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%\textwidth162mm \textheight24cm \hoffset-16mm \voffset-20mm


\begin{document}

\title[Singular elliptic problems]{The influence of the distance function in some singular elliptic problems}


\author{Marius Ghergu}
%    Address of record for the research reported here
\address{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,
P.O. Box 1-764, 014700 Bucharest, Romania}
\email{marius.ghergu@imar.ro}

\author{Vicen\c tiu R\u adulescu}
\address{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,
P.O. Box 1-764, 014700 Bucharest, Romania}\email{vicentiu.radulescu@imar.ro}
\address{Department of Mathematics, University of Craiova,
200585 Craiova, Romania} \email{vicentiu.radulescu@math.cnrs.fr}
\date{}

\subjclass[2000]{Primary 35J60; Secondary 35B25, 35B50, 58J55}

\date{}

\dedicatory{This paper is dedicated to the memory of Aurel Cornea}

\keywords{nonlinear elliptic equation, singular nonlinearity, bifurcation problem, asymptotic behavior of solutions}

\begin{abstract}
We present some existence and nonexistence results for classical
solutions to singular elliptic problems of the form $$-\Delta u\pm
\delta(x)^{-\alpha}u^{-\beta}=\la f(x,u)\quad \mbox{\rm in }
\Omega,$$ subject to homogeneous Dirichlet boundary condition. Here
$\o\subset\RR^N(N\geq 1)$ is a smooth and bounded domain,
$\delta(x)=$dist$(x,\partial\Omega)$, $\alpha$, $\beta$ and $\la$ are positive real numbers, while $f$
has either a linear or a sublinear growth with respect to the second
variable.
\end{abstract}



\maketitle

\section{Introduction and the main results}

Let $\Omega\subset \RR^N$ $(N\geq 1)$ be a bounded domain with a
smooth boundary. We are concerned in this paper with singular
elliptic problems of the following type
$$\left\{\begin{aligned}
-&\Delta u\pm  \delta(x)^{-\alpha}u^{-\beta}=\la f(x,u)&&
\quad \mbox{\rm in}\ \Omega,\\
&u>0&& \quad \mbox{\rm in}\ \Omega,\\
&u=0&& \quad \mbox{\rm on}\ \partial\Omega,
\end{aligned} \right.\eqno(P_\la)^\pm$$
where $\delta(x)=\mbox{dist}(x,\partial\Omega),$ $\alpha,\beta,\la>0$.
Throughout this paper we suppose that
$f:\overline{\Omega}\times[0,\infty)\rightarrow[0,\infty)$ is a
H\"{o}lder continuous function which is nondecreasing with respect
to the second variable and such that $f$ is positive on
$\overline\Omega\times (0,\infty).$ The analysis we develop in
this paper concerns the cases where $f$ is either linear or $f$ is
sublinear with respect to the second variable. This last case
means that $f$ fulfills the hypotheses
\medskip

\noindent $\di(f1)\qquad$ the mapping $\di (0,\infty)\ni
t\longmapsto\frac{f(x,t)}{t}\quad\mbox{is nonincreasing for
all}\;\, x\in\overline{\Omega};$
\medskip

\noindent $\di(f2)\qquad \lim_{t\ri 0^+}
\frac{f(x,t)}{t}=+\infty\quad\mbox{and}\;\;
\lim_{t\rightarrow+\infty}\frac{f(x,t)}{t}=0,\;\;\mbox{uniformly
for}\;\,x\in\overline{\Omega}.$

\medskip

Such singular boundary value problems arise in the context of
chemical heterogeneous catalysts and chemical catalyst kinetics (such as the Langmuir--Hinshelwool model),
in the theory of heat conduction in electrically conducting
materials, singular minimal surfaces, as well as in the study of
non-Newtonian fluids or boundary layer phenomena for viscous
fluids (we refer for more details to \cite{caf,cn,cp,coh,
diaz,dmo} and the more recent papers
\cite{crg,grjde,grcras,grbook, her,mea,sy1,sy2}). We also point out
that, due to the meaning of the unknowns (concentrations,
populations, etc.), only the positive solutions are relevant in
most cases. For instance, problems of this type
characterize some reaction-diffusion processes where $u\geq 0$ is
viewed as the density of a reactant and the region where $u=0$ is
called the {\it dead core}, where no reaction takes place (see
Aris \cite{aris} for the study of a single, irreversible steady-state
reaction). Nonlinear singular elliptic equations are also
encountered in glacial advance, in transport of coal slurries down
conveyor belts and in several other geophysical and industrial
contents (see Callegari and Nachman \cite{cn}  for the case of the
incompressible flow of a uniform stream past a semi-infinite flat
plate at zero incidence).

To the best of our knowledge, there does not exist a qualitative
theory for the study of singular boundary value problems with
nonlinearities in the Kato class $K_N^{\rm loc}(\RR^N)$. This
theory was introduced by Aizenman and Simon in \cite{aize} to
describe wide classes of functions arising in Potential Theory. We
refer to the recent paper \cite{rhouma} for existence and
bifurcation results on Dirichlet boundary value problems with
indefinite nonlinearities.

In this paper we aim to study the influence of the distance
function $\delta(x)$ in such singular elliptic problems. We first
establish the following result related to problem
$(P_\la)^+$.

\begin{theorem}\label{nonexpminus}
Assume that $f$ satisfies $(f1),(f2)$.
\begin{enumerate}
\item[(i)] If $\alpha+\beta\leq 1$ then $(P_\la)^+$ has no
classical solutions;

\item[(ii)]  If $\alpha+\beta>1$, then there exists $\la^*>0$ such
that $(P_\la)^+$ has at least one classical solution for all
$\la>\la^*$ and no solution exists if $\;0<\la<\la^*.$


\end{enumerate}

\end{theorem}



In contrast to the results in Theorem \ref{nonexpminus}, the study
of  $(P_\la)^-$ is quite different. The one dimensional case,
namely the problem
\begin{equation}\label{dunu}
\left\{\begin{aligned}
-&H''(t)=\delta(t)^{-\alpha} H(t)^{-\beta}, H>0&& \quad \mbox{ in } (0,1),\\
&H(0)=H(1)=0,
\end{aligned} \right.
\end{equation}
was discussed in \cite{tal} and \cite[Section 2]{ar}. It has been
shown that \eq{dunu} has solutions if and only if $\alpha<2$. We first
obtain that condition $\alpha<2$ is also necessary and sufficient to obtain a
solution in higher dimension provided $f$ has a sublinear growth. More precisely we have:


\begin{theorem}\label{nonexplus}
Assume that $f$ satisfies $(f1),(f2)$.
\begin{enumerate}
\item[(i)] If $\alpha\geq 2$ then $(P_\la)^-$ has no
classical solutions;

\item[(ii)]  If $0<\alpha<2$, then for all $\la>0$ problem $(P_\la)^-$ has al least one solution.

Moreover,  there exist $0<\eta<1$ and $C_1,
C_2>0$ such
that $u_\la$ satisfies

\begin{enumerate}
\item[{\rm (ii1)}] If  $\alpha+\beta>1,$ then
\neweq{sp1}
C_1 \delta (x)^{\frac{2-\alpha}{1+\beta}}\leq u_\la(x)\leq
C_2 \delta(x)^{\frac{2-\alpha}{1+\beta}}, \quad\mbox{ for all
}\;x\in\Omega;
\endeq
\item[{\rm (ii2)}] If  $\alpha+\beta=1,$ then
\neweq{sp2}
C_1 d(x)(-\ln \delta (x))^{\frac{1}{2-\alpha}}\leq u_\la (x)\leq C_2
d(x)(-\ln \delta (x))^{\frac{1}{2-\alpha}},
\endeq
for all $x\in\Omega$ with $\delta(x)<\eta;$
\item[{\rm (ii3)}] If  $\alpha+\beta<1,$ then
\neweq{sp3}
C_1\delta (x)\leq u_\la(x)\leq C_2\delta(x), \quad\mbox{ for all }\;x\in\Omega.
\endeq
 \end{enumerate}
\end{enumerate}
\end{theorem}

If $\alpha+\beta<1$, we are able to obtain the uniqueness and to
provide the regularity of solution to $(P_\la)^-$ by means of the associated Green function.
\begin{theorem}\label{th3} Assume that $f$ satisfies $(f1),(f2)$
and $\alpha+\beta<1$. Then, for all $\la>0$ problem $(P_\la)^-$
has a unique solution $u_\la$ which in addition satisfies
 \begin{enumerate}
\item[(i)] $(0,\infty)\ni \la\longmapsto u_\la$ is increasing in
$\o$;

\item[(ii)]  $u_\la\in C^2(\o)\cap C^{1,1-\alpha-\beta}(\oo)$.
\end{enumerate}

\end{theorem}



We now consider the case where $f$ is asymptotically linear. More precisely, we assume that
$f$ satisfies $(f1)$ and

\noindent $\di(f3)\qquad m:=\lim_{t\ri 0^+}
\frac{f(x,t)}{t}\in (0,\infty).$

In case $\alpha+\beta<1$ we have the following result.

\begin{theorem}\label{t4}
Assume that $\alpha+\beta<1$ and $f$ satisfies $(f1),(f3)$. Then, problem $(P_\la)^-$ has solutions if and only if $0<\la<\la_1/m$.
Moreover, for all $0<\la<\la_1/m$, there exists a unique solution $u_\la$ of $(P_\la)^-$ such that
\begin{enumerate}
\item[(i)] $u_\la\in C^2(\o)\cap C^{1,1-\alpha-\beta}(\oo)$;

\item[(ii)]  $\lim_{\la\nearrow \la_1/m} u_\la=\infty$ uniformly
on compact subsets of $\o$.
\end{enumerate}
\end{theorem}

The next sections contains the proofs of the above results.

\section{Proof of Theorem \ref{nonexpminus}}





Several times in this paper we apply the following comparison
result (we refer to \cite[Lemma 2.1]{grjmaa} for a complete
proof).

\begin{lem}\label{l1}
Let $\Phi:{\Omega}\times(0,\infty)\rightarrow\RR$ be a
H\"older continuous function such that the mapping
$\di(0,\infty)\ni t\longmapsto {\Phi(x,t)}/t$ is strictly
decreasing for each $x\in\Omega.$ Assume that there exist  $v$,
$w\in C^2(\Omega)\cap C({\overline{\Omega}})$ such that

$(a)\qquad \Delta w+\Phi(x,w)\leq 0\leq \Delta v+\Phi(x,v)$ in
$\Omega;$

$(b)\qquad v,w>0$ in $\Omega$ and $v\leq w$ on $\partial\Omega;$

$(c)\qquad\Delta v\in L^1(\Omega)\;\mbox{ or }\; \Delta w\in
L^1(\Omega).$

Then $v\leq w$ in $\Omega.$
\end{lem}


\noindent{\bf Proof of Theorem \ref{nonexpminus}.} (i) Assume that $\alpha+\beta\geq 1$ and there exists
$\la>0$ such that $(P_\la)^+$ has a classical solution $u$ and
let $C=\max_{\overline\Omega}\la f(u)>0$. Let also $v\in
C^2(\overline\Omega)$ be the unique solution of
\neweq{Fi}
\left\{\begin{aligned}
-&\Delta v=C && \quad \mbox{\rm in}\ \Omega,\\
& v>0&& \quad \mbox{\rm in}\ \Omega,\\
&v=0 && \quad \mbox{\rm on}\ \partial\Omega.
\end{aligned} \right.
\endeq
Moreover, there exist $c_1,c_2>0$ such that
\neweq{udst}
c_1 d(x)\leq v\leq c_2 d(x), \quad\mbox{for all} \;x\in\Omega.
\endeq
By maximum principle, it follows that $u\leq v$ in $\Omega$. Next
we consider the perturbed problem
\begin{equation}\label{epsilon}
 \left\{\begin{aligned}
-&\Delta u+(\delta(x)+\ep)^{-\alpha} (u+\ep)^{-\beta}=C&& \quad
{\rm in}\ \Omega,\\
&u>0 && \quad {\rm in}\ \Omega,\\
&u=0 && \quad {\rm on}\ \partial\Omega.
\end{aligned} \right.
\end{equation}
Then, $u$ and $v$ are, respectively, sub- and super-solution of
(\ref{epsilon}). By standard arguments and elliptic regularity
(see \cite{gt}), there exists $u_{\ep}\in
C^{2}(\overline{\Omega})$ a solution of (\ref{epsilon}) such that
$u\leq u_{\ep}\leq v$ in $\Omega$. Integrating in (\ref{epsilon})
we obtain
$$\di -\int_{\Omega}\Delta
u_{\ep}dx+\int_{\Omega}(\delta(x)+\ep)^{-\alpha}
(u_{\ep}+\ep)^{-\beta} dx=C|\Omega|.
$$
Hence
\begin{equation}\label{treitrei}
\di-\int_{\partial\Omega}\frac{\partial u_{\ep}}{\partial n}ds+
\int_{\Omega} (\delta(x)+\ep)^{-\alpha} (u_{\ep}+\ep)^{-\beta}
dx\leq M,
\end{equation}
where $M$ is a positive constant. Taking into account that $\di
\partial u_{\ep}/ \partial n\leq 0$ on $\partial\Omega$ and
$v\leq u_\ep$ in $\Omega$ we have $\di \int_{\Omega}
(\delta(x)+\ep)^{-\alpha} (u_{\ep}+\ep)^{-\beta} dx\leq M.$ Thus,
for any compact subset $\omega\subset\subset\Omega$ we have
$$\di \int_{\omega}(\delta(x)+\ep)^{-\alpha}
(u_{\ep}+\ep)^{-\beta} dx\leq M.$$ Passing to the limit with
$\ep\ri 0^+$ we obtain $\di \int_{\omega} \delta(x)^{-\alpha}
u_{\ep}^{-\beta} dx\leq M$, for all $\omega\subset\subset\Omega$.
Therefore
\begin{equation}\label{treitpatru}
\di \int_{\Omega} \delta(x)^{-\alpha} u_{\ep}^{-\beta} dx\leq M.
\end{equation}
On the other hand, using \eq{udst} and the hypothesis
$\alpha+\beta\geq 1$, it follows that
$$\di M\geq \int_{\Omega}\delta(x)^{-\alpha}
u_{\ep}(x)^{-\beta} dx\geq c \int_{\Omega}
\delta(x)^{-\alpha-\beta} dx=+\infty,$$ which is a contradiction.
Hence, problem $(P_\la)^+$ has no classical solutions and the
proof of Theorem \ref{nonexpminus} is complete.


(ii)  We first establish the existence of solutions to $(P_\la)^+$ for $\la>0$ large.
By virtue of \cite[Lemma 2.4]{sy1} (see also (\cite[Theorem 2.2]{sy2}),
the problem
\neweq{UU}
\left\{\begin{aligned}
-&\Delta U=\la f(x,U) && \quad \mbox{\rm in}\ \Omega,\\
&U>0 && \quad \mbox{\rm in}\ \Omega,\\
& U=0 && \quad \mbox{\rm on}\ \partial\Omega,
\end{aligned} \right.
\endeq
has at least one classical solution $U_\la$, for all $\la>0$ and
$U_\la$ is a super-solution of
$(P_\la)^+.$ The main point is to find a
sub-solution $\underline u_\la$ of $(P_\la)^+$ such that $\underline
u_\la\leq
U_\la$ in $\Omega.$
To this aim, let
$$\di \Psi:[0,\infty)\ri[0,\infty),\quad
\Psi(t)=\int_0^t  \left[ 2\left( 1-\frac{1}{1-\alpha-\beta} s^{1-\alpha-\beta}\right)\right]^{-1/2} ds.$$
Since $\alpha+\beta<1$, $\Psi$ is well defined. Moreover, $\Psi$ is a bijective map.
Let $h:[0,\infty)\rightarrow[0,\infty)$ be the inverse of $\Psi.$
Then $h\in C^2(0,\infty)\cap C^1[0,\infty)$ and $h$ satisfies
\begin{equation}\label{doicinci}
\left\{\begin{tabular}{ll}
$h''(t)=h(t)^{-\alpha-\beta}, h>0$&$ \quad\mbox{ in
}\;(0,\infty),$\\
$\di h(0)=h'(0)=0.$
\end{tabular} \right.
\end{equation}
The following technical result provides a suitable sub-solution for our problem.
\begin{lem}{\rm (see \cite{dgrjmpa})}\label{lstep}
There exist two positive constants $c>0$ and $M>0$ such that
$\underline{u}_{\la}:=Mh(c\vp)$
is a sub-solution of $(P_\la)^+$ provided $\la>0$ is large enough.
\end{lem}
Using Lemma \ref{l1}, it follows that
$\underline u_\la\leq U_\la$ in $\Omega$ and by standard elliptic
arguments (see \cite{gt}) we obtain a classical solution $u_\la$ of
$(P_\la)^+$
such that $\underline u_\la\leq u_\la\leq U_\la$ in $\Omega.$


Next we prove that $(P_\la)^+$ has no solutions for $\la>0$ small.
since $f$ satisfies $(f1)-(f2)$, we can find $m>0$ such that
\neweq{n3}
\di f(x,t)-p(d(x))g(t)<mt, \quad \mbox{ for all }\,
(x,t)\in\Omega\times(0,+\infty).
\endeq
Set $\la_0=\min\left\{1,\la_1/2m\right\}.$ We claim that problem
$(P_\la)^+$ has no classical solution for $0<\la\leq \la_0.$ Indeed,
assume by contradiction that $u_0$ is a classical solution of
$(P_\la)^+$ with $\la\in(0,\la_0].$ Then, according to \eq{n3}, $u_0$ is
a sub-solution of
\neweq{n4}
\left\{\begin{aligned} -&\Delta v=\frac{\la_1}{2}v && \quad
{\rm in}\ \Omega,\\
&v>0 && \quad {\rm in}\ \Omega,\\
&v=0 && \quad {\rm on}\ \partial\Omega.
\end{aligned} \right.
\endeq
By Lemma \ref{l1} we have
$u_0\leq U_\la$ in $\Omega$ which yields $c u_0\leq \vp$ in $\Omega$ for some positive constant $c>0.$
Note that $cu_0$ is still
a sub-solution of \eq{n4} while $\vp$ is a super-solution of \eq{n4}.
By standard elliptic arguments, it follows that problem \eq{n4} has a solution
$v\in
C^2(\overline\Omega).$ Multiplying by $\vp$ in \eq{n4} and
integrating on $\Omega$ we have
$$\di -\int_\Omega\vp\Delta vdx=\frac{\la_1}{2}\int_\Omega
v\vp dx,$$ that is,
$$\di \la_1\int_\Omega v\varphi_1 dx =-\int_\Omega v
\Delta\vp dx=\frac{\la_1}{2}\int_\Omega v\vp dx.$$
The above equality yields $\int_\Omega v\vp dx=0,$ but
this is clearly a contradiction, since $v$ and $\vp$ are both
positive in $\Omega.$ It follows that $(P_\la)^+$ has no classical
solutions for $0<\la\leq \la_0.$

Next we define
$$ \di A=\left\{\la>0; \mbox{ problem }(P_\la)^+
\mbox{ has at least one classical solution}\right\}.$$
>From the above
arguments we deduce that $A$ is nonempty and $\la^*:=\inf A$ is
positive. We only need to show that if $\la\in A,$ then $(\la,\infty)\subseteq
A$ but this follows by the sub and super-solution method. Hence $(\la^*,\infty)\subseteq A \subseteq [\la^*,\infty)$
and the proof is now complete.
\qed







\section{Proof of Theorem \ref{nonexplus}}


(i) We proceed similarly as in the proof of Theorem \ref{nonexpminus}.

(ii) Remark first that  for all $\la>0$, the solution $U_\la$ of \eq{UU} is a sub-solution for $(P_\la)^-$.
However, we need a suitable sub-solution for $(P_\la)^-$ that will
provide the asymptotic behavior for $u_\la$ as describet in \eq{sp1}-\eq{sp3}.


\begin{lem}{\rm (see \cite{dgrjmpa})}\label{lstepp}
Assume $0<\alpha<2$ and let $H$ be the solution of \eq{dunu}. Then, for all $\la>0$
there exist  positive constants $C,c>0$ and $M,m>0$ (depending on $\la$) such that
$\underline{u}_{\la}:=mH(c\vp)$, $\overline{u}_{\la}:=MH(C\vp)$ is a sub-solution and a super-solution of  $(P_\la)^-$ such that
$\underline u_\la\leq \overline u_\la$ in $\Omega$..
\end{lem}
Hence, for all $\la>0$ there exists an ordered pair of  sub- and super-solution of $(P_\la)^-$.
By standard elliptic arguments, problem $(P_\la)^-$ has at least one
solution $u_\la$ such that
\neweq{uus}
mH(c\vp) \leq u_\la\leq M H(C\vp) \quad\mbox{ in }\Omega.
\endeq




(ii1) Remark that
$$\di
H(t)=\left(\frac{(1+\beta)^2}{(2-\alpha)(\alpha+\beta-1)}\right)^{1/(1+\beta)}
t^{\frac{2-\alpha}{1+\beta}},\qquad t>0,\;$$ is a solution of
\eq{dunu} provided $\alpha+\beta>1.$ The conclusion in this case
follows now from \eq{uus}.
\medskip

(ii2) Since $H$ is concave,  there exists $H'(0^+)\in(0, \infty]$. Taking $0<b<1$ small enough,
we can assume that $H'>0$ on $(0,b]$. Hence $H$ satisfies
\begin{equation}\label{dunus}
\left\{\begin{tabular}{ll}
$H''(t)=-t^{-\alpha}H^{\alpha-1}(t),\quad \mbox{ for all } 0<t\leq
b<1,$\\
$H(0)=0,$\\
$H, H'>0\quad\mbox{\rm in}\  (0,b].$
\end{tabular} \right.
\end{equation}

Since $H$ is concave, it follows that
\neweq{HH}
H(t)>tH'(t),\quad\mbox{for all }\;0<t\leq b.
\endeq
Relations \eq{dunus} and \eq{HH} yield
$$\di -H''(t)<t^{-1}(H'(t))^{\alpha-1},\quad\mbox{for all }\;0<t\leq
b.$$
Hence
\neweq{HH1}
-H''(t)(H'(t))^{1-\alpha}\leq  \frac{1}{t},\quad\mbox{for all
}\;0<t\leq b.
\endeq
Integrating in \eq{HH1} over $[t,b]$ we get
$$\di (H')^{2-\alpha}(t)-(H')^{2-\alpha}(b)\leq
(2-\alpha)(\ln b-\ln
t), \quad\mbox{for all }\;0<t\leq b.$$ Hence, there exist $c_1>0$ and
$\delta_1\in(0,b)$ such that
\neweq{HH2}
H'(t)\leq c_1(-\ln t)^{\frac{1}{2-\alpha}},\quad\mbox{for all
}\;0<t\leq \delta_1.
\endeq
Fix $t\in(0,\delta_1].$ Integrating over $[\ep,t]$, $0<\ep<t,$ in
\eq{HH2} we have
\neweq{HH3}
\di H(t)-H(\ep)\leq c_1 t(-\ln t)^{\frac{1}{2-\alpha}}+
\frac{c_1}{2-\alpha}\int_{\ep}^t (-\ln
s)^{\frac{\alpha-1}{2-\alpha}}ds.
\endeq
Note that
\neweq{HH0}
\di \int_{0}^t (-\ln
s)^{\frac{\alpha-1}{2-\alpha}}ds<+\infty\qquad\mbox{ and }\;\;
\lim_{t\ri 0^+}\frac{\int_{0}^t (-\ln
s)^{\frac{\alpha-1}{2-\alpha}}ds}{t(-\ln
t)^{\frac{1}{2-\alpha}}}=0.
\endeq
Therefore, taking $\ep\ri 0^+$ in
\eq{HH3} we deduce that there exist $c_2>0$ and
$\delta_2\in(0,\delta_1)$ such that
\neweq{HH4}
\di H(t)\leq c_2t(-\ln t)^{\frac{1}{2-\alpha}},\quad\mbox{for all
}\;0<t\leq \delta_2.
\endeq
>From \eq{dunus} and \eq{HH4} we obtain
$$\di -H''(t)\geq c_2^{\alpha-1}t^{-1}(-\ln
t)^{\frac{\alpha-1}{2-\alpha}},\quad\mbox{for all
}\;0<t\leq \delta_2.$$ Integrating over $[t,\delta_2]$ in the
above inequality we get
$$\di H'(t)\geq (2-\alpha)c_2^{\alpha-1}
\left[(-\ln t)^{\frac{1}{2-\alpha}}-(-\ln
\delta_2)^{\frac{1}{2-\alpha}}\right],\quad\mbox{for all
}\;0<t\leq \delta_2.$$
Therefore, there exist $c_3>0$ and $\delta_3\in(0,\delta_2)$ such that
$$\di H'(t)\geq c_3 (-\ln t)^{\frac{1}{2-\alpha}},
\quad\mbox{for all
}\;0<t\leq \delta_3.$$
With the same arguments as in \eq{HH2}-\eq{HH4}
we obtain $c_4>0$ and
$\delta_4\in(0,\delta_3)$ such that
\neweq{HH5}
\di H(t)\geq c_4t(-\ln t)^{\frac{1}{2-\alpha}},\quad\mbox{for all
}\;0<t\leq \delta_4.
\endeq
The conclusion of (ii2) in Theorem \ref{nonexplus} follows now from
\eq{HH4} and \eq{HH5}.

(ii3) Using the fact that $H'(0+)\in(0,\infty]$ and the
inequality
\eq{HH}, we get the existence of $c>0$ such that
$$\di H(t)>ct,\quad\mbox{for all
}\;0<t\leq b.$$ This yields
$$-H''(t)\leq c^{-\beta} t^{-(\alpha+\beta)},\quad\mbox{for all
}\;0<t\leq b.$$ Since $\alpha+\beta<1,$ it follows that
$H'(0+)<+\infty,$ that is, $H\in C^1[0,b].$ Thus, there exists
$c_1,c_2>0$ such that
\neweq{HH6}
c_1t\leq H(t)\leq c_2t,\quad\mbox{for all }\;0<t\leq b.
\endeq
The conclusion in Theorem \ref{nonexplus} (ii3) is now immediately by
\eq{HH6}. This finishes the proof of Theorem \ref{nonexplus}.
\qed


\section{Proof of Theorem \ref{th3}}

(i) From Theorem \ref{nonexplus}, for all $\la>0$ there exists a
solution $u_\la$ of $(P_\la)^-$ such that \eq{sp3} holds.
Since $\alpha+\beta<1$ this
implies
$$\di 0\leq -\Delta u_\la=\delta(x)^{-\alpha} u_\la^{-\alpha+
\beta}+\la f(x,u_\la)\leq C\delta(x)^{-\alpha-\beta}\in L^1(\o).$$
 Therefore, by Lemma \ref{l1} we derive
that $u_\la$ is the unique solution of $(P_\la)^-$.

(ii) Let $0<\la_1<\la_2$ and $u_{\la_1}$, $u_{\la_2}$ be the
corresponding solutions to $(P_{\la_1})^-$ and $(P_{\la_2})^-$
respectively. From the above arguments we have that $\Delta
u_{\la_i}\in L^1(\o)$, $i=1,2$. We only have to apply Lemma
\ref{l1} for the mapping $\Phi(x,t)=\delta(x)^{-\alpha}
t^{-\beta}+\la_1 f(x,t)$, $(x,t)\in \o\times (0,\infty)$ in order
to deduce $u_{\la_1}\leq u_{\la_2}$ in $\o$. By maximum principle
in follows that $u_{\la_1}<u_{\la_2}$ in $\o$.

(iii) For the proof of regularity  we use a similar method to that developed by Gui and Lin \cite{gl}.
Let ${\mathcal G}$ be the Green function
 associated with the Laplace operator in $\o$. We shall use
the following estimates, which are due to Widman \cite{widman}.
\begin{lem}\label{greenreg}
There exists a positive constant $c>0$ such that for all $x,y\in\o$, $x\neq y$ we have
\begin{enumerate}
\item[\rm(i)] $\di |{\mathcal G}_x(x,y)|\leq c\,
\frac{\min\{|x-y|,d(y)\}}{|x-y|^{N}}$; \item[\rm(ii)] $\di
|{\mathcal G}_{xx}(x,y)|\leq c\,
\frac{\min\{|x-y|,d(y)\}}{|x-y|^{N+1}}$.
\end{enumerate}
\end{lem}


Then
$$u_{\la}(x)=-\int_{\o}{\mathcal G}(x,y)\Phi(y)dy\quad\mbox{ for all } x\in\o,$$
where $\Phi(y)=\delta(y)^{-\alpha} u_{\la}(y)^{-\beta}+\la
f(y,u_{\la}(y)).$ Hence
\neweq{wwsbb09}
\di \nabla u_{\la}(x)=-\int_{\o}{\mathcal
G}_x(x,y)\Phi(y)dy\quad\mbox{ for all } x\in\o.
\endeq

A very useful tool in our approach is the following technical
result.
\begin{lem}\label{greentool}
There exists $c_0=c_0(\Omega)>0$ and $\delta_0=\delta_0(\Omega)>0$
such that for all $x_1,x_2\in \Omega$, $0<|x_1-x_2|<\delta_0$
there exists a $C^1$ path $\xi:[0,1]\rightarrow \Omega$ with the
properties
\begin{enumerate}
\item[\rm(i)] $\di \xi(0)=x_1$ and $\xi(1)=x_2$; \item[\rm(ii)]
$\di |\xi'(t)|\leq c_0|x_1-x_2|$, for all $0\leq t\leq 1$.
\end{enumerate}
\end{lem}
Fix $x_1,x_2\in\o,$ $0<|x_1-x_2|<\delta_0$ and let
$\xi:[0,1]\ri\o$ be the corresponding path in Lemma
\ref{greentool}.

By \eq{wwsbb09} we have
\neweq{evalnablac}
\begin{aligned}
\di |\nabla u_{\la}(x_1)-\nabla u_{\la}(x_2)|\leq&\int_{\o}|{\mathcal G}_x(x_1,y)-{\mathcal G}_x(x_2,y)|\Phi(y)dy\\
\leq& \underbrace{ \int_{B_r(x_1)}|{\mathcal G}_x(x_1,y)-{\mathcal G}_x(x_2,y)|\Phi(y)dy }_{I}\\
& +\underbrace{ \int_{\o\setminus B_r(x_1)}|{\mathcal G}_x(x_1,y)-{\mathcal G}_x(x_2,y)|\Phi(y)dy }_{II}\;,\\
\end{aligned}
\endeq
where $r=(c_0+1)|x_1-x_2|$ and $c_0$ is the constant appearing in
Lemma \ref{greentool}.

Before evaluating $I$ and $II$, let us remark that there exists
$c_1>0$ such that $\Phi(y)\leq c_1\delta(y)^{-\alpha-\beta}$ for
all $y\in\o$. Then
$$
\begin{aligned}
I&\leq c_1\int_{B_r(x_1)}|{\mathcal G}_x(x_1,y)-{\mathcal G}_x(x_2,y)|\delta(y)^{-\alpha-\beta}dy\\
&\leq c_1\int_{B_r(x_1)}|{\mathcal G}_x(x_1,y)|\delta(y)^{-\alpha-\beta}dy+c_1\int_{B_R(x_2)}|{\mathcal G}_x(x_2,y)|\delta(y)^{-\alpha-\beta}dy,\\
\end{aligned}
$$
where $R=r+|x_1-x_2|$. Let $y\in B_r(x_1)$.\\
If $d(y)\geq |x_1-y|$ then, by Lemma \ref{greenreg} (i), we have
$$\di |{\mathcal G}_x(x_1,y)|\delta(y)^{-\alpha-\beta}\leq c|x_1-y|^{-N+1}\delta(y)^{-\alpha-\beta}\leq c|x_1-y|^{-N+1-\alpha-\beta}.$$
If $\delta(y)<|x_1-y|$ then, by Lemma \ref{greenreg} (i), we
obtain
$$\di |{\mathcal G}_x(x_1,y)|\delta(y)^{-\alpha-\beta}\leq
c|x_1-y|^{-N}\delta(y)^{1-\alpha-\beta}\leq
c|x_1-y|^{-N+1-\alpha-\beta}.$$ Therefore, for all $y\in B_r(x_1)$
we have
$$\di |{\mathcal G}_x(x_1,y)|\delta^{-\\lpha-\beta}(y)\leq c|x_1-y|^{-N+1-\alpha-\beta},$$
and similarly
$$\di |{\mathcal G}_x(x_2,y)|\delta(y)^{-\alpha-\beta}\leq c|x_2-y|^{-N+1-\alpha-\beta}\quad\text{ for all }y\in B_R(x_2).$$
Hence,
\neweq{evalIconv}
\begin{aligned}
I&\leq c_2\int_{B_r(x_1)}|x_1-y|^{-N+1-\alpha-\beta}dy+c_2\int_{B_R(x_2)}|x_2-y|^{-N+1-\alpha-\beta}dy\\
&\leq c_3\int_0^{R} t^{-\alpha-\beta}dt\leq c_4|x_1-x_2|^{1-\alpha-\beta}.\\
\end{aligned}
\endeq


To evaluate $II$, we first apply the mean value theorem.
\index{mean value theorem} \index{theorem!mean value} We have
$$\begin{aligned}
II&\leq c_1\int_{\o\setminus B_r(x_1)}|{\mathcal G}_x(\xi(0),y)-{\mathcal G}_x(\xi(1),y)|\delta(y)^{-\alpha-\beta}dy\\
&\leq c_1\int_{\o\setminus B_r(x_1)}\int_0^1|{\mathcal G}_{xx}(\xi(t),y)||\xi'(t)|\delta(y)^{-\alpha-\beta}dtdy\\
&\leq c_5|x_1-x_2|\int_{\o\setminus B_r(x_1)}\int_0^1|{\mathcal G}_{xx}(\xi(t),y)|\delta(y)^{-\alpha-\beta}dtdy.\\
\end{aligned}$$
As shown earlier, by Lemma \ref{greenreg} (ii) we obtain
$$\di |{\mathcal G}_x(\xi(t),y)|\delta(y)^{-\alpha-\beta}\leq c|\xi(t)-y|^{-N-\alpha-\beta}\quad\text{ for all }y\in B_r(x_1) \text{ and } 0\leq t\leq 1.$$


Let $c_6=1/(1+c_0)$, where $c_0$ is the constant from Lemma
\ref{greentool} that depends only on $\Omega$ and not on
$x_1,x_2$. Then for all $y\in\o\setminus B_r(x_1)$,
$$\begin{aligned}
\di |\xi(t)-y| & \di \geq |x_1-y|-|\xi(t)-x_1|=
|x_1-y|-|\xi(t)-\xi(0)|\\
\di  & \di \geq |x_1-y|-t|\xi'(c_t)|\geq |x_1-y|-c_0|x_1-x_2|\\
\di & \di \geq c_6|x_1-y|.\\
\end{aligned}$$
Combining the last two estimates we obtain
$$\di |{\mathcal G}_x(\xi(t),y)|d^{-\alpha-\beta}(y)\leq c_7|x_1-y|^{-N-\alpha-\beta}
\quad\text{ for all }y\in B_r(x_1) \text{ and } 0\leq t\leq 1.$$

Hence, we may write
\neweq{evalIIconv}
\begin{aligned}
II&\leq c_7|x_1-x_2|\int_{\o\setminus B_r(x_1)}|x_1-y|^{-N-\alpha-\beta}dy\\
&\leq c_7|x_1-x_2|\int_r^{\infty}t^{-1-\alpha-\beta}dt\leq c_8|x_1-x_2|r^{-\alpha-\beta}\\
&\leq c_{9}|x_1-x_2|^{1-\alpha-\beta}.\\
\end{aligned}
\endeq
The conclusion follows now from \eq{evalnablac}, \eq{evalIconv},
and \eq{evalIIconv}. The proof of Theorem \ref{th3} is now
complete. \qed





\section{Proof of Theorem \ref{t4}}


Let $\la^*=\la_1/m$. We first prove that $(P_\la)^-$ has no
classical solutions for $\la\geq \la^*$. Indeed, assume that there
exists $\la\geq \la^*$ such that $(P_\la)^-$ has a solution $u$
and let $c=\max_{\oo}\delta(x)^{-\alpha} u(x)^{-\beta}$. Since the
mapping $t\longmapsto f(x,t)/t$ is nonincreasing, it follows that
$\la f(x,u)\geq \la_1u$ in $\o$. Then, $u$ satisfies $-\Delta
u\geq c+\la_1 u$ in $\o$. Multiplying by $\vp$ in the last
inequality and integrating by parts, we have
$$\di \la_1\int_{\o} u\vp dx\geq \int_{\o} (c+\la_1 u)\vp dx,$$
which is clearly a contradiction.

Let $0<\la<\la^*$. Since $\alpha+\beta<1$, by \cite[Theorem
2]{crg} there exists $v\in C^2(\o)\cap C(\oo)$ such that
\neweq{pps}
\left\{
\begin{aligned}
-&\Delta v=v^{-\alpha-\beta}+\la f(v)&&\quad\mbox{ in }\o,\\
&v>0&&\quad\mbox{ in }\o,\\
&v=0&&\quad\mbox{ on }\partial\o,
\end{aligned}\right.
\endeq
and $c_1\delta (x)\leq v\leq c_2 \delta(x)$ in $\o$, for some $c_1,c_2>0$.
Therefore, $v$ satisfies
\neweq{edde}
-\Delta v\geq c\delta(x)^{-\alpha}v^{-\beta}+\la f(v)\quad\mbox{ in }\o,
\endeq
where $c>0$. Let now $M>1$ large such that $M^{1+\beta}c>1$. Since $Mf(v)\geq f(Mv)$ in $\o$,
from \eq{edde} we deduce that $\overline u_\la:=Mv$ satisfies
$$-\Delta \overline u_\la\geq \delta(x)^{-\alpha}\overline u_\la^{-\beta}+\la f(\overline u_\la)\quad\mbox{ in }\o.$$
Hence $\overline u_\la$ is a super-solution of $(P_\la)^-$ and
obviously $\overline u_\la\geq \underline u_\la$ in $\o$. Thus,
$(P_\la)^-$ has at least one solution $u_\la$. Furthermore, from
the last inequality we also have $u_\la\geq c \delta(x)$ in $\o$,
for some $c>0$.  This implies that $\Delta u_\la\in L^1(\o)$ and
by Lemma \ref{l1} we obtain the uniqueness. The regularity in (i)
follows exactly in the same way as in Theorem \ref{th3}. We now
prove (ii).

Remark that $\{u_\la\}_{0<\la<\la^*}$ is a sequence of positive
increasing super-harmonic functions. By \cite[Theorem
3.7.3]{gardiner} it follows that
$$\di u^*(x):=\lim_{\la\nearrow \la^*}u_\la(x)\quad x\in\Omega,$$
is either identically $\infty$ or a superharmonic function. Assume
by contradiction that $u^*\not\equiv \infty$, so that $u^*$ is
superharmonic. Hence, $u^*\in L_{\rm loc}^1(\o)$ (see, e.g.,
Theorem 3.1.3 in \cite{gardiner}).



We claim that $\{u_{\la}\}_{0<\la<\la^*}$ is bounded in
$L^2(\Omega)$. We argue by contradiction.  Thus, passing
eventually at a subsequence we have $u_{\la}=k(\la)w_{\la},$ where
\begin{equation} \label{cdoi} k(\la)=\|u_{\la}\|_{2}\ri\infty
\mbox{ as } \la\nearrow\la^* \mbox{ and  } w_{\la}\in
L^2(\Omega),\;\; \|w_{\la}\|_{2}=1.
\end{equation}

Notice that $f(x,t)\leq at+b$ for all $(x,t)\in \oo\times
(0,\infty)$, where $a,b>0$. This implies
$$ \frac{1}{k(\la)}(\delta(x)^{-\alpha}u_\la^{-\beta}+\la f(x,u_\la)) \ri 0 \quad \text{ in } L^1_{\rm
loc}(\Omega)\ \mbox{as }\la\nearrow \la^*.$$ That is,
\begin{equation} \label{cpatru} -\Delta
w_{\la}\ri 0 \quad\mbox{in } L^1_{\rm loc}(\Omega)\ \mbox{as
}\la\nearrow \la^*.\end{equation} By Green's first identity,
\index{Green!identity} for all $\phi\in C_0^\infty(\Omega)$ we
have
\begin{equation} \label{car1}
\int_\Omega \nabla w_\lambda\cdot \nabla \phi \, dx=-\int_\Omega
\phi\, \Delta w_\lambda dx=-\int_{ {\rm Supp}\, \phi} \phi\,
\Delta w_\lambda dx.
\end{equation}
Using (\ref{cpatru}) we obtain
\begin{equation} \label{car2}
\begin{aligned}
\left|\int_{{\rm Supp}\,\phi} \phi\, \Delta w_\la\,dx\right| &
\leq
\int_{{\rm Supp}\,\phi}|\phi||\Delta w_\lambda|\,dx\\
& \leq \|\phi\|_{\infty}\int_{{\rm Supp}\,\phi}|\Delta
w_\lambda|\,dx\to 0\quad \mbox{as }\la\nearrow \la^*.
\end{aligned}
\end{equation}
Combining (\ref{car1}) and (\ref{car2}) we derive that for all
$\phi\in C_0^\infty(\Omega)$ we have
\begin{equation} \label{car3}
\int_{\Omega} \nabla w_\lambda\cdot \nabla \phi \,dx \to 0 \quad
\mbox{ as }\la\nearrow \la^*.
\end{equation} By definition, the sequence
$(w_{\la})_{0<\la<\la^*}$ is bounded in $L^2(\Omega)$. We claim
that $(w_{\la})_{0<\la<\la^*}$ is bounded in $H^1_0(\Omega)$.
Indeed, using Sobolev-Hardy and H\"older inequality, we have
$$\begin{aligned}
\int_{\Omega}|\nabla w_{\la}|^2dx& =-\int_{\Omega}w_{\la}\Delta
w_{\la}dx\\
&=\frac{1}{k(\la)}\int_{\Omega}w_{\la}(\delta(x)^{-\alpha}u_\la^{-\beta}+\la
f(x,u_\la))dx\\
&=\frac{1}{k(\la)^{1+\beta}}\int_{\Omega}\frac{w_\la^{1-\beta}}{\delta(x)^{\alpha}}dx+
a\la\int_{\o} w_\la^2 dx+\frac{\la}{k(\la)}\int_{\o} w_\la dx\\
&\leq \frac{1}{k(\la)^{1+\beta}}\left(\int_{\Omega}|\nabla
w_\la|^{2}dx\right)^{(1-\beta)/2}+
a\la^*+\frac{\la^*}{k(\la)}|\o|^{1/2}.\\
\end{aligned} $$
 From the above
estimates it follows that $\{w_{\la}\}_{0<\la<\la^*}$ is bounded
in $H^1_0(\Omega)$, so the claim follows. Thus, there exists $w\in
H^1_0(\Omega)$ such that up to a subsequence and as $\la\nearrow
\la^*$ we have
\neweq{ccinci}
\begin{aligned}
w_\la& \weak w&&  \text{ weakly in }
H_0^1(\Omega),\\
\di w_\la &\to w&& \text{ strongly in } L^2(\Omega).
\end{aligned}\endeq
On the one hand, by (\ref{cdoi}) and \eq{ccinci}, we derive that
$\|w\|_{2}=1$. Furthermore, using \eq{car3} and \eq{ccinci}, we
infer that
$$ \int_{\Omega} \nabla w\cdot \nabla \phi \,dx=0\quad \mbox{ for
all } \phi\in C^\infty_0 (\Omega). $$ Because $w\in
H_0^1(\Omega)$, using the previous relation and the definition of
$H_0^1(\Omega)$, we find $w=0$. This contradiction shows that
$\{u_{\la}\}_{0<\la<\la^*}$ is bounded in $L^2(\Omega)$. As noted
earlier for $\{w_{\la}\}_{0<\la<\la^*}$, we derive that
$\{u_{\la}\}_{0<\la<\la^*}$ is bounded in $H_0^1(\Omega)$. Hence,
there exists $u^*\in H_0^1(\Omega)$ such that, up to a subsequence
and as $\la\nearrow \la^*$, there holds
\neweq{car4}
\begin{aligned} u_\la& \weak u^*&&  \text{ weakly in }
H_0^1(\Omega),\\
\di u_\la &\to u^*&&\text{ strongly in } L^2(\Omega),\\
u_\la& \to u^*&& \text{ almost everywhere in }\Omega.
\end{aligned}
\endeq

Now we obtain the desired contradiction in the same manner as in
the proof of (i)$-$(ii). By $(f1)$ we have $f(x,u_\la)\geq m
u_\la$ in $\o$. We next multiply by $\vp$ in $(P_\la)^-$ and then
we integrate over $\o$. We obtain
\neweq{copt}
\di \la_1\int_{\Omega}u_{\la}\varphi_1
dx=\int_{\Omega}(\delta(x)^{-\alpha}u_\la^{-\beta}+\la
f(u_{\la}))\varphi_1 dx \geq \int_{\Omega}(\delta(x)^{-\alpha}
u_{\la}^{-\beta}+\la m u_\la)\varphi_1 dx,
\endeq
for all $0<\la<\la^*$. Passing to the limit in \eq{copt} with
$\la\nearrow\la^*,$ by virtue of Lebesgue's theorem on dominated
convergence we find
$$\di \la_1\int_{\Omega}u^*\varphi_1=
\int_{\Omega}(\delta(x)^{-\alpha} {u^*}^{-\beta}+\la_1
u^*)\varphi_1 dx,$$  which is a contradiction. Hence $u^*\equiv
\infty$. It is easy now to see, using the monotonicity of
$\{u_\la\}_{0<\la<\la^*}$ that $ \lim_{\la\nearrow
\la^*}u_{\la}=\infty$ uniformly on compact subsets of $\Omega$,
and the proof is now complete. \qed



\medskip



\noindent {\bf Acknowledgments.} The authors are partially
supported by Grants 2-CEx06-11- 18/2006 and CNCSIS PN II-79/2007
{\it Procese Neliniare Degenerate \c si Singulare}.









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