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\AtBeginDocument{ {\noindent\small
2006 International Conference in Honor of Jacqueline Fleckinger.
\newline {\em Electronic Journal of Differential Equations},
Conference 15, 2007, pp. 137--154.
\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{9mm}}
\begin{document} \setcounter{page}{137}
\title[\hfilneg EJDE/Conf/16 \hfil A minimax formula for the principal eigenvalues]
{A minimax formula for the principal eigenvalues\\
of Dirichlet problems and its applications}
\author[T. Godoy, J.-P. Gossez, S. R. Paczka \hfil EJDE/Conf/16 \hfilneg]
{Tomas Godoy, Jean-Pierre Gossez, Sofia R. Paczka} % in alphabetical order
\address{Tomas Godoy \newline
FAMAF, Univ.~Nacional C\'ordoba,
Ciudad Universitaria,
5000 C\'ordoba, Argentina}
\email{godoy@mate.uncor.edu}
\address{Jean-Pierre Gossez \newline
D\'epartement de Math\'ematique, C.~P.~214,
Universit\'e Libre de Bruxelles,
B--1050 Bruxelles, Belgium}
\email{gossez@ulb.ac.be}
\address{Sofia Rosalia Paczka \newline
FAMAF, Univ.~Nacional C\'ordoba,
Ciudad Universitaria,
5000 C\'ordoba, Argentina}
\email{paczka@mate.uncor.edu}
\thanks{Published May 15, 2007.}
\subjclass[2000]{35J20, 35P15}
\keywords{ Nonselfadjoint elliptic problem; principal eigenvalue;
\hfill\break\indent
indefinite weight; minimax formula; weighted Sobolev spaces;
degenerate elliptic equations; \hfill\break\indent
antimaximum principle}
\dedicatory{Dedicated to Jacqueline Fleckinger on the occasion of \\
an international conference in her honor}
\begin{abstract}
A minimax formula for the principal eigenvalue of a nonselfadjoint
Dirichlet problem was established in \cite{D-V,Ho}. In this paper we
generalize this formula to the case where an indefinite weight is
present. Our proof requires less regularity and, unlike that in
\cite{D-V,Ho}, does not rely on semigroups theory nor on stochastic
differential equations. It makes use of weighted Sobolev spaces.
An application is given to the study of the uniformity of the
antimaximum principle.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
The main purpose of this paper is to
establish a variational formula of minimax type for the principal
eigenvalues of the (generally nonselfadjoint) Dirichlet problem
\begin{equation}\label{eq1.1}
\begin{gathered}
Lu = \lambda m(x) u \quad \mbox{in } \Omega, \\
u=0 \quad \mbox{on } \partial \Omega.
\end{gathered}
\end{equation}
Here $\Omega$ is a bounded domain in $\mathbb{R}^N$, $L$ is a
second order elliptic operator of the form
\begin{equation}\label{eq1.2}
Lu := -\mathop{\rm div} (A(x)\nabla u)+\langle a(x), \nabla u \rangle
+ a_0(x) u
\end{equation}
with $\langle,\rangle$ the scalar product in
$\mathbb{R}^N$, and $m(x)$ is a possibly indefinite weight.
Calculating the principal eigenvalues of a selfadjoint operator
via minimization of the Rayleigh quotient is a classical matter.
Problem \eqref{eq1.1} above is generally nonselfadjoint and this
Euler-Lagrange technique does not apply anymore. Other approaches
were introduced in \cite{D-V}, \cite{Ho}. In \cite{D-V} $m \equiv
1$ and a minimax formula was derived through the consideration of
an associated semigroup of positive operators. In \cite{Ho} the
weight is definite (i.e. $m(x) \geq \varepsilon > 0$ in $\Omega$)
and a similar minimax formula was derived by using results on
stochastic differential equations. Both \cite{D-V}, \cite{Ho}
assume $C^{\infty}$ smoothness for the coefficients of $L$, and
\cite{Ho} assume $m$ of class $C^{2}$.
The formula we obtain (cf. Theorem \ref{theo3.1} and Theorem \ref{theo3.5})
is rather similar to that in \cite{D-V}, \cite{Ho}. Our
contribution is triple. First we deal with the general case where
the weight $m$ may vanish or change sign in $\Omega$. Secondly
much less regularity on the coefficients and on the weight is
required. Finally our proof does not rely on semigroups theory nor
on stochastic differential equations.
Our proof follows the general approach initially introduced in
\cite{Ho} and further developed in \cite{Go-Go-Pa} in the case of
the Neumann-Robin problem. The main difficulty in adapting this
approach to the case of the Dirichlet problem comes from the fact
that several auxiliary equations which in the Neumann-Robin case
are uniformly elliptic now degenerate on $\partial \Omega$ (cf.
equations (\ref{eq3.2}) and (\ref{eq3.3})). A large part of the
present paper is devoted to the study of these degenerate
equations, to which the classical results of \cite{Tru} do not
apply. Our study is carried out in the context of weighted Sobolev
spaces, and Moser's iteration technique is in particular used in
that context to derive a crucial $L^\infty$ bound (cf. Lemma
\ref{lem4.7}).
The second part of this paper briefly deals with an application to
the antimaximum principle (in short AMP). This principle concerns
the problem
\begin{equation}\label{eq1.3}
Lu = \lambda m (x) u + h(x) \quad\mbox{in } \Omega, \quad
u=0 \quad \mbox{on } \partial \Omega
\end{equation}
and says roughly the following : if $\lambda^{\ast}$ denotes the
largest principal eigenvalue of \eqref{eq1.1}
, then for any $h \geq 0$, $h
\not \equiv 0$, there exists $\delta > 0$ such that for $\lambda
\in ] \lambda^{\ast},\lambda^{\ast} + \delta [$, the solution $u$
of \eqref{eq1.3} is $< 0$. This AMP was first established in
\cite{Cl-Pe} in the case where there is no weight, i.e. $m(x)
\equiv 1$ in $\Omega$. It was later extended in \cite{He} to the
case of an indefinite weight $m \in C({\bar{\Omega}})$. In this
paper we extend it further to the case of an indefinite weight $m
\in L^\infty(\Omega)$ (cf. Theorem \ref{theo5.1}). Our method of
proof differs from that in \cite{Cl-Pe}, \cite{He} and is more in
the line of the approach introduced in \cite{F-G-T-dT} to deal
with nonlinear operators. It was also proved in \cite{Cl-Pe} that
for $L=-\Delta$ and $m(x) \equiv 1$, the AMP is nonuniform, in the
sense that a $\delta > 0$ cannot be found which would be valid for
all $h$. This was derived in \cite{Cl-Pe} from considerations
involving the associated Green function. In this paper we use our
minimax formula to prove that this nonuniformity still holds in
the general case of \eqref{eq1.3}. For further recent results
involving the uniformity of the AMP, see \cite{Cl-Sw},
\cite{Go-Go-Pa}. See also \cite{Ejde} in the selfadjoint case.
The plan of the paper is the following. In section 2, which has a
preliminary character, we collect some known results on the
existence of principal eigenvalues for \eqref{eq1.1} in the
presence of an indefinite weight. Section 3 deals with the minimax
formula itself, while the study of the auxiliary degenerate
equations is postponed to section 4. Section 5 deals with the AMP
and section 6 with its nonuniformity.
\section{Principal eigenvalues}
Let us start by stating the assumptions to be imposed on the
operator $L$ and the domain $\Omega$ in \eqref{eq1.1}. $\Omega$ is
a bounded $C^{1,1}$ domain in $\mathbb{R}^N$, $N \geq 1$, and the
coefficients of $L$ satisfy: $A$ is a symmetric uniformly positive
definite $N \times N$ matrix, with $A \in C^{0,1}(\bar{\Omega})$,
$a$ and $a_0 \in L^\infty(\Omega)$. The weight $m$ in
\eqref{eq1.1} belongs to $L^\infty(\Omega)$, with $m \not \equiv
0$. These conditions will be assumed throughout the paper. More
restrictions on $\Omega$ and $a$ will be imposed later.
Our purpose in this preliminary section is to collect some known
results on the existence of principal eigenvalues of
\eqref{eq1.1}, with some indications of proofs in order to allow
later use. Standard references include \cite{He-Ka}, \cite
{HeBook}, \cite{Lo-Go}, \cite{Da}, \cite{Fl-He-dT}.
By a principal eigenvalue we mean $\lambda \in \mathbb{R}$ such
that \eqref{eq1.1} admits a solution $u \not \equiv 0$ with $u
\geq 0$. Unless otherwise stated, solutions are understood in the
strong sense, i.e. $u \in W^{2,p}(\Omega)$ for some $1 < p <
\infty$, the equation is satisfied a.e. in $\Omega$ and the
boundary condition is satisfied in the sense of traces. We will
denote by $W(\Omega)$ the intersection of all $W^{2,p}(\Omega)$
spaces for $1 < p < \infty$.
A fundamental tool is the following form of the maximum principle,
which can be derived from \cite[Theorem 9.6 and Lemma 3.4]{Gi-Tr}.
\begin{proposition}\label{prop2.1}
Assume $a_0 \geq 0$. Let $u \in W^{2,p}(\Omega)$ satisfy
$$
Lu = f \quad\text{in } \Omega, \quad u = g \quad\text{on } \partial \Omega
$$
where $p > N$, $f \geq 0$, $g \geq 0$ and $f$ or $g \not \equiv
0$. Then $u> 0$ in $\Omega$. Moreover, if $u(x_0)=0$ for some $x_0 \in
\partial \Omega$, then $\partial u/\partial \eta(x_0) < 0$ for
any exterior direction $\eta$ at $x_0$.
\end{proposition}
Another tool is the following existence, unicity and regularity
result, which follows e.g. from \cite[Theorem 2.4, 2.5]{Gr}.
\begin{proposition}\label{prop2.2}
Let $1 < p < \infty$. If $l \in \mathbb{R}$ is sufficiently
large, then the problem
\begin{equation}\label{eq2.1}
(L+l)u = f \quad\text{in } \Omega, \quad u = 0 \quad\text{on }
\partial \Omega
\end{equation}
has a unique solution $u \in W^{2,p}(\Omega)$ for any $f \in
L^p(\Omega)$. Moreover, the solution operator $S_l : f\to u$ is
continuous from $L^p(\Omega)$ into $W^{2,p}(\Omega)$. In addition,
the above holds with $l=0$ if the problem $Lu=0$ in $\Omega$,
$u=0$ on $\partial \Omega$ has only the trivial solution $u \equiv
0$. This is the case in particular if $a_0 \geq 0$.
\end{proposition}
The solution operator $S_l$ provided by Proposition \ref{prop2.2}
will be mainly looked at as an operator from $C_0^1(\bar{\Omega})$
into itself (and then denoted by $S_{lC}$). Here
$C_0^1(\bar{\Omega})$ denotes the space of the $C^1$ functions on
$\bar{\Omega}$ which vanish on $\partial \Omega$; it is endowed
with its natural ordering and norm. Note that the interior of the
positive cone $P$ in $C_0^1(\bar{\Omega})$ is nonempty and made of
those $u \in C_0^1(\bar{\Omega})$ such that $u > 0$ in $\Omega$
and $\partial u/\partial \nu < 0$ on $\partial \Omega$, where
$\nu$ denotes the unit exterior normal.
Combining the above two propositions with the Krein-Rutman theorem
for strongly positive operators (cf. e.g. \cite{Am}), one easily
gets the following
\begin{lemma}\label{lem2.3}
Assume $l$ sufficiently large. Then: (i) $S_{lC}$ is compact and
strongly positive (i.e. $f \geq 0$ with $f \not \equiv 0$ implies
$u \in$ int $P$). (ii) The spectral radius $\rho_l$ of $S_{lC}$ is
$>0$ and $\rho_l$ is an algebraically simple eigenvalue of
$S_{lC}$, having an eigenfunction $u$ in int $P$; in addition,
there is no other eigenvalue having a nonnegative eigenfunction.
(iii) For every $f \in C_0^1(\bar{\Omega})$ such that $f \geq 0$,
$f \not \equiv 0$, the equation $\rho u - S_{lC}u=f$ has exactly
one solution $u$, which belongs to int $P$, if $\rho > \rho_l$,
and has no solution $u \geq 0$ if $\rho \leq \rho_l$.
\end{lemma}
The above considerations apply in particular to the operator
$L-\lambda m$. It follows that for each $\lambda \in \mathbb{R}$
there is a unique $\mu=\mu(\lambda) \in \mathbb{R}$ such that
\begin{equation}\label{eq2.2}
Lu - \lambda m u = \mu u \quad\text{in } \Omega, \quad u=0
\quad\text{on } \partial \Omega
\end{equation}
has a solution $u=u_\lambda$ with $u \geq 0$, $u \not \equiv 0$.
Moreover this solution $u$ belongs to $W(\Omega) \cap$
int$P$,
and the space of solutions of (\ref{eq2.2}) is one dimensional.
This function $\mu : \mathbb{R} \to \mathbb{R}$ is directly
related with the principal eigenvalues of \eqref{eq1.1} since
$\lambda \in \mathbb{R}$ is a principal eigenvalue of
\eqref{eq1.1} if and only if $\mu(\lambda)=0$. Various properties
of this function are collected in the following lemma, whose proof
can for instance be adapted from that \cite[Lemma 2.5]{Go-Go-Pa}.
(Note that under further assumptions on $L$ and
$m$, the concavity of $\mu(\lambda)$ could also be derived from
Holland's formula of \cite{Ho} or from Kato's result of \cite{Ka}
on the concavity of the spectral radius).
\begin{lemma}\label{lem2.4}
(i) If $a_0 \geq 0$, then $\mu(0) > 0$. (ii) If $m^+ \not \equiv
0$, then $\mu(\lambda) \to -\infty$ as $\lambda \to +\infty$; if
$m^- \not \equiv 0$, then $\mu(\lambda) \to -\infty$ as $\lambda
\to -\infty$. (iii) $\lambda \to \mu(\lambda)$ is concave and real
analytic.
\end{lemma}
We are now in a position to state the main result of this section,
whose proof easily follows from Lemma \ref{lem2.4} and
Proposition \ref{prop2.2}.
\begin{proposition}\label{prop2.5}
Assume $a_0 \geq 0$. (i) If $m$ changes sign,
then \eqref{eq1.1} admits exactly two
principal eigenvalues, one is $>0$, the other is $< 0$. (ii) If $m
\geq 0$, $m \not \equiv 0$, then \eqref{eq1.1} admits exactly one
principal eigenvalue, which is $>0$. (iii) If $m \leq 0$, $m \not
\equiv 0$, then \eqref{eq1.1} admits exactly one principal
eigenvalue, which is $< 0$.
\end{proposition}
We now turn to the case where the condition $a_0 \geq 0$ of
Proposition \ref{prop2.5} does not hold.
\begin{proposition}\label{prop2.6}
If $m$ changes sign, then \eqref{eq1.1} may have zero, one or two principal
eigenvalues. If $m$ does not change sign, then \eqref{eq1.1} may have zero
or one principal eigenvalue.
\end{proposition}
\begin{proof} If $m$ changes sign, then there exists $l_{0}$ such
that the problem
\begin{equation}\label{eq2.3}
Lu + lu = \lambda m(x)u \quad \mbox{in} \quad \Omega, \quad u = 0
\quad \mbox{on} \quad \partial\Omega
\end{equation}
has two (resp. one, zero) principal eigenvalues for $l > l_{0}$
(resp. $l = l_{0}, l < l_{0}$). This is easily deduced from Lemma
\ref{lem2.4} since the function $\mu_{l}(\lambda)$ associated to
\eqref{eq2.3} is given by $\mu_{0}(\lambda) + l$.
Suppose now that $m$ does not change sign, say $m \geq 0$ in
$\Omega$. Then \eqref{eq1.1} has at most one principal eigenvalue. Indeed
if it had two, then by Lemma \ref{lem2.4},
$\mu_{0}(\lambda)\to -\infty$ not only as $\lambda
\to +\infty$ but also as $\lambda\to -\infty$.
This implies that $\mu_{l}(\lambda)$ has two distinct zeros for $l
\geq 0$; taking $l$ such that $a_{0}(x)+ l \geq 0 \quad \mbox{in}
\quad \Omega$, one gets a contradiction with part (ii) of
Proposition \ref{prop2.5}.
We finally give a simple example showing that \eqref{eq1.1} with $m \geq
0$ may have no principal eigenvalue. (More refined results in this
direction can be found in \cite{Lo-Go}, \cite {Da},
\cite{Fl-He-dT}). We will show that if $m \in L^{\infty}(\Omega)$,
$m \not\equiv 0$ vanishes on a ball $B \subset \Omega$, then
\begin{equation}\label{eq2.4}
-\Delta u - lu = \lambda m(x)u \quad \mbox{in } \Omega, \quad
u = 0 \quad \mbox{on } \partial\Omega
\end{equation}
has no principal eigenvalue for $l > \lambda_{1}^{B}$, where
$\lambda_{1}^{B}$ is the principal eigenvalue of $-\Delta$ on
$H_{0}^{1}(B)$. Indeed, for such a value of $l$, there exists $v \in
H_{0}^{1}(B)$ such that $\int_{B}(|\nabla v|^{2} - lv^{2})< 0$.
Using the fact that if $u \in H_{0}^{1}(\Omega)$ satisfies
$\int_{\Omega}mu^{2}= 1$, then $\int_{\Omega}m (u +
r\widetilde{v})^{2}= 1$ for any $r \in \mathbb{R}$ ($\widetilde{v}$
denotes $v$ extended by $0$ on $\Omega \backslash B$), one deduces
that
\begin{equation}\label{eq2.5}
\inf\{{\int_{\Omega}(|\nabla u|^{2}- l u^{2}): u \in
H_{0}^{1}(\Omega)\quad \mbox{and}\quad \int_{\Omega}m u^{2} = 1}\} =
-\infty.
\end{equation}
Suppose now by contradiction that (2.4) admits a principal
eigenvalue $\lambda^{\ast}$. Applying the classical Rayleigh
formula to $-\triangle u - lu + ku = (\lambda^{\ast} m (x) + k)u$
with $k$ taken $> l,$ one gets
\begin{equation}\label{eq2.6}
1= \inf \{ \frac{\int_{\Omega}(|\nabla u|^{2} - l u^{2} + k
u^{2})}{\int_{\Omega}(k + \lambda^{\ast} m)u^{2}} :u \in
H_{0}^{1}(\Omega) \quad\mbox{and}\quad \int_{\Omega}(k + \lambda^{\ast}
m)u^{2} > 0\} .
\end{equation}
Choosing $k$ larger if necessary so that $k > \lambda^{\ast}
\|m\|_\infty$, one observes that any nonzero $u \in
H_{0}^{1}(\Omega)$ satisfies the constraint in (2.6).
Consequently, by (2.6),
$$
\lambda^{\ast} \leq \int_{\Omega}(|\nabla u|^{2} - l u^{2})
$$
for all $u \in H_{0}^{1}(\Omega)$ with $\int_{\Omega}m u^{2}= 1$.
But this contradicts (2.5).
\end{proof}
\section{Minimax Formula}
The operator $L$ and the weight $m$ in this section are assumed to
satisfy the conditions indicated at the beginning of section 2, with
in addition $a \in C^{0,1} (\bar{\Omega})$ and $\Omega$ of class
$C^2$. Our purpose is to give a formula of minimax type for the
principal eigenvalues of \eqref{eq1.1}.
Let us define the distance function to the boundary $d(x) : =$
dist $(x,\partial \Omega)$ and call
\begin{align*}
D(\Omega) &:= \{u : \Omega \to \mathbb{R} \mbox{ measurable } :
\exists c_i = c_i(u) > 0 \\
&\quad \mbox{ such that } c_1d \leq u \leq c_2 d \mbox{ a.e. in } \Omega \}.
\end{align*}
Note that the positive eigenfunctions associated to the principal
eigenvalues to $D(\Omega)$. Let us also define, for $\sigma \in
\mathbb{R}$, the weighted Sobolev space
$$
H^1(\Omega,d^\sigma) :=\{u \in H^1_{\rm loc} (\Omega) :
\int_\Omega d^\sigma(u^2+|\nabla u|^2) < \infty \},
$$
which is endowed with the norm given by the square root of the
above integral.
\begin{theorem}\label{theo3.1}
Suppose $a_{0} \geq 0, m^+ \not \equiv 0$ and let $\lambda^*$ be
the largest principal eigenvalue of \eqref{eq1.1}
(cf. Proposition \ref{prop2.5}). Then
\begin{equation}\label{eq3.1}
\lambda^* =\inf_{u \in U} \sup_{v \in H^1(\Omega,d^2)}
\frac{\Lambda(u)-Q_u (v)}{\int_\Omega mu^2}
\end{equation}
where
\begin{gather*}
U : = \{u \in H^1(\Omega) \cap D(\Omega) : \int_\Omega mu^2 >0\},\\
\Lambda(u) := \int_\Omega (\langle A \nabla u, \nabla
u\rangle + \langle a, \nabla u \rangle u + a_0 u^2 ),\\
Q_u (v) := \int_\Omega u^2 ( \langle A \nabla v, \nabla
v\rangle - \langle a, \nabla v\rangle).
\end{gather*}
Moreover, the infimum and the supremum in \eqref{eq3.1} are
achieved.
\end{theorem}
Note that the smallest principal eigenvalue can be handled via
Theorem \ref{theo3.1}, after changing $m$ into $-m$.
The following two lemmas will be used in the proof of Theorem
\ref{theo3.1}. They concern auxiliary equations which degenerate
on $\partial \Omega$ and which will be considered in a suitable
weak sense. The proof of these two lemmas will be given in section
4. The first one deals with $Q_u$. The second one introduces a
function $G$ whose role is the following : in the selfadjoint
case, the minimum of the Rayleigh quotient is achieved at an
eigenfunction; it will turn out that in the present nonselfadjoint
situation, the infimum in \eqref{eq3.1} is achieved for $u$ equal
to an eigenfunction multiplied by $\sqrt{G}$.
\begin{lemma}\label{lem3.2}
For any $u \in D(\Omega)$, the infimum of $Q_u$ on $H^1(\Omega,
d^2)$ is achieved at some $W_u$. This $W_u$ is unique up to an
additive constant and can be characterized as the solution of
\begin{equation}\label{eq3.2}
\begin{gathered}
W_u \in H^1(\Omega,d^2),\\
\int_\Omega u^2 \langle 2A \nabla W_u - a, \nabla \varphi
\rangle = 0 \quad \forall \varphi \in H^1(\Omega,d^2).
\end{gathered}
\end{equation}
Moreover
$$
Q_u (W_u) = -\int_\Omega u^2 \langle A \nabla W_u, \nabla W_u
\rangle = - \frac{1}{2} \int_\Omega u^2 \langle a, \nabla W_u
\rangle.
$$
\end{lemma}
\begin{lemma}\label{lem3.3}
Let $u \in D(\Omega) \cap C^1(\bar{\Omega})$. Then the problem
\begin{equation}\label{eq3.3}
\begin{gathered}
G \in H^1(\Omega,d^2),\\
\int_\Omega u^2 \langle A \nabla G + a G, \nabla \varphi
\rangle = 0 \quad \forall \varphi \in H^1(\Omega,d^2)
\end{gathered}
\end{equation}
has a non trivial solution $G$, which is unique up to a
multiplicative constant and satisfies
\begin{equation}\label{eq3.4}
c_1 \leq G \leq c_2 \quad \text{ a.e. in } \Omega
\end{equation}
for some constants $c_i > 0$.
\end{lemma}
Note that by Lemma \ref{lem3.2}, formula \eqref{eq3.1} can be stated
equivalently as
\begin{equation}\label{eq3.5}
\lambda^* = \inf_{u \in U} \frac{\Lambda(u)-Q_u(W_u)}{\int_\Omega
mu^2}.
\end{equation}
Once these two lemmas are accepted, the proof of \eqref{eq3.1} can
be carried out by following the same general lines as in
\cite{Go-Go-Pa}, and we will only indicate below the main
differences. In this adaptation of \cite{Go-Go-Pa}, special care
must be taken to the boundary behaviour of the functions involved,
and the introduction of $D(\Omega)$, $H^1(\Omega,d^2)$ plays in this
respect a central role.
\begin{proof}[Proof of Theorem \ref{theo3.1}]
Let $u^*$ be an
eigenfunction associated to $\lambda^{\ast}$ and satisfying $u^* \in
W(\Omega) \cap \quad\text{int } P$. We will first prove that inequality
$\leq$ holds in (\ref{eq3.5}), i.e.
\begin{equation}\label{eq3.6}
\lambda^* \int_\Omega mu^2 \leq \Lambda(u) - Q_u(W_u)
\end{equation}
for all $u \in U$.
Call $v^*:= -\log u^*$. Then $v^* \in W_{\rm loc}^{2,p}(\Omega)$
for all $1 < p < \infty$ and satisfies
\begin{equation}\label{eq3.7}
-\mathop{\rm div} \; (A \nabla v^*) = -\langle A \nabla v^*, \nabla
v^* \rangle - \langle a, \nabla v^* \rangle + a_0 - \lambda^*
m \quad\text{in } \Omega .
\end{equation}
Note that, unlike (3.10) from \cite{Go-Go-Pa}, no boundary
condition appears here since $v^*=+\infty$ on $\partial \Omega$.
Now one takes $u \in U$, multiply both sides of equation
(\ref{eq3.7})
by $u^2$, integrate and use as in formula (\ref{eq3.11}) of \cite{Go-Go-Pa} an
argument based on the idea of completing a square to obtain
\begin{equation}\label{eq3.8}
\int_\Omega \langle A \nabla u^2 - u^2 w_u, \nabla v^* \rangle
+ \lambda^* \int_\Omega mu^2 \leq \frac{1}{4} \int_\Omega u^2
\langle a+w_u, A^{-1}(a+w_u)\rangle + \int_\Omega a_0 u^2
\end{equation}
where $w_u := -a + 2A ((\nabla u/u) + \nabla W_u)$. In this
process one should verify that all the integrals involved do make
sense in the usual $L^1(\Omega)$ sense, which is easy by using the
regularity of $u^*$ and the fact that $u^* \in D(\Omega)$, $u\in
D(\Omega) \cap H^1(\Omega)$ and $W_u \in H^1(\Omega, d^2)$. One
should also justify the use of the divergence theorem to write
\begin{equation}\label{eq3.9}
\int_\Omega [\mathop{\rm div}\, (A \nabla v^*)]u^2 = - \int_\Omega
\langle A \nabla v^*, \nabla u^2\rangle.
\end{equation}
This latter formula follows by applying Lemma \ref{lem3.4} below
to the vector field $V=A(\nabla v^*)u^2$.
Once (\ref{eq3.8}) is obtained, the calculation on page 96 from
\cite{Go-Go-Pa} can be pursued without any change to derive
(\ref{eq3.6}) above. The only point to be observed at this stage
is the (easily verified) fact that $\log u \in H^1(\Omega,d^2)$,
which allows the use of equation (\ref{eq3.2}) for $W_u$ with
$\varphi = \log u$ as testing function.
We will now show that if we put $\tilde{u} := u^* \sqrt{G^*}$,
where $G^*$ is a function provided by Lemma \ref{lem3.3} for
$u=u^*$, then $\tilde{u} \in U$ and equality holds in
(\ref{eq3.6}). This will conclude the proof of Theorem
\ref{theo3.1}.
One first observes that $\tilde{u} \in H^1(\Omega)\cap D(\Omega)$
and then argue as on page 97 from \cite{Go-Go-Pa}, multiplying
both sides of equation (\ref{eq3.7}) by $\tilde{u}^2$ and
integrating to reach now
\begin{equation}\label{eq3.10}
\int_\Omega \langle A \nabla \tilde{u}^2 - \tilde{u}^2 \eta,
\nabla v^* \rangle + \lambda^* \int_\Omega m\tilde{u}^2
=\frac{1}{4} \int_\Omega \tilde{u}^2 [\langle a + \eta,
A^{-1}(a+\eta)+a_0]
\end{equation}
where $\eta := -a-2A \nabla v^*$. The rest of the calculation on
page 97 from \cite{Go-Go-Pa} can be pursued without any change. It
uses in particular the fact that $W_{\tilde{u}}=-(\log G^*)/2$ up to
an additive constant, which follows from (\ref{eq3.2}) and
(\ref{eq3.3}) above. Proceeding in this way, one reaches equality in
(\ref{eq3.6}) for $u=\tilde{u}$.
It remains to see that $\tilde{u} \in U$, i.e. that
${\int_\Omega m \tilde{u}^2 > 0}$. For this purpose one
deduces as in formula \eqref{eq3.16} from \cite{Go-Go-Pa} that
$$
\lambda^* \int_\Omega m\tilde{u}^2 = \int_\Omega \tilde{u}^2
[\langle A \nabla v^*, \nabla v^* \rangle + a_0 ],
$$
and the conclusion follows since $\lambda^* > 0, \quad v^*$ is not a
constant and $a_{0} \geq 0$.
\end{proof}
\begin{lemma}\label{lem3.4}
Let $\Omega$ be a bounded $C^2$ domain in $\mathbb{R}^N$. Let $V
: \Omega \to \mathbb{R}^N$ be a vector field in
$L^\infty(\Omega)$ such that $\mathop{\rm div}V \in L^1(\Omega)$ and
$\|V\|_{L^\infty(\Gamma_\epsilon)}\to 0$ as $\epsilon \to 0$,
where
$$
\Gamma_\epsilon := \{ x \in \bar{\Omega} : d(x) < \epsilon\}.
$$
Then ${\int_\Omega \mathop{\rm div} V = 0}$.
\end{lemma}
The proof of the above lemma is an easy adaptation of the proof in
\cite[Lemma A.1]{Cu-Ta}.
We now turn to the case where the condition $a_{0} \geq 0$ of
Theorem \ref{theo3.1} does not hold.
\begin{theorem}\label{theo3.5}
Assume $m^{+}\not \equiv 0$. Assume also the existence of a
principal eigenvalue for \eqref{eq1.1} and let $\lambda^{\ast}$ be the
largest of these principal eigenvalues (cf. Proposition \ref{prop2.6} ).
Then formula \eqref{eq3.1} holds for $\lambda^{\ast}$ .
\end{theorem}
Note that the smallest principal eigenvalue can be handled by
Theorem \ref{theo3.5}, after changing $m$ into $-m$.
\begin{proof}[Proof of Theorem \ref{theo3.5}]
Applying formula \eqref{eq3.1} to
$Lu - \lambda m u +lu = (\mu(\lambda)+ l)u$ with $l$ sufficiently
large, one deduces that
\begin{equation}\label{eq3.11}
\mu(\lambda)= \inf_{u \in H^{1}(\Omega)\bigcap D(\Omega)} \quad
\frac{\Lambda(u) - \lambda \int_{\Omega} m u^{2} - \inf
Q_{u}}{\int_{\Omega} u^{2}},
\end{equation}
where here and below $\inf \hspace{1mm} Q_{u}$ denotes $\inf
\hspace{1mm} \{{{Q_{u} (v): v \in H^{1}(\Omega, d^{2})}}\}$.
Consequently $\mu(\lambda)$ is $\geq 0$ at a given $\lambda$ if and
only if the following three conditions hold:
\begin{gather}\label{eq3.12}
\lambda \leq \Lambda(u) - \inf Q_{u} \quad \mbox{for all} \quad u \in
D (\Omega)\cap H^{1}(\Omega) \quad \mbox{with} \int_{\Omega}m u^{2}
= 1, \\
\label{eq3.13}
\lambda \geq - \Lambda(u) + \inf Q_{u} \quad \mbox{for all} \quad u
\in D (\Omega)\cap H^{1}(\Omega) \quad \mbox{with} \int_{\Omega}m
u^{2} = -1, \\
\label{eq3.14}
0 \leq \Lambda(u) - \inf Q_{u} \quad \mbox{for all} \quad u \in D
(\Omega)\cap H^{1}(\Omega) \quad \mbox{with} \int_{\Omega}m u^{2} =
0.
\end{gather}
Note that the class of u's in \eqref{eq3.13} and \eqref{eq3.14} may be empty.
\noindent\textbf{Claim.} If \eqref{eq3.12} holds for some $\lambda$, then
\eqref{eq3.14} also holds.
\begin{proof}[Proof of the claim]
Let $u \in D(\Omega)\cap
H^{1}(\Omega)\quad \mbox{with}\quad \int_{\Omega} m u^{2} = 0$. Take
$\psi \in C_{c}^{\infty}(\Omega)$ such that $\int_{\Omega} m (u +
\varepsilon\psi)^{2} > 0$ for $\varepsilon > 0$ sufficiently small
and call $u_{\varepsilon} = u + \varepsilon\psi$. Condition \eqref{eq3.12}
gives
$$
\lambda \int_{\Omega} m u_{\varepsilon}^{2} \leq \Lambda
(u_{\varepsilon})- \inf {Q_{u}}_{\varepsilon}
$$
and so, since $u_{\varepsilon} \to u \quad \mbox{in} \quad
H^{1}(\Omega)$ as $\varepsilon \to 0$, the conclusion \eqref{eq3.14}
will follow if we show that
\begin{equation}\label{eq3.15}
\inf Q_{u_{\varepsilon}} \to \inf Q_{u} \quad \mbox{as }
\varepsilon \to 0.
\end{equation}
To prove \eqref{eq3.15} fix a ball $\bar{B} \subset \Omega$ and recall that
by Lemma \ref{lem3.2},
\begin{equation}\label{eq3.16}
\inf Q_{u_{\varepsilon}} = Q_{u_{\varepsilon}}
(W_{u_{\varepsilon}})= -\int_{\Omega} u_{\varepsilon}^{2} \langle A
\nabla W_{u_{\varepsilon}}, \nabla W_{u_{\varepsilon}} \rangle =
-\frac{1}{2} \int_{\Omega} u_{\varepsilon}^{2} \langle a, \nabla
W_{u_{\varepsilon}} \rangle
\end{equation}
for a unique $W_{u_{\varepsilon}} \in H_{B}^{1} (\Omega, d^{2}),$
where this later space is defined below in Lemma \ref{lem4.2}. Using
\eqref{eq3.16}, the ellipticity of $L$ and the fact that $c_{1} d \leq
u_{\varepsilon} \leq c_{2} d$ for some positive constants $c_{1},
c_{2}$ and all $\varepsilon > 0$ sufficiently small, one gets
$$
\int_{\Omega}d^{2} |\nabla W_{u_{\varepsilon}}|^{2} \leq c_{3}
\int_{\Omega} d^{2} \langle a, \nabla W_{u_{\varepsilon}} \rangle \leq c_{4}
(\int_{\Omega} d^{2} |\nabla
W_{u_{\varepsilon}}|^{2})^{\frac{1}{2}}
$$
for some other constants $c_{3}, c_{4}$. This implies that $\nabla
W_{u_{\varepsilon}}$ remains bounded in $L^{2}(\Omega, d^{2})$,
and consequently, by Lemma \ref{lem4.2} below,
$W_{u_{\varepsilon}}$ remains bounded in the space
$H_{B}^{1} (\Omega, d^{2})$. It follows that for some subsequence,
$W_{u_{\varepsilon}} \to W$ weakly in $H_{B}^{1}(\Omega,
d^{2})$. Going to the limit in equation \eqref{eq3.2} for
$W_{u_{\varepsilon}}$ and using the fact that
$(\frac{u_{\varepsilon}}{d})^{2} \to (\frac{u}{d})^{2}$ in
$L^{2}(\Omega, d^{2})$, one then sees that $W = W_{u}$. Finally
one deduces \eqref{eq3.15}
from the last equality in \eqref{eq3.16}. This completes the proof of the
claim.
\end{proof}
Recall that by Lemma \ref{lem2.4}, the existence of a principal
eigenvalue is equivalent to the existence of $\lambda$ with $\mu
(\lambda) \geq 0$. It then follows from \eqref{eq3.12}, \eqref{eq3.13}
and \eqref{eq3.14},
using the above claim and Lemma \ref{lem2.4}, that \{${{\lambda
\in \mathbb{R}: \mu (\lambda) \geq 0}}$\} is a nonempty closed
interval with left and right extremities given respectively by
\begin{gather*}
\sup \{{{- \Lambda(u) + \inf Q_{u}: u \in D (\Omega)\cap
H^{1}(\Omega)\quad \mbox{with} \int_{\Omega}m u^{2}= -1}}\},
\\
\inf \{{{\Lambda(u) - \inf Q_{u}: u \in D (\Omega)\cap
H^{1}(\Omega)\quad \mbox{with} \int_{\Omega}m u^{2} = 1}}\},
\end{gather*}
where the above supremum is $-\infty$ in case $m \geq 0 \quad
\mbox{in} \quad \Omega$; moreover the largest principal eigenvalue
$\lambda^{\ast}$ is the right extremity of this interval, i.e. the
infimum above. This is exactly saying that formula \eqref{eq3.1} holds for
$\lambda^{\ast}$.
\end{proof}
\begin{remark} \label{rem3.6} \rm
In the context of Theorem \ref{theo3.5}, it is not clear whether the infimum
in \eqref{eq3.1} is achieved. This is however so when $m(x) \geq
\varepsilon > 0 $ since then, by writing \eqref{eq1.1} as $Lu + lmu =
(\lambda + l) m u$, one can reduce to Theorem \ref{theo3.1}.
\end{remark}
\begin{remark}\rm \label{rem3.7}
The proof of Theorem \ref{theo3.5} shows that by using Lemmas \ref{lem2.4}
and \ref{lem3.2}, formula \eqref{eq3.1} for a problem with weight can be
deduced from formula \eqref{eq3.1} for a problem without weight.
\end{remark}
\begin{remark}\rm \label{rem3.8}
Formula \eqref{eq3.1} in the presence of an indefinite weight was
considered recently in \cite{Be} in the particular case where
$a_{0} = div \hspace{1mm} a$. Beside $C^{\infty}$ smoothness of
the coefficients and of the weight, \cite{Be} requires an extra
hypothesis on the principal eigenvalue $\lambda^{\ast}$, namely
$\int_{\Omega}m (u^{\ast})^{2} G^{\ast} > 0$. Theorem \ref{theo3.5} shows
that this extra hypothesis is not needed for formula \eqref{eq3.1} to
hold. The proof in \cite{Be} relies as in \cite{Ho} on stochastic
differential equations.
\end{remark}
\begin{remark}\label{rem3.9} \rm
When $A^{-1}a$ in \eqref{eq1.2} is a gradient, then
\eqref{eq3.1} reduces to a formula of Rayleigh quotient type.
Indeed, if $-A^{-1}a=\nabla \alpha$, then \eqref{eq1.1} can be
rewritten as
$$
\tilde{L}u := - \mathop{\rm div} \, (\tilde{A}(x)\nabla u) +
\tilde{a}_0(x)u = \lambda \tilde{m} (x) u \quad\text{in } \Omega, \;\;
u=0 \quad\text{on } \partial \Omega,
$$
where $\tilde{A} = e^\alpha A$, $\tilde{a}_0 = e^\alpha a_0$ and
$\tilde{m} = e^\alpha m$. So by the usual Rayleigh formula,
\begin{equation}\label{eq3.17}
\lambda^* = \min \{ \int_\Omega (\langle \tilde{A} \nabla u,
\nabla u \rangle + \tilde{a}_0 u^2) : u \in H_0^1 (\Omega)
\mbox{ and } \int_\Omega \tilde{m} u^2 = 1\}.
\end{equation}
Since the minimum in (\ref{eq3.17}) is achieved at one $u$ which
belongs to $D(\Omega)$, one can limit oneself in (\ref{eq3.17}) to
taking $u$ in $H_0^1(\Omega)\cap D(\Omega)$; moreover, writing $u$
as $e^{-\alpha/2}w$, (\ref{eq3.17}) becomes
\begin{equation}\label{eq3.18}
\begin{aligned}
\lambda^* = \min \{& \int_\Omega (\langle A \nabla w, \nabla w
\rangle + \langle a, \nabla w \rangle w + a_0 w^2 + \frac{1}{4}
\langle a, A^{-1}a \rangle w^2) :\\
&w \in H_0^1 (\Omega)\cap
D(\Omega) \mbox{ and } \int_\Omega m w^2 = 1\}.
\end{aligned}
\end{equation}
But, by completing a square,
\begin{align*}
&\max \big\{-Q_w(v) : v \in H^1(\Omega,d^2)\}\\
&= \max \{-\int_\Omega w^2 \langle A(\nabla v - \frac{1}{2}
A^{-1} a), \nabla v - \frac{1}{2} A^{-1} a \rangle+ \frac{1}{4}
\int_\Omega w^2 \langle a, A^{-1}a\rangle \\
&\qquad : v \in H^1(\Omega,d^2)\big\}\\
&= \frac{1}{4} \int_\Omega w^2 \langle a, A^{-1} a \rangle,
\end{align*}
which implies that (\ref{eq3.18}) reduces to the minimax formula
\eqref{eq3.1}.
\end{remark}
\begin{remark} \label{rem3.10} \rm
Minimax formulas of a different nature, in the line of the
classical formula of Barta, can be found in \cite{Bere}.
\end{remark}
\section{Two degenerate elliptic equations}
In this section we give a proof of Lemmas \ref{lem3.2} and
\ref{lem3.3}. The assumptions on $L$ and $\Omega$ are the same as
in section 3. Beside the weighted Sobolev space
$H^1(\Omega,d^\sigma)$, we will use for $\sigma \in \mathbb{R}$
the weighted Lebesgue space
$$
L^p(\Omega,d^\sigma) := \{ u \mbox{ measurable on } \Omega :
\int_\Omega d^\sigma |u|^p < \infty\},
$$
which is endowed with the norm given by the $p$-th root of the
above integral.
The following three lemmas concern these spaces. Lemma
\ref{lem4.1} is a particular case of imbedding results in
\cite[Theorems 2.4 and 2.5]{Ne}. See also \cite[Theorem 8.2]{Ku}
and \cite[Theorem 19.5]{Op-Ku}. Lemma \ref{lem4.2} is a
Poincar\'e type inequality, which follows easily from Lemma
\ref{lem4.1}. Lemma \ref{lem4.3} is a particular case of another
imbedding result in \cite[Theorem 19.9]{Op-Ku}.
\begin{lemma}\label{lem4.1}
$H^1(\Omega,d^2)$ is continuously imbedded into $L^2(\Omega)$, and
compactly imbedded into $L^2(\Omega,d^\epsilon)$ for any
$\epsilon > 0$.
\end{lemma}
\begin{lemma}\label{lem4.2}
Fix a ball $B$ such that $\bar{B} \subset \Omega$ and let
$\epsilon> 0$. Then there exists $c=c(\Omega, B, \epsilon)$ such that
$$
\| u \|_{L^2(\Omega,d^\epsilon)} \leq c \|\nabla
u\|_{L^2(\Omega,d^2)} \quad \forall u \in H_B^1(\Omega,d^2),
$$
where $H_B^1(\Omega,d^2)$ denotes the subspace of
$H^1(\Omega,d^2)$ made of those $u$ such that $\int_B u=0$.
\end{lemma}
\begin{proof} It clearly suffices to consider the case
where $\epsilon \leq 2$. Assume by contradiction that for each
$k=1,2,\ldots$ there exists $u_k \in H_B^1(\Omega,d^2)$ such that
$$
\| u_k \|_{L^2 (\Omega,d^\epsilon)} > k \| \nabla
u_k\|_{L^2(\Omega,d^2)}.
$$
One can assume $\|u_k\|_{H^1(\Omega,d^2)}=1$ and so, for a
subsequence, $u_k$ converges weakly to some $u$ in
$H_B^1(\Omega,d^2)$. By Lemma \ref{lem4.1}, $u_k \to u$ in
$L^2(\Omega, d^\epsilon)$, and by the inequality above, $\nabla
u_k \to 0$ in $L^2(\Omega,d^2)$. So $u_k \to u$ in $H^1_B(\Omega,
d^2)$, $\|u\|_{H_B^1(\Omega,d^2)} = 1$, and $u \equiv$ constant. But
this is impossible, since $\int_B u = 0$.
\end{proof}
\begin{lemma}\label{lem4.3}
$H^1(\Omega, d^2)$ is continuously imbedded into $L^p(\Omega,d^2)$
for $p \leq 2 +4/N$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem3.2}]
Let $u \in D(\Omega)$
and fix $B \subset \Omega$ as in Lemma \ref{lem4.2}. It is clear
that $Q_u$ is continuous on $H^1(\Omega,d^2)$; moreover, by
ellipticity, one has
$$
Q_u(v) \geq c_1 \| \nabla v\|^2_{L^2(\Omega,d^2)} - c_2\| \nabla
v\|_{L^2(\Omega,d^2)}
$$
for some constants $c_1 > 0$ and $c_2 \geq 0$ and all $v \in
H^1(\Omega,d^2)$. Combining with Lemma \ref{lem4.2} yields that
$Q_u$ is coercive on $H_B^1(\Omega,d^2)$. It follows that the
strictly convex functional $Q_u$ achieves its minimum on
$H_B^1(\Omega,d^2)$ at a unique $W_u \in H_B^1(\Omega,d^2)$.
Moreover, this $W_u$ is characterized by
\begin{equation}\label{eq4.1}
\int_\Omega u^2 \langle 2A \nabla W_u - a, \nabla
\varphi\rangle =0 \quad \forall \varphi \in H_B^1(\Omega,d^2).
\end{equation}
Clearly the minimum of $Q_u$ on $H_B^1(\Omega,d^2)$ coincides with
its minimum on $H^1(\Omega,d^2)$; moreover (\ref{eq4.1}) holds for
all $\varphi \in H_B^1(\Omega,d^2)$ if and only if it holds for
all $\varphi \in H^1(\Omega,d^2)$. It follows that $W_u$ is
characterized up to an additive constant as the solution of
(\ref{eq3.2}). Finally taking $\varphi=W_u$ in (\ref{eq3.2}), one
deduces the formulas for $Q_u(W_u)$.
\end{proof}
The proof of Lemma \ref{lem3.3} will be more involved. Writing the
equation in (\ref{eq3.3}) as
\begin{equation}\label{eq4.2}
{\mathcal L} G := -\mathop{\rm div} (u^2(A \nabla G+aG))=0,
\end{equation}
we will first show that for $l$ sufficiently large, some sort of
inverse of $({\mathcal L}+lu^2)$ is well-defined and compact (cf.
Lemma \ref{lem4.4}), and enjoys a rather strong positivity
property (cf. Lemma \ref{lem4.5}). This allows the application of
a version of the Krein-Rutman theorem for irreducible operators,
which yields a positive solution $G$ of (\ref{eq3.3}) (cf. Lemma
\ref{lem4.6}). The remaining parts of the proof of Lemma
\ref{lem3.3} consist in proving that $G$ belongs to
$L^\infty(\Omega)$ (cf. Lemma \ref{lem4.7}) and is bounded away
from zero (cf. Lemma \ref{lem4.8}).
\begin{lemma}\label{lem4.4}
Let $u \in D(\Omega)$. Then for $l$ sufficiently large, the problem
\begin{equation}\label{eq4.3}
\left\{ \begin{array}{l} g \in H^1(\Omega,d^2),\\
{\int_\Omega u^2 \langle A \nabla g + ag, \nabla
\varphi \rangle + \int_\Omega lu^2 g \varphi = \int_\Omega u^2
f\varphi}\quad
\forall \varphi \in H^1(\Omega, d^2)
\end{array} \right.
\end{equation}
has for each $f \in L^2(\Omega,d^2)$ a unique solution $g$.
Moreover the solution operator $T_l : f \to g$ is continuous from
$L^2(\Omega,d^2)$ into $H^1(\Omega,d^2)$, and compact from
$L^2(\Omega, d^2)$ into itself.
\end{lemma}
\begin{proof} The left-hand side of (\ref{eq4.3})
defines a bilinear form $B_l(g,\varphi)$ which is clearly
continuous on $H^1(\Omega, d^2)$. It is also coercive for $l$
sufficiently large. Indeed, using the inequality $2rs \leq
(\varepsilon r)^2 +(s/\varepsilon)^2$, one easily obtains, for $l$
sufficiently large,
\begin{equation}\label{eq4.4}
B_l(\varphi, \varphi) \geq c \|\varphi\|^2_{H^1(\Omega,d^2)}
\end{equation}
for some constant $c > 0$ and all $\varphi \in H^1(\Omega, d^2)$.
The right-hand side of (\ref{eq4.3}) defines for $f \in
L^2(\Omega, d^2)$ a continuous linear form on $H^1(\Omega,d^2)$.
It thus follows from the Lax-Milgram lemma that (\ref{eq4.3}) has
a unique solution $g$, with moreover the continuous dependance of
$g \in H^1(\Omega,d^2)$ with respect to $f \in L^2(\Omega,d^2)$.
Finally the compactness of the solution operator $T_l$ in
$L^2(\Omega,d^2)$ follows from Lemma \ref{lem4.1}.
\end{proof}
\begin{lemma}\label{lem4.5}
Let $u \in D (\Omega) \cap C^1(\bar{\Omega})$. Then for $l$
sufficiently large, the solution operator $T_l$ of Lemma
\ref{lem4.4} enjoys the following positivity property : if $f \in
L^2 (\Omega, d^2)$ is $\geq 0$ and $\not \equiv 0$, then for any
$\Omega' \subset\subset \Omega$,
\begin{equation}\label{eq4.5}
\mathop{\rm ess\,inf}_{x \in \Omega'} (T_l f)(x) > 0.
\end{equation}
\end{lemma}
\begin{proof} Let $f$ be as in the statement of the lemma
and call $g=T_l f$. Taking $-g^-$ as testing function in
(\ref{eq4.3}), one obtains $B_l(g^-, g^-) \leq 0$ and
consequently, by \eqref{eq4.4}, $g$ is $\geq 0$, with clearly $g
\not \equiv 0$. It remains to prove (\ref{eq4.5}).
To do so, we will first consider the particular case where the
vector field $a$ in \eqref{eq1.2} satisfies
\begin{equation}\label{eq4.6}
\langle a,\nu\rangle > 0 \quad\text{on } \partial \Omega.
\end{equation}
Since $u \in D(\Omega) \cap C^1(\bar{\Omega})$, $\nabla u$ on
$\partial \Omega$ is a strictly negative multiple of $\nu$ and
consequently, by continuity, (\ref{eq4.6}) implies
\begin{equation}\label{eq4.7}
\langle a, \nabla u \rangle < 0 \quad\text{on } \Gamma_\varepsilon
\end{equation}
for some $\varepsilon = \varepsilon(a,u) > 0$, where
$\Gamma_\varepsilon$ was defined in Lemma \ref{lem3.4}. Consider now
the zero order coefficient of ${\mathcal L} + lu^2$, where
${\mathcal L}$ is defined in (\ref{eq4.2}). It is equal to $u(-2
\langle a, \nabla u \rangle - (\mathop{\rm div} a)u+lu)$ and so, using
(\ref{eq4.7}) and the fact that $a \in C^{0,1}(\bar{\Omega})$, one
easily sees that taking $l$ larger if necessary (depending on $u$
and $a$), this coefficient can be made $\geq 0$ on $\Omega$. Since
the solution $g$ of (\ref{eq4.3}) belongs to $H_{\rm loc}^1(\Omega)$
and is a weak solution of $({\mathcal L}+lu^2)g=fu^2$ in $\Omega$,
the strong maximum principle can be applied on any $\Omega'' \subset
\subset \Omega$ (cf. \cite[Theorem 8.19]{Gi-Tr}), which yields the
conclusion (\ref{eq4.5}).
Let us now consider the general case where (\ref{eq4.6}) possibly
does not hold. Let us write $g$ as $hw$, where $w$ is a (fixed)
function with the following properties~:
\begin{equation}\label{eq4.8}
w \in C^{1,1}(\bar{\Omega}), \; w > 0 \quad\text{on }
\bar{\Omega}, \quad \langle A \frac{\nabla w}{w}+a, \nu\rangle > 0
\quad\text{on } \partial \Omega.
\end{equation}
The existence of such a function $w$ will be shown later. Clearly
$h=g/w \in H_{\rm loc}^1 (\Omega)$ and is a weak solution of
\begin{equation}\label{eq4.9}
-\mathop{\rm div} \left[ (u^2w) (A\nabla h +(A \frac{\nabla
w}{w}+a)h)\right] + l(u^2w)h = \frac{f}{w} (u^2w) \quad\text{in }
\Omega.
\end{equation}
Equation (\ref{eq4.9}) is of the same type as $({\mathcal
L}+lu^2)g=fu^2$ : $u$ is replaced by $u\sqrt{w}$ (which still
belongs to $D(\Omega) \cap C^1(\bar{\Omega})$), $a$ is replaced by
$A (\nabla w/w)+a$ (which still belongs to $C^{0,1}(\bar{\Omega})$
but now satisfies (\ref{eq4.6})), and $f$ is replaced by $f/w$
(which still belongs to $L^2(\Omega,d^2)$). It follows that the
preceding argument can be repeated for (\ref{eq4.9}), which yields
that
$$
\mathop{\rm ess\,inf}_{x \in \Omega'} h(x) > 0 \mbox{ for any } \Omega'
\subset \subset \Omega.
$$
The conclusion (\ref{eq4.5}) for $g=hw$ then follows.
It remains to show the existence of a function $w$ satisfying
(\ref{eq4.8}). Putting $w=e^v$, it suffices to construct $v \in
C^{1,1}(\bar{\Omega})$ such that
\begin{equation}\label{eq4.10}
\langle A \nabla v + a, \nu \rangle > 0 \quad\text{on } \partial
\Omega.
\end{equation}
By the regularity of $\Omega$, any point in $\partial \Omega$
belongs to an open set $U$ such that there exists a $C^{1,1}$
diffeomorphism $X$ from $U$ onto the unit ball $B \subset \mathbb{R}^N$ with the properties that $B \cap \{x_N > 0\}$ corresponds to
$\Omega \cap U$ and $B \cap \{x_N = 0\}$ corresponds to $\partial
\Omega \cap U$. We take an open covering $\{V^j : j = 1, \ldots,
m\}$ of $\partial \Omega$ such that $V^j \subset \subset U^j$ with
$(U^j, X^j)$ as $(U,X)$ above. We also take functions $\Psi^j \in
C^{1,1}(\mathbb{R}^N)$ such that supp $\psi^j \subset U^ j,
\psi^j \equiv 1$ on $V^j$ and $0 \leq \Psi^j \leq 1$. Define for
$P \in \bar{\Omega}$
$$
v(P) = r \sum_j \Psi^j(P) X_N^j(P)
$$
where $X_N^j$ is the $N^{\rm th}$ component of $X^j$ and $r$ is a
constant to be chosen later. Clearly $v \in C^{1,1}(\mathbb{R}^N)$, and for $P \in \partial \Omega$,
\begin{equation}\label{eq4.11}
\nabla v(P) = r \sum_j \Psi^j(P)c^j(P)\nu(P)
\end{equation}
since $\nabla X_N^j(P)=c^j(P)\nu(P)$ where $\nu(P)$ is the
exterior normal at $P$ and $c^j(P) < 0$. Calling $rf(P)$ the
coefficient of $\nu(P)$ in the right-hand side of (\ref{eq4.11}),
one has $f \in C^{0,1}(\partial \Omega)$ and $f < 0$ on $\partial
\Omega$ (since the $V^j$'s cover $\partial \Omega$). One also has
$$
\langle A \nabla v + a,\nu \rangle = rf \langle A \nu, \nu \rangle
+ \langle a, \nu \rangle,
$$
which is $>0$ on $\partial \Omega$ if the constant $r$ is chosen
sufficiently large $( < 0)$. Inequality (\ref{eq4.10}) thus
follows.
\end{proof}
\begin{lemma}\label{lem4.6}
Let $u \in D(\Omega) \cap C^1(\bar{\Omega})$. Then problem
(\ref{eq3.3}) has a solution $G$, which is unique up to a
multiplicative constant and which satisfies
\begin{equation}\label{eq4.12}
\mathop{\rm ess\,inf}_{x \in \Omega'}G(x) > 0
\end{equation}
for any $\Omega' \subset \subset \Omega$.
\end{lemma}
\begin{proof} We recall that in the context of a Lebesgue
space $L^p(E,d\mu)$ with $1 \leq p < \infty$, the irreducibility
of a positive operator $T$ can be characterized by the property
that $E$ does not admit any nontrivial subset $F$ which is
invariant for $T$ (cf. \cite{Ze}, \cite{Sc}); invariant here means
that $f=0$ a.e. on $F$ implies $Tf = 0$ a.e. on $F$. Lemmas
\ref{lem4.4} and \ref{lem4.5} thus imply that the Krein-Rutman
theory for compact positive irreducible operators (cf. e.g.
\cite{Ze}, \cite{Sc}) can be applied to $T_l$ in $L^2(\Omega,d^2)$
for $l$ sufficiently large. This yields that the spectral radius
$\bar{\rho}_l$ of $T_l$ is $> 0$ and is a simple eigenvalue of
$T_l$ having an eigenfunction $G \geq 0$, $G \not \equiv 0$. Now
$T_l G = \bar{\rho}_l G$ implies that $G$ satisfies (\ref{eq4.12})
and that
\begin{equation}\label{eq4.13}
\int_\Omega u^2 \langle A \nabla
(\bar{\rho}_lG)+a(\bar{\rho}_lG), \nabla \varphi \rangle +
\int_\Omega lu^2 (\bar{\rho}_lG)\varphi = \int_\Omega u^2 G
\varphi
\end{equation}
for all $\varphi \in H^1(\Omega,d^2)$. Taking $\varphi \equiv 1$
yields $\bar{\rho}_l = 1/l$, which shows that (\ref{eq4.13}) reduces
to (\ref{eq3.3}). So $G$ solves (\ref{eq3.3}). Finally the statement
about unicity in Lemma \ref{lem4.6} follows from the fact that
(\ref{eq3.3}) can now be rewritten as $T_l G = \bar{\rho}_l G$.
\end{proof}
\begin{lemma}\label{lem4.7}
Let $u\in D(\Omega) \cap C^1(\bar{\Omega})$. Then the function $G$
provided by Lemma \ref{lem4.6} belongs to $L^\infty(\Omega)$.
\end{lemma}
\begin{proof} It is inspired from Moser's iteration
technique as given for instance in \cite[Theorem 8.15]{Gi-Tr}.
For $\beta \geq 1$ and $M > 0$, let $H \in C^1[0,+\infty[$ be
defined by setting $H(r) = r^\beta$ for $r \in [0,M]$ and taking $H$
to be linear for $r \geq M$. Put $v(x) := \int_0^{G(x)}[H'(s)]^2ds$.
One has that $v \in H^1(\Omega,d^2)$ since
$$
v(x) = \begin{cases}
\frac{\beta^2}{2\beta-1}G(x)^{2\beta-1} &\mbox{if } G(x) \leq
M,\\
\frac{\beta^2}{2\beta-1} M^{2\beta-1} + \beta^2 M^{2\beta
-2}(G(x)-M) &\mbox{if } G(x) > M,
\end{cases}
$$
$\nabla v = (H'(G))^2 \nabla G$ and $G \in H^1(\Omega,d^2)$. So $v$
is an admissible test function in (\ref{eq3.3}) and consequently
$$
\int_\Omega u^2 \langle A \nabla G + aG, \nabla v \rangle = 0.
$$
Using the inequality $2rs \leq (\varepsilon r)^2 +
(s/\varepsilon)^2$, one obtains from the above that
\begin{equation}\label{eq4.14}
\int_\Omega u^2 |\nabla (H(G))|^2 \leq c_1 \int_\Omega
u^2(H'(G))^2G^2
\end{equation}
where $c_1=c_1(A,a)$. On the other hand $H(G)\in H^1(\Omega,d^2)$
and so, fixing $p$ with $2 < p \leq 2+4/N$, one has by Lemma
\ref{lem4.3}
%
\begin{equation}\label{eq4.15}
\|H(G)\|_{L^p(\Omega,d^2)} \leq c_2 \|
H(G)\|_{H^1(\Omega,d^2)}
\end{equation}
%
where $c_2=c_2(\Omega,p)$.
Combining \eqref{eq4.14} and \eqref{eq4.15}
and using the fact that $u \in D(\Omega)$, it follows
\begin{equation}\label{eq4.16}
\|H(G)\|_{L^p(\Omega,d^2)} \leq c \left( \|
H(G)\|_{L^2(\Omega,d^2)}+\|H'(G)G\|_{L^2(\Omega,d^2)}\right)
\end{equation}
where $c$ depends on $L, \Omega, u, p$ but does not depend on
$G,\beta,M$. The function $H$ above depends on $M$, i.e. $H=H_M$,
and when $M \to +\infty$, one has that for each $r \geq 0$, $H_M(r)
\to \tilde{H}(r)$ and $H'_M(r)\to \tilde{H}'(r)$ in a nondecreasing
way, where $\tilde{H}(r)=r^\beta$. The monotone convergence theorem
can thus be applied to (\ref{eq4.16}), which shows that
(\ref{eq4.16}) still holds with $H$ replaced by $\tilde{H}$. This
means that
$$
\| G^\beta\|_{L^p(\Omega,d^2)} \leq
c(1+\beta)\|G^\beta\|_{L^2(\Omega,d^2)};
$$
i.e.,
\begin{equation}\label{eq4.17}
\|G\|_{L^{p\beta}(\Omega,d^2)} \leq [c(1+\beta)]^{1/\beta}
\|G\|_{L^{2\beta}(\Omega,d^2)},
\end{equation}
where $c$ is the same constant as in (\ref{eq4.16}). A priori the
above quantities might be $+\infty$, but a simple iteration of
(\ref{eq4.17}), where one takes successively $\beta=1$ (for which
the right-hand side of (\ref{eq4.17}) is finite), $\beta = p/2$,
$\beta=p^{2}/4$,$\ldots$, $\beta=(p/2)^j$, $\ldots$ $\to +\infty$
shows that $G \in L^q(\Omega,d^2)$ for all $1 \leq q < \infty$.
We now consider another iteration of (\ref{eq4.17}) for which the
constants will be controlled. Take $\beta=(p/2)^j/2$ for $j=j_0,
j_0+1,\dots $ with $j_0 \in {\mathbb N}$ chosen so that
$(p/2)^{j_{0}}/2 \geq 1$. One gets
\begin{equation}\label{eq4.18}
\|G\|_{L^{(p/2)^{j+1}}(\Omega,d^2)} \leq c^{2
\sum_{i=j_0}^j(2/p)^i} \prod_{i=j_0}^j
[1+(p/2)^i/2]^{\frac{1}{(p/2)^i/2}}\|G\|_{L^{(p/2)^{j_0}}(\Omega,d^2)}.
\end{equation}
Note that $G \in L^{(p/2)^{j_0}}(\Omega,d^2)$ as previously
observed. The exponent of $c$ in (\ref{eq4.18}) converges as $j
\to +\infty$ since it is part of a convergent geometric series.
Calling $q_j$ the product $\prod_{i=j_0}^j \ldots$ in (4.18), one
has
$$
\log q_j = \sum_{i=j_0}^j 2(2/p)^i \log [1+(p/2)^i/2];
$$
since
$$
\log [1+(p/2)^i/2]=i \log (p/2)+\log[(2/p)^i+(1/2)]\leq i \log (p/2)
$$
for $i$ sufficiently large, and since the series
${\sum_{i=1}^\infty (2/p)^i i}$ converges, one sees
that there exists $\bar{q}$ such that $q_j \leq \bar{q}$ for all
$j \geq j_0$. It thus follows from (\ref{eq4.18}) that
$$
\|G\|_{L^{(p/2)^{j+1}}(\Omega,d^2)} \leq \bar{c}
\|G\|_{L^{p/2}(\Omega,d^2)} < + \infty
$$
for all $j \geq j_0$, with a constant $\bar{c}$ independent of $j$.
Letting $j \to +\infty$, one deduces that $G$ belongs to
$L^\infty(\Omega)$.
\end{proof}
\begin{lemma}\label{lem4.8}
Let $u \in D(\Omega) \cap C^1(\bar{\Omega})$. Then the function
$G$ provided by Lemma \ref{lem4.6} satisfies
\begin{equation}\label{eq4.19}
G \geq \delta \quad \mbox{ a.e. in } \Omega
\end{equation}
for some constant $\delta > 0$.
\end{lemma}
\begin{proof} We will first consider the particular case
where the vector field $a$ in \eqref{eq1.2} satisfies
\begin{equation} \label{eq4.20}
\langle a, \nu \rangle < 0 \quad\text{on } \partial \Omega.
\end{equation}
\noindent \textbf{Claim.} For any $l \geq 0$ there exists
$\varepsilon = \varepsilon(a,u,l) > 0$ such that
$$
\int_\Omega u^2 \langle A \nabla (G-c) + a(G-c), \; \nabla \varphi
\rangle + \int_\Omega lu^2(G-c)\varphi \geq 0
$$
for all constants $c \geq 0$ and all $\varphi \in
H^1(\Omega,d^2)\cap L^{\infty} (\Omega)$ with $\varphi \geq 0$ and
supp $\varphi \subset \Gamma_\varepsilon$ ($\Gamma_\varepsilon$ was
defined in Lemma \ref{lem3.4}).
\begin{proof}[Proof of the Claim]
Using equation (\ref{eq3.3}) for
$G$ and the divergence theorem from Lemma \ref{lem3.4}, one obtains
$$
\int_\Omega u^2 \langle A \nabla (G-c) + a(G-c), \nabla \varphi
\rangle + \int_\Omega lu^2(G-c)\varphi \geq \int_\Omega
cu[2\langle \nabla u,a\rangle+u\mathop{\rm div} a-lu]\varphi.
$$
Since \eqref{eq4.20} implies $\langle a, \nabla u \rangle > 0$ on $\partial
\Omega$ and since $u$ vanishes on $\partial \Omega$, the bracket in
the last integral is $> 0$ on $\partial \Omega$, and consequently is
$\geq 0$ on $\Gamma_\epsilon$ for some $\varepsilon =
\varepsilon(a,u,l) > 0$. The inequality of the claim thus follows.
\end{proof}
We now turn to the proof of \eqref{eq4.19} in the particular case
where \eqref{eq4.20} holds. Let us fix $l$ sufficiently large so that
\eqref{eq4.4} holds on $H^1(\Omega,d^2)$ and let $\varepsilon =
\varepsilon(a,u,l)$ be given by the above claim. Call
$$
\delta = \mathop{\rm ess\,inf}_{x \in \Omega_{\varepsilon/2}} G(x),
$$
which is $> 0$ by Lemma \ref{lem4.6}, and let
$\varphi=(G-\delta)^-$. Clearly $\varphi \geq 0$, with supp $\varphi
\subset \Gamma_\varepsilon$ since $G \geq \delta$ on
$\Omega_{\varepsilon/2}$; moreover $\varphi \in H^1(\Omega,d^2)\cap
L^{\infty}(\Omega)$ since $G$ belongs to that space (by Lemma
\ref{lem4.7}). Applying the inequality of the claim with $c=\delta$
and $\varphi$ as above gives
$$
0 \leq B_l (G-\delta, \varphi) = B_l(-\varphi,\varphi)
$$
Inequality \eqref{eq4.4} then implies $\varphi=0$ a.e. in $\Omega$,
i.e. $G \geq \delta$ a.e. in $\Omega$, and the lemma is proved (in
the case where \eqref{eq4.20} holds).
Let us now consider the general case where \eqref{eq4.20} possibly does not
hold. Call $\varphi_1$ a positive eigenfunction associated to the
principal eigenvalue $\lambda_1$ of $-\Delta$ on $H_0^1(\Omega)$.
Put $w=r\varphi_1+1$ where $r \geq 0$ is chosen so that $\langle a,
\nu \rangle + r \langle A \nabla \varphi_1, \nu \rangle$ is $< 0$ on
$\partial \Omega$, which is clearly possible since $\langle A \nabla
\varphi_1, \nu\rangle$ is $< 0$ on $\partial \Omega$. Write $G$ as
$Hw$. It follows that $H=G/w \in H^1(\Omega,d^2)\cap
L^{\infty}(\Omega)$ and satisfies
\begin{equation}\label{eq4.21}
\int_\Omega (u^2 w) \langle A \nabla H + (A\frac{\nabla
w}{w}+a) H, \nabla \varphi \rangle = 0
\end{equation}
for all $\varphi \in H^1(\Omega,d^2)$, with moreover $H > 0$ a.e. in
$\Omega$. Equation \eqref{eq4.21} is of the same type as
(\ref{eq3.3}) : $u$ is replaced by $u \sqrt{w}$ (which still belongs
to $D(\Omega) \cap C^1(\bar{\Omega}))$ and $a$ is replaced by $A
(\nabla w/w)+a$ (which still belongs to $C^{0,1}(\bar{\Omega})$ but
now satisfies \eqref{eq4.20} by the choice of $r$ and the fact that $w\equiv
1$ on $\partial \Omega$). It follows that $H$ is the solution
provided by applying Lemma \ref{lem4.6} to this new equation
\eqref{eq4.21}. By that part of Lemma \ref{lem4.8} which has already
been proved, one deduces that $H$ satisfies \eqref{eq4.19} for some
$\delta > 0$. Since $w \geq 1$, one gets that $G = Hw$ also
satisfies \eqref{eq4.19}. This completes the proof of Lemma
\ref{lem4.8}.
\end{proof}
Lemma \ref{lem3.3} clearly follows from the previous Lemmas
\ref{lem4.6}, \ref{lem4.7} and \ref{lem4.8}.
\section{Antimaximum principle}
It is our purpose in this section to present the AMP in the previous
framework, i.e. for some nonselfadjoint problems with a weight in
$L^\infty(\Omega)$. The assumptions on $L, m$ and $\Omega$ are the
same as in section 2. We directly deal with the general case where
$a_{0}$ may not be $\geq 0$.
\begin{theorem}\label{theo5.1}
Suppose $m^+ \not \equiv 0$. Assume also the existence of a
principal eigenvalue for \eqref{eq1.1} and let $\lambda^{\ast}$ be the
largest of these principal eigenvalues (cf. Proposition \ref{prop2.6}). Take
$h \in L^p(\Omega)$ with $p
> N$ and $h \geq 0$, $h \not \equiv 0$. Then there exists $\delta =
\delta (h) > 0$ such that for $\lambda \in ]\lambda^*,
\lambda^*+\delta[$, any solution $u$ of \eqref{eq1.3} satisfies $u <
0$ in $\Omega$ and $\partial u/\partial \nu > 0$ on $\partial
\Omega$.
\end{theorem}
The proof of Theorem \ref{theo5.1} is based on a preliminary
nonexistence result, which reads as follows.
\begin{lemma}\label{lem5.2}
Let $\lambda^*$ be as above and take $h \in L^p(\Omega)$ with $1 <
p < \infty$ and $h \geq 0$, $h \not \equiv 0$.Then problem
\eqref{eq1.3} has no solution $u\geq 0$ if $\lambda > \lambda^*$,
and no solution at all if $\lambda = \lambda^*$.
\end{lemma}
The proof of the above two results can be carried out through a
rather standard adaptation to the present Dirichlet situation of
the arguments developed in \cite{Go-Go-Pa} in the case of the
Neumann-Robin boundary conditions, and we will omit it. The
general philosophy of this adaptation consists in replacing the
space $C(\bar{\Omega})$ by the space $C_0^1(\Omega)$, the
condition $u > 0$ on $\bar{\Omega}$ by the condition $u > 0$ in
$\Omega$ and $\partial u/\partial \nu < 0$ on $\partial \Omega$,
and the restriction $h \in L^p(\Omega)$ with $p > N/2$ by the
restriction $h \in L^p(\Omega)$ with $p > N$. One should also
remark that the assumption $a_{0} \geq 0$ in \cite{Go-Go-Pa} is
used there only to guarantee the existence of a principal
eigenvalue.
As before the case of the smallest principal eigenvalue can be
reduced to the case covered by Theorem \ref{theo5.1} by changing $m$
into $-m$.
\section{Nonuniformity of the antimaximum principle}
The assumptions on $L,m$ and $\Omega$ in this section are those of
section 3. Our purpose is to show that the AMP of Theorem
\ref{theo5.1} is not uniform, i.e. that a $\delta > 0$ independent
of $h$ cannot be found.
\begin{proposition}\label{prop6.1}
Assume $m^+ \not \equiv 0$ and let $\lambda^* > 0$ be as in Theorem
\ref{theo5.1}. Suppose that $\lambda \in \mathbb{R}$ enjoys the
following property : (*) for any $h \in C_c^\infty(\Omega)$, $h \geq
0$, $h \not \equiv 0$, problem \eqref{eq1.3} has a solution $u$
which satisfies $u < 0$ in $\Omega$. Then $\lambda \le \lambda^*$.
\end{proposition}
The proof of Proposition \ref{prop6.1} uses the minimax formula of
section 3. It is again an adaptation of arguments developed in
\cite{Go-Go-Pa} in the case of the Neumann-Robin boundary
conditions. However the adaptation here is not so standard as in
section 5 since it involves the introduction of spaces with weights
and of the set $D(\Omega)$. It seems consequently useful to sketch
part of the arguments, and the rest of this section will be devoted
to that.
Recall that by Theorem \ref{theo3.5},
\begin{equation}\label{eq6.1}
\lambda^* = \min_{u \in U} \frac{\Lambda (u)-\inf
Q_u}{\int_\Omega mu^2}
\end{equation}
where as before $\inf Q_u$ stands for $\inf \{Q_u(w) : w \in
H^1(\Omega,d^2)\}$.
We start with the following lemma whose proof is similar to that
of inequality (\ref{eq3.6}) in section 3. In fact, with respect to
(\ref{eq3.6}), (\ref{eq6.2}) below involves $\lambda$ instead of
$\lambda^*$ and has an extra term ${-\int_\Omega \frac{h}{v}u^2}$.
\begin{lemma}\label{lem6.2}
Let $\lambda \in \mathbb{R}$ be such that for some $h \in
C_c^\infty(\Omega)$, the problem $Lv = \lambda m v + h$ in
$\Omega$, $v=0$ on $\partial \Omega$ has a solution $v$ with $v >
0$ in $\Omega$. Then
\begin{equation}\label{eq6.2}
\lambda \int_\Omega mu^2 \leq \Lambda(u) - \inf Q_u -
\int_\Omega \frac{h}{v} u^2
\end{equation}
for any $u \in U$.
\end{lemma}
The objective is to prove that if $\lambda$ enjoys property (*),
then (\ref{eq6.2}) holds without the extra term
${-\int_\Omega \frac{h}{v} u^2}$. Once this is done,
the conclusion of Proposition \ref{prop6.1} follows by using
(\ref{eq6.1}). As an intermediate step towards this objective one
has the following
\begin{lemma}\label{lem6.3}
Suppose $\lambda$ enjoys property (*). Then
\begin{equation}\label{eq6.3}
\lambda \int_\Omega mu^2 \leq \Lambda(u) - \inf Q_u
\end{equation}
for any $u \in H^1(\Omega)$ such that $0 \leq u(x) \leq c_u d(x)$
in $\Omega$ for some constant $c_u$,
${\int_\Omega mu^2 > 0}$, and $u$ vanishes on some ball $B_u \subset \Omega$.
\end{lemma}
The proof of Lemma \ref{lem6.3} can be adapted from
\cite[Lemma 5.5]{Go-Go-Pa}. The main modifications consist in using now
as approximates for $u$ the functions
$$
u_j = \max \{u(x), \frac{d(x)}{j}\}
$$
for $j=1,2,\dots$ and in replacing \cite[Lemma 5.4]{Go-Go-Pa}
by the following
\begin{lemma}\label{lem6.4}
For any $u$ such that
\begin{equation}\label{eq6.4}
u(x) \leq c_u d(x)
\end{equation}
for some constant $c_u$ and a.e. $x \in \Omega$, one has
$\inf Q_u > - \infty$. Moreover if $u$ with \eqref{eq6.4} and $w \in
H^1(\Omega,d^2)$ vary in such a way that $\|u\|_\infty$ remains
bounded and $Q_u(w)$ remains bounded from above, then
$\|u \nabla w\|_2$ remains bounded.
\end{lemma}
The idea now to prove Proposition \ref{prop6.1} is to approximate
any $u \in U$ by functions as those in Lemma \ref{lem6.3} and go to
the limit in (\ref{eq6.3}). Here are some details. Given $u \in U$,
there exists $u_k \in H^1(\Omega)$ such that $0 \leq u_k \leq u$,
$u_k \to u$ in $H^1(\Omega)$, $u_k = 0$ on some hall $B_k$, with in
addition, for any $\Omega' \subset \subset \Omega$, $u_k \equiv u$
on $\Omega'$ for $k$ sufficiently large. One can for instance take
$u_k=u\psi_k$ where the functions $\psi_k$ are given by Lemma
\ref{lem6.6} below. Note that the proof that $u_k \to u$ in
$H^1(\Omega)$ uses the fact that $u$ satisfies an estimate near
$\partial \Omega$ of the type \eqref{eq6.4}. Note also that it is at
the moment of this approximation that in the case of the
Neumann-Robin boundary conditions, one had to impose in
\cite{Go-Go-Pa} the restriction $N \geq 2$, restriction which is not
necessary here. With these approximations $u_k$ at our disposal, the
proof of Proposition \ref{prop6.1} can be completed by following the
same lines as on pages 106-107 from \cite{Go-Go-Pa}. The main
modifications in this last part consist in introducing the weight
$d^2$ in the spaces to which the functions $w_k, w, \nabla w_k,
\nabla w$ from \cite{Go-Go-Pa} belong, and in replacing ultimately
\cite[Lemma 5.6]{Go-Go-Pa} by the following
\begin{lemma}\label{lem6.5}
Let $u \in L^2_{\rm loc}(\Omega)$ with $\nabla u \in
L^2(\Omega,d^2)$. Then $u \in L^2(\Omega,d^2)$.
\end{lemma}
The proof of lemma \ref{lem6.5} uses the Poincar\'e inequality of
Lemma \ref{lem4.2}.
\begin{lemma}\label{lem6.6}
There exists a sequence $\psi_k \in C_c^1(\Omega)$ such that (i) $0
\leq \psi_k \leq 1$ in $\Omega$, (ii) $\psi_k \equiv 1$ on
$\Omega_{2/k}$, (iii) supp $\psi_k \subset \Omega_{1/k}$, (iv)
$d|\nabla \psi_k| \leq K$ on $\Omega$ for some constant $K$ and all
$k$, where $\Omega_\eta : = \{x \in \Omega : d (x,\partial \Omega) >
\eta\}$.
\end{lemma}
\begin{proof}Take $\psi \in C^\infty (\mathbb{R})$ such
that $0 \leq \psi \leq 1$, $\psi(x) =0$ for $x \leq 1$, $\psi(x) =1$
for $x \geq 2$. For $k=1,2,\ldots$, define $\psi_k(x) = \psi(kd(x))$
for $x \in \bar{\Omega}\setminus \Omega_{2/k}$ and $\psi_k(x)=1$ for
$x \in \Omega_{2/k}$. Since $\Omega$ is of class $C^2$, it follows
from \cite[Lemma 14.16]{Gi-Tr} that $\psi_k \in C^2(\bar{\Omega})$
for $k$ sufficiently large. Properties (i), (ii), (iii) clearly
hold, and (iv) is easily certified using the fact that $2/k \leq
d(x) \leq 1/k$ where $\nabla \psi_k(x) \neq 0$.
\end{proof}
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