(m) < \lambda \leq
\bar{\lambda}(m)$ or
$-\bar{\lambda}(-m) \leq \lambda < 0$, then any solution $u$ of (\ref{2.1})
with $h
\gneqq 0$ satisfies $u < 0$ in $\overline{\Omega}$. (ii)
$\bar{\lambda}(m)$
and $-\bar{\lambda}(-m)$ are respectively the largest and the smallest
numbers such that the
preceding implications hold.
\end{theo}
\begin{theo}
Assume (\ref{3.2}) and $N=1$. (i) Given $h \gneqq 0$, there exists
$\delta = \delta(h) > 0$ such that if $\bar{\lambda}(m) < \lambda <
\bar{\lambda}(m) +
\delta$ or $-\bar{\lambda}(-m)-\delta < \lambda < -\bar{\lambda}(-m)$, then any
solution $u$ of (\ref{2.1}) satisfies $u < 0$ in $\overline{\Omega}$. (ii) No such
$\delta$ independent of $h$ exists (either at the right of
$\bar{\lambda}(m)$ or at
the left of $-\bar{\lambda}(-m)$).
\end{theo}
Our final result makes precise the statement in the introduction that the
AMP cannot
hold far away to the right of $\bar{\lambda}(m)$ or to the left of
$-\bar{\lambda}(-m)$.
\begin{theo}
Assume (\ref{3.2}). (i) Given $\varepsilon > 0$, there exists $h \gneqq 0$
such that for any $\lambda \geq \bar{\lambda}(m)+\varepsilon$,
(\ref{2.1}) has no
solution $u$ satisfying $u < 0$ in $\overline{\Omega}$. (ii) Given $\varepsilon
> 0$,
there exists $h \gneqq 0$ such that for any $\lambda \leq
-\bar{\lambda}(-m)-\varepsilon$, (\ref{2.1}) has no solution $u$ satisfying
$u < 0$
in $\overline{\Omega}$.
\end{theo}
\begin{rem}
{\rm Assume (\ref{3.2}). The following four numbers
$$-\bar{\lambda}(-m) \leq -\lambda^*(-m) =0 \leq
\lambda^*(m)\leq\bar{\lambda}(m)
$$
thus control the domains of validity of the maximum principle and of the
antimaximum
principle. }
\end{rem}
\section{Proofs}
\setcounter{equation}{0}
\paragraph{Proof of Lemma 3.1} We start with the case $N \geq 2$ and introduce
the following
functions: for $N \geq 3$
$$
v_k(x) := \left\{ \begin{array}{l}
1 \mbox{ if } |x| \geq 1/k,\\[1.5mm]
2k|x|-1 \mbox{ if } 1/2k < |x| < 1/k,\\[1.5mm]
0 \mbox{ if } |x| \leq 1/2k,
\end{array} \right.
$$
while for $N=2$,
$$
v_k(x) :=
\left\{
\begin{array}{l}
1-2/k \mbox{ if } |x| \geq 1/k,\\[1.5mm]
|x|^{\delta_k}-1/k \mbox{ if } (1/k)^{1/\delta_k} < |x| < 1/k,\\[1.5mm]
0 \mbox{ if } |x| < (1/k)^{1/\delta_k},
\end{array}
\right.
$$
where $\delta_k \in ]0,1[$ is chosen so that $(1/k)^{\delta_k}=1-1/k$. A simple
calculation shows that $v_k$ converges to the constant function 1 in
$H_{\mbox{loc}}^1({\mathbb R}^N)$ as $k \to \infty$. Fix now $x_0 \in \Omega$.
Then, for any
given $u \in H^1(\Omega) \cap L^\infty(\Omega)$, the function $u(x) v_k(x-x_0)$
vanishes on some ball in $\Omega$ and converges to $u$ in $H^1(\Omega)$ as
$k \to
\infty$. This easily yields the conclusion $\lambda^*(m)=\bar{\lambda}(m)$.
We now turn to the proof that for $N=1$,
\begin{equation}
\label{4.1}
\lambda^*(m) < \bar{\lambda}(m).
\end{equation}
As observed at the beginning of Section 3, when $N=1$,
\begin{equation} \label{4.2}
\bar{\lambda}(m) = \inf \left\{ \begin{array}{l}
a(u,u) : \ u \in H^1(\Omega)\,,\ \int_\Omega mu^2 =1\,, \\
\mbox{and $u$ vanishes somewhere in } \overline{\Omega} \end{array}\right\}.
\end{equation}
Since $H^1(\Omega)$ is compactly imbedded in $C(\overline{\Omega})$, the infimum in
(\ref{4.2}) is achieved. Replacing $u$ by $|u|$ if necessary, we can assume
that it
is achieved at $u \geq 0$.
\begin{claim}
$u$ vanishes at exactly one point $x_0$ in $\overline{\Omega}=[a,b]$. Moreover
$u \in
C^1[a,x_0] \cap C^1[x_0,b]$ and
\begin{equation}
u'(x_0-) < 0 < u'(x_0^+)
\label{4.3}
\end{equation}
(where (\ref{4.3}) is modified into $0 < u'(x_0^+)$ if $x_0=a$, and
similarly if
$x_0=b$).
\end{claim}
\paragraph{Proof.} Part of the argument here is adapted from \cite{Ar-Ca-Go}. We
first show
that if $u$ vanishes at some $x_0 \in \overline{\Omega}$, then
\begin{equation}
\label{4.4}
a(u,v)=\bar{\lambda}(m) \int_\Omega m uv
\end{equation}
for all $v \in V_{x_0}=\{ v \in H^1(\Omega); \, v(x_0)=0\}$. Indeed one has
$$
\bar{\lambda}(m) = \inf \big\{ a(v,v) ; \, v \in V_{x_0} \quad \mbox{and}\quad
\int_\Omega mv^2=1\big\},
$$
where the latter infimum is also achieved at $u$. Applying the standard
theorem on Lagrange
multipliers in the Hilbert space $V_{x_0}$,
$u$ solves an equation like (\ref{4.4}) with some multiplier $\lambda$
instead of
$\bar{\lambda}(m)$. But taking
$v=u$, one gets
$\lambda=\bar{\lambda}(m)$, which yields (\ref{4.4}). Assume now by
contradiction that $u$
vanishes at at least two points $x_1$ and $x_2 \in \overline{\Omega}$. So $u$
satisfies
(\ref{4.4}) for all $v
\in V_{x_1}$ and also for all $v \in V_{x_2}$. Since any $v \in
H^1(\Omega)$ can be written
as $v_1+v_2$ with $v_1 \in V_{x_1}$ and $v_2 \in V_{x_2}$, we conclude that
$u$ satisfies
(\ref{4.4}) for all $v \in H^1(\Omega)$, i.e. that $u$ is a solution of
$$
Lu = \bar{\lambda}(m)m u \quad\mbox{in } \Omega, \quad
\partial u/\partial \nu_L = 0 \quad\mbox{on } \partial \Omega.
$$
Proposition 2.1 then implies $u > 0$ in $\overline{\Omega}$, contradiction. So $u$
vanishes at exactly one point $x_0$.
Finally, assuming for instance $a < x_0 < b$, equation (\ref{4.4}) implies
that $u$
solves the mixed problem
$$
Lu=\bar{\lambda}(m)m u \quad\mbox{in } ]a,x_0[, \, u'(a)=u(x_0)=0\,.
$$
This implies $u \in C^1[a,x_0]$. Moreover $u'(x_0) < 0$ because otherwise
$u$ solves
the corresponding Neumann problem on $]a,x_0[$ and consequently, by
Proposition 2.1,
$u > 0$ on $[a,x_0]$, a contradiction. A similar argument on $[x_0,b]$
completes the
proof of the claim.
The idea of the proof of (\ref{4.1}) is now the following. Define, for
$\varepsilon
\geq 0$, $u_\varepsilon(x)=\max(u(x),\varepsilon)$. Clearly $u_\varepsilon
\to u$ in
$H^1(\Omega)$ as $\varepsilon \to 0$, and so $\int_\Omega m u_\varepsilon^2
> 0$ for
$\varepsilon$ sufficiently small. Putting
$J(\varepsilon)=a(u_\varepsilon,u_\varepsilon)/\int_\Omega m
u_\varepsilon^2$ and
$J(0)=a(u,u)/\int_\Omega mu^2$, we will show below that
\begin{equation}
\label{4.5}
\limsup_{\varepsilon \to 0, \varepsilon > 0}
\frac{J(\varepsilon)-J(0)}{\varepsilon}
< 0\,.
\end{equation}
This implies in particular $J(\varepsilon) < J(0)$ for $\varepsilon > 0$
sufficiently
small. Consequently
$$
\lambda^*(m) \leq J(\varepsilon) a$. So $u \in C^1[a,b]$, $u > 0$ in $]a,b]$, $u(a)=0$, $u'(a+) > 0$;
moreover
$L$ is of the form $Lu=-(p(x)u')'$, with $p(x) \geq p_0 > 0$ on $[a,b]$.
Writing
$u'(a+)=\alpha$, we first fix $c > a$ such that $u'(x) \geq \alpha/2$ and
$\alpha/2$ $(x-a) \leq u(x) \leq 2 \alpha$ $(x-a)$ on $[a,c]$. In the
following,
$\varepsilon > 0$ will be taken sufficiently small so that $u(x) \geq
\varepsilon$ sur $[c,b]$. We have
\begin{eqnarray*}
a(u_\varepsilon, u_\varepsilon) &=& a(u,u) - \int_{u < \varepsilon} p(u')^2\\
&\leq& a(u,u)-(\varepsilon/2\alpha)p_0(\alpha/2)^2,\\
\int_\Omega mu_\varepsilon^2 &=& \int_\Omega mu^2 + \int_{u < \varepsilon}
(mu_\varepsilon^2-mu^2)\\
&=& \int_\Omega mu^2 + O(\varepsilon^3),
\end{eqnarray*}
and consequently
$$
\frac{J(\varepsilon)-J(0)}{\varepsilon} \leq -p_0(\alpha/8) \left(\int_\Omega
mu_\varepsilon^2 \right)^{-1} +O(\varepsilon^2),
$$
which yields (\ref{4.5}).
To conclude the proof of Lemma 3.1, it remains to see that when $N=1$,
there is no
eigenvalue in $]\lambda^*(m),\bar{\lambda}(m)]$. Let $\lambda >
\lambda^*(m)$ be an
eigenvalue, with associated eigenfunction $u$. By Proposition 2.2, $u$
changes sign
and consequently vanishes somewhere in $\overline{\Omega}$. Taking $u$ as
testing function
in $Lu=\lambda mu$, one gets
\begin{equation}
\label{4.6}
a(u,u)=\lambda \int_\Omega mu^2,
\end{equation}
which implies $\lambda \geq \bar{\lambda}(m)$. Suppose now by contradiction
that
$\lambda=\bar{\lambda}(m)$. Since, by (\ref{4.6}),
${\int_\Omega m u^2 >
0}$, we have ${\int_\Omega m(u^+)^2>0}$ or
${\int_\Omega
m(u^-)^2 >0}$. Consider the first case (the argument is similar in the
second case).
Taking $u^+$ as testing function in $Lu=\bar{\lambda}(m)mu$, one gets
$$
a(u^+,u^+)=\bar{\lambda}(m) \int_\Omega m(u^+)^2,
$$
which shows that $u^+$ is a nonnegative minimizer in (\ref{4.2}). The claim
above
then implies that $u^+$ vanishes at exactly one point, which is impossible
since $u$
changes sign. Q. E. D.
\begin{rem}
{\rm Functions like $v_k$ in the proof above were used in \cite{DF-Go} in
the study of
the asymptotic behavior of the first curve in the Fu\v cik spectrum. Note
that no
approximation such as that considered at the beginning of the proof of
Lemma 3.1 for $N \geq
2$ is possible when $N=1$ since, in that case, $H^1$ convergence implies
uniform convergence.}
\end{rem}
\begin{rem}
{\rm Lemma 3.1 still holds, with the same proof, if $m$ does not change
sign, with $m
\gneqq 0$. }
\end{rem}
\paragraph{Proof of Theorem 3.2.} We first prove part (i) at the right of
$\lambda^*(m)$ (the
argument at the left of $0$ is similar). Assume by contradiction the
existence for
some $h \gneqq 0$ of sequences $\lambda_k > \lambda^*(m)$ and
$u_k$ such
that $\lambda_k \to \lambda^*(m)$,
\begin{equation}
\label{4.7}
Lu_k=\lambda_k mu_k + h \quad\mbox{in } \Omega, \quad \partial u_k/\partial \nu_L
= 0 \quad\mbox{on } \partial \Omega
\end{equation}
and
\begin{equation}
\label{4.8}
u_k \geq 0 \mbox{ somewhere in } \overline{\Omega}.
\end{equation}
We distinguish two cases: either $||u_k||_{L^2}$ remains bounded, or, for a
subsequence, $||u_k||_{L^2} \to + \infty$. In the first case, one derives from
(\ref{4.7}) that $u_k$ remains bounded in $W^{2,p}(\Omega)$. Going to the
limit in
(\ref{4.7}), one gets a solution $u$ of
$$
Lu=\lambda^*(m)mu+h \quad\mbox{in } \Omega, \quad \partial u/\partial \nu_L = 0
\quad\mbox{on }\partial \Omega,
$$
which is impossible by Proposition 2.6. In the second case, one considers
$v_k=u_k/||u_k||_{L^2}$, and arguing in a similar way from
$$
Lv_k=\lambda_k mv_k + h/||u_k||_{L^2} \quad\mbox{in } \Omega,\quad \partial
v_k/\partial \nu_L = 0\quad\mbox{on } \partial \Omega,
$$
one gets that, for a subsequence, $v_k \to v$ in $C^1(\overline{\Omega})$ where
$||v||_{L^2}=1$ and
$$
Lv=\lambda^*(m) mv \quad\mbox{in } \Omega, \quad \partial v/\partial \nu_L = 0
\quad\mbox{on }\partial \Omega.
$$
Consequently $v$ is an eigenfunction associated to $\lambda^*(m)$ and so
either $v >
0$ in $\overline{\Omega}$ or $v < 0$ in $\overline{\Omega}$. In the first case, we
deduce $v_k > 0$ in $\overline{\Omega}$ for $k$ sufficiently large, which leads to a
contradiction
with Proposition 2.4. In the second case we deduce $v_k < 0$ in
$\overline{\Omega}$ for
$k$ sufficiently large, which leads to a contradiction with (\ref{4.8}). (This
argument to derive the AMP is adapted from \cite{Fl-Go-Ta-DT}).
Part (ii) of Theorem 3.2 is a consequence of Theorem 3.6 since (\ref{3.2})
and $N
\geq 2$ imply $\bar{\lambda}(m)=\lambda^*(m)$ and
$\bar{\lambda}(-m)=\lambda^*(-m)=0$. Q. E. D.
\paragraph{Proof of Theorem 3.4.} We start with part (i) in the case $\lambda^*(m) <
\lambda \leq \bar{\lambda}(m)$ (the case $-\bar{\lambda}(-m) \leq \lambda <
0$ can be
treated similarly). Let $u$ be a solution of (\ref{2.1}) for some $h \gneqq 0$.
By Proposition 2.4, $u$ can not be $\geq 0$, i.e. $u^- \not
\equiv 0$.
Taking $-u^-$ as testing function in (\ref{2.1}), we get
\begin{equation}
\label{4.9}
a(u^-,u^-)=\lambda \int_\Omega m(u^-)^2-\int_\Omega hu^-.
\end{equation}
If $a(u^-,u^-)=0$, then $u=$Cst$<0$ and we are finished. If $a(u^-,u^-) >
0$, then
(\ref{4.9}) implies ${\int_\Omega m(u^-)^2 > 0}$ and so
\begin{equation}
\label{4.10}
a(u^-,u^-)/\int_\Omega m(u^-)^2 \leq \lambda.
\end{equation}
Suppose first $\lambda < \bar{\lambda}(m)$. Then (\ref{4.10}) implies that
$u^-$ is
not admissible in the definition (\ref{1.2}) of $\bar{\lambda}(m)$.
Consequently
$u^-$ does not vanish in $\overline{\Omega}$, i.e. $u < 0$ in $\overline{\Omega}$.
Suppose now
$\lambda = \bar{\lambda}(m)$. If $u^-$ does not vanish in $\overline{\Omega}$,
we are
finished as above. If $u^-$ vanishes somewhere in $\overline{\Omega}$, then
(\ref{4.10})
(with $\lambda = \bar{\lambda}(m)$) implies that $u^-$ is a nonnegative
minimizer in
the definition of $\bar{\lambda}(m)$. By the claim in the proof of Lemma
3.1, $u^-$
vanishes at exactly one point. But (\ref{4.9}) (with $\lambda =
\bar{\lambda}(m)$)
implies $\int_\Omega hu^-=0$, so that $u^-$ vanishes on the set of positive
measure
where $h > 0$, a contradiction.
Part (ii) of Theorem 3.4 is a consequence of Theorem 3.6. Q. E. D.
\paragraph{Proof of Theorem 3.5.} We first prove part (i) at the right of
$\bar{\lambda}(m)$ (the argument is similar at the left of
$-\bar{\lambda}(-m)$).
Assume by contradiction the existence for some $h \gneqq 0$ of
sequences
$\lambda_k > \bar{\lambda}(m)$ and $u_k$ such that $\lambda_k \to
\bar{\lambda}(m)$,
\begin{equation}
\label{4.11}
Lu_k = \lambda_k m u_k + h \quad\mbox{in } \Omega\,,
\quad \partial u_k/\partial \nu_L=0 \quad\mbox{on } \partial \Omega
\end{equation}
and
\begin{equation}
\label{4.12}
u_k \geq 0 \mbox{ somewhere in } \overline{\Omega}.
\end{equation}
As in the proof of Theorem 3.2, we distinguish two cases: either
$||u_k||_{L^2}$
remains bounded, or, for a subsequence, $||u_k||_{L^2} \to \infty$. In the
first
case, one obtains that, for a subsequence, $u_k$ converges in
$C^1(\overline{\Omega})$ to
a solution $u$ of
$$
Lu=\bar{\lambda}(m)mu+ h \quad\mbox{in } \Omega\,,\quad
\partial u/\partial \nu_L = 0\quad\mbox{on } \partial \Omega\,;
$$
moreover, by (\ref{4.12}), $u \geq 0$ somewhere in $\overline{\Omega}$. But this
contradicts the fact that the AMP holds for $\lambda=\bar{\lambda}(m)$ (cf.
Theorem
3.4). In the second case, one considers $v_k=u_k/||u_k||_{L^2}$ and obtains
that, for
a subsequence, $v_k$ converges in $C^1(\overline{\Omega})$ to a nonzero
solution $v$ of
$$
Lv=\bar{\lambda}(m) mv \quad\mbox{in } \Omega,\quad \partial v/\partial \nu_L=0
\quad\mbox{on }
\partial \Omega.
$$
This again yields a contradiction since by Lemma 3.1, $\bar{\lambda}(m)$ is
not an
eigenvalue.
Part (ii) of Theorem 3.5 clearly follows from the sharpness of
$\bar{\lambda}(m)$ and
$-\bar{\lambda}(-m)$ in Theorem 3.4. Q. E. D.
\paragraph{Proof of Theorem 3.6.} We prove part (i) (part (ii) is proved
similarly). Assume
by contradiction that there exists $\varepsilon > 0$ such that for any
$h \gneqq 0$ there exists $\lambda$ with $\lambda \geq \bar{\lambda}(m) +
\varepsilon$
such that (\ref{2.1}) has a solution $u < 0$ in $\overline{\Omega}$. We start
with $w \in
H^1(\Omega) \cap L^\infty(\Omega)$ satisfying $\int_\Omega mw^2 > 0$ and
vanishing on
some ball in $\overline{\Omega}$, as in the definition (\ref{3.1}) of
$\bar{\lambda}(m)$.
Then we choose $h \gneqq 0$ with supp$h \, \cap$ supp$w =
\emptyset$, and
finally we consider $\lambda=\lambda_w$ and $u=u_w$ as provided by the above
contradictory hypothesis. So $-u > 0$ in $\overline{\Omega}$ and consequently can be
written as $-u=e^{-v}$ with $v \in C^1(\overline{\Omega})$. We then take $e^vw^2$ as
testing function in (\ref{2.1}). A simple calculation using the idea of
``completing
a square'', as in the proof of Proposition 2.4, yields a relation analogous to
(\ref{2.9}):
$$
\lambda \int_\Omega mw^2=a(w,w) + \int_\Omega h e^v w^2 - \int_\Omega
\langle A(\nabla w + w
\nabla v),(\nabla w + w \nabla v) \rangle \,.
$$
Here the integral involving $h$ vanishes since $h$ and $w$ have disjoint
supports.
Consequently
$$
\bar{\lambda}(m)+\epsilon \leq \lambda_w \leq a(w,w) \big( \int_\Omega
mw^2\big)^{-1}
$$
for all $w$ as above. Taking the infimum with respect to $w$ yields
$\bar{\lambda}(m)+\epsilon \leq \bar{\lambda}(m)$, a contradiction. Q. E. D.
\begin{rem}
{\rm It is clear from the above proof that the function $h$ in Theorem 3.6
can be
taken in $C_c^\infty(\Omega)$, with support of arbitrary small diameter.
Similarly
the statements (ii) in each of Theorems 3.2, 3.4 and 3.5 still hold if one
restricts
$h \gneqq 0$ to vary in $C_c^\infty(\Omega)$ with support of
arbitrarily
small diameter. }
\end{rem}
\begin{rem}
{\rm
The above arguments can easily be adapted to the case where $m$ does not
change sign
in $\Omega$, say $m \gneqq 0$, as in Remark 2.8. In this case
the AMP
holds at the right of 0. It is nonuniform when $N \geq 2$ and uniform when
$N=1$. In
this latter case the interval of uniformity is exactly $0 < \lambda \leq
\bar{\lambda}(m)$, with $\bar{\lambda}(m)$ given by (\ref{1.2}); moreover
the AMP
still holds at the right of $\bar{\lambda}(m)$, in a nonuniform way.
Finally, as in
Theorem 3.6, the AMP cannot hold far away to the right of $\bar{\lambda}(m)$. }
\end{rem}
\begin{rem}
{\rm
The above results can also be adapted to the case where $L$ is replaced by
$$
L_0u:=Lu+a_0(x)u,
$$
where $a_0 \in L^\infty(\Omega)$ satisfies $a_0 \gneqq 0$. In
this case 0
is not anymore an eigenvalue. If $m$ changes sign, the principal
eigenvalues are
$\lambda^*(m)$ and $-\lambda^*(-m)$, with $\lambda^*(m)$ defined by
(\ref{2.8})
where $a(u,v)$ now stands for the Dirichlet form associated to $L_0$.
$\bar{\lambda}(m)$ is defined similarly, and the domains of validity of the
maximum
principle and the antimaximum principle are again controlled by the
following four
numbers:
$$
-\bar{\lambda}(-m) \leq -\lambda^*(-m) <0 <\lambda^*(m)\leq \bar{\lambda}(m).
$$
If $m$ does not change sign, say $m \gneqq 0$, then only the
following two
numbers play a role:
$$
0< \lambda^*(m)\leq \bar{\lambda}(m).
$$ }
\end{rem}
\section{Dirichlet boundary conditions}
\setcounter{equation}{0}
In this section we briefly consider the Dirichlet problem
\begin{equation}
\label{5.1}
L_0u=\lambda m(x)u+h(x) \quad\mbox{in } \Omega, \quad u = 0\quad\mbox{on } \partial
\Omega\,,
\end{equation}
where $L_0u=Lu+a_0(x)u$, with $a_0\in L^\infty(\Omega)$, $a_0 \geq 0$ in
$\Omega$.
The assumptions on $\Omega, L, m$ and $h$ are the same as in Section 2.
The basic spectral theory for (\ref{5.1}) is well described in \cite{DF}.
There are
two principal eigenvalues: $\lambda_1(m) > 0$ and $-\lambda_1(-m) < 0$, where
$$
\lambda_1(m) := \inf \{a(u,u); u \in H_0^1(\Omega) \mbox{ and } \int_\Omega
mu^2=1\},
$$
with $a(u,v)$ the Dirichlet form associated to $L_0$. If $-\lambda_1(-m) <
\lambda <
\lambda_1(m)$ and $h \gneqq 0$, then (\ref{5.1}) has a (unique)
solution
$u$, which satisfies $u > 0$ in $\Omega$ and $\partial u/\partial \nu < 0$ on
$\partial \Omega$ (where $\partial/\partial \nu$ represents exterior normal
derivation). With respect to the AMP, we have the following two results.
\begin{theo}
(i) Given $h \gneqq 0$, there exists $\delta = \delta(h) > 0$
such that if
$\lambda_1(m) < \lambda < \lambda_1(m) + \delta$ or $-\lambda_1(-m)-\delta
< \lambda
< -\lambda_1(-m)$, then any solution $u$ of (\ref{5.1}) satisfies $u < 0$
in $\Omega$
and $\partial u/\partial \nu > 0$ on $\partial \Omega$. (ii) No such $\delta$
independent of $h$ exists (either at the right of $\lambda_1(m)$ or at the
left of
$-\lambda_1(-m)$).
\end{theo}
\begin{theo}
(i) Given $\epsilon > 0$ there exists $h \gneqq 0$ such that for
any $\lambda
\geq \lambda_1(m) + \epsilon$, (\ref{5.1}) has no solution $u$ satisfying
$u < 0$ in
$\Omega$. (ii) Similar statement at the left of $-\lambda_1(-m)$.
\end{theo}
As indicated in the introduction, part (i) of Theorem 5.1 in the case of a
weight $m$
in $C(\overline{\Omega})$ was proved in \cite{He1}.
The proof of part (i) of Theorem 5.1 can be carried out by contradiction in
a way
similar to the proof of Theorem 3.2. Part (ii) of Theorem 5.1 follows from
Theorem
5.2. Let us sketch the proof of the latter.
\paragraph{Proof of Theorem 5.2.} We only consider part (i). Assume by
contradiction that
there exists $\epsilon > 0$ such that for any $h \gneqq 0$
there exists
$\lambda$ with $\lambda \geq \lambda_1(m) +\epsilon$ such that (\ref{5.1})
has a
solution $u$ satisfying $u < 0$ in $\Omega$. We start with $w \in
C_c^\infty(\Omega)$
satisfying $\int_\Omega mw^2 > 0$. Then we choose $h \gneqq 0$
with supp
$h \, \cap$ supp $w=\emptyset$, and finally we consider $\lambda=\lambda_w$
and $u=u_w$
as provided by the above contradictory hypothesis. So $-u > 0$ in $\Omega$ and
consequently can be written as $-u = e^{-v}$ with $v \in C^1(\Omega)$. We take
$e^vw^2\in C^1_c(\Omega)$ as testing function in (\ref{5.1}). By a calculation
identical to that in the proof of Theorem 3.6, we get
$$
\lambda_1(m)+\epsilon \leq \lambda_w \leq a(w,w) \big( \int_\Omega mw^2
\big)^{-1}
$$
for all $w$ as above. Since the infimum of the right-hand side with respect
to $w$ is
equal to $\lambda_1(m)$, we reach a contradiction. Q. E. D.
\begin{rem}
{\rm Similar results hold when the weight does not change sign in $\Omega$.
}
\end{rem}
\begin{rem}
{\rm
We insist on the fact that the nonuniformity of the AMP in Theorem 5.1
holds for {\it
any} weight. This should be compared with the recent result in \cite{Al-Go}
that if
$N=1$ and $m$ has compact support in $\Omega$, then the first curve in the
corresponding Fu\v cik spectrum is {\it not} asymptotic to the horizontal and
vertical lines through $(\lambda_1(m),\lambda_1(m))$ (even if $m$ does not
change
sign). It follows that the qualitative and quantitative connections between
``uniformity
of the AMP'' and ``existence of a gap at infinity in the Fu\v cik spectrum
between
the first curve and the horizontal and vertical lines through $(\lambda_1(m),
\lambda_1(m))$'', which were observed in \cite{DF-Go} when $m(x)\equiv 1$,
do not
hold any more in general in the presence of a weight. }
\end{rem}
\begin{thebibliography}{10}
\bibitem{Al-Go} M. ALIF and J.-P. GOSSEZ, On the Fu\v cik spectrum with
weight, to
appear.
\bibitem{Ar-Ca-Go} M. ARIAS, J. CAMPOS and J.-P. GOSSEZ, On the
antimaximum principle and the Fu\v cik spectrum for the Neumann
$p$-Laplacian, Diff. Int. Eqns., to appear.
\bibitem{Bi} I. BIRINDELLI, Hopf's lemma and antimaximum in general
domains, J. Diff. Eqns. 119 (1995), 450-472.
\bibitem{Br-Li} K. BROWN and S. LIN, On the existence of positive
eigenfunctions for
an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75
(1980), 112-120.
\bibitem{Cl-Pe} P. CLEMENT and L. PELETIER, An antimaximum principle
for second order elliptic operators, J. Diff. Eqns., 34 (1979), 218-229.
\bibitem{DF} D. DE FIGUEIREDO, Positive solutions of semilinear elliptic
equations,
Lect. Notes Math., Springer, 957 (1982), 34-87.
\bibitem{DF-Go} D. DE FIGUEIREDO and J.-P. GOSSEZ, On the first
curve of the Fu\v cik spectrum of an elliptic operator, Diff. Int.
Equat., 7 (1994), 1285-1302.
\bibitem{Fl-Go-Ta-DT} J. FLECKINGER, J.-P. GOSSEZ, P. TAKA\v C and
F. DE THELIN, Existence, nonexistence et principe de l'antimaximum
pour le $p$-laplacien, C. R. Ac. Sc. Paris, 321 (1995), 731-734.
\bibitem{Gi-Tr} D. GILBARG and N. TRUDINGER, Elliptic Partial
Differential Equations
of Second Order, Springer, 1983.
\bibitem{He1} P. HESS, An antimaximum principle for linear elliptic
equations with
an indefinite weight function, J. Diff. Eqns., 41 (1981), 369-374.
\bibitem{He2} P. HESS, On positive solutions of semilinear periodic-parabolic
problems, Lect. Notes Math., Springer, 1076 (1984), 101-114.
\bibitem{Se-He} S. SENN and P. HESS, On positive solutions of a linear
elliptic
eigenvalue problem with Neumann boundary conditions, Math. Ann., 258
(1982), 459-470.
\bibitem{Sw1} G. SWEERS, $L^n$ is sharp for the antimaximum principle, J.
Diff. Eqns., 134 (1997), 148-153.
\bibitem{Sw2} G. SWEERS, to appear.
\end{thebibliography} \bigskip
{\sc T. Godoy \& S. Paczka}\\
FAMAF, Univ. Nacional C\'ordoba\\
Ciudad universitaria, 5000 C\'ordoba, Argentina\\
e-mail: godoy@mate.uncor.edu \&
paczka@mate.uncor.edu \medskip
{\sc J.-P. Gossez} \\
D\'ep. Math\'ematique, C.P. 214,\\
Universit\'e Libre de Bruxelles\\
1050 Bruxelles, Belgique\\
e-mail: gossez@ulb.ac.be \medskip
\end{document}