# Estimates of Nonnegative Solutions to Semilinear Elliptic Equations††thanks:
This work is supported by the Research Laboratory in Mathematics:
Deterministic and Stochastic Modeling - LAMMDA. Code: LR16ES13.

Khalifa El Mabrouk E-mail address<EMAIL_ADDRESS>Basma
Nayli Corresponding author<EMAIL_ADDRESS>
###### Abstract

Let $L$ be a second order uniformly elliptic differential operator in a domain
$D$ of $\mathbb{R}^{d}$, $\psi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a
nondecreasing continuous function and let $\xi,g:D\to\mathbb{R}_{+}$ be
locally bounded Borel measurable functions. Under appropriate conditions, we
determine a function $\varphi$ with values in $]0,1]$ such that for every
nonnegative solution to inequality $-Lu+\xi\psi(u)\geq g$ in $D$ and for every
$x\in D$,

$u(x)\geq p(x)\,\varphi\left(\frac{G_{D}(\xi\psi(p))(x)}{p(x)}\right)$

where $p=G_{D}g$ is the Green function of $g$. The function $\varphi$ is
completely determined by $\psi$ and does not depend on $L,D,\xi$ or $g$.

Keywords: Green function, Semilinear Dirichlet problem, $L-$harmonic function.

Mathematics Subject Classification: 35J61, 35B45, 31B10.

##  1 Introduction

Consider a domain $D$ of $\mathbb{R}^{d},d\geq 1,$ and a second order
uniformly elliptic differential operator

$Lu=\sum_{i,j=1}^{d}a_{ij}\partial_{ij}u+\sum_{i=1}^{d}b_{i}\partial_{i}u,$

where coefficients $a_{ij}$ and $b_{i}$ are sufficiently smooth. We are
concerned with nonnegative continuous solutions, in the distributional sense,
to inequalities of type

$-Lu+\xi\psi(u)\geq g\quad\mbox{in }D,$ (1.1)

where $g,\xi:D\rightarrow\mathbb{R}_{+}$ are locally bounded Borel measurable
functions and $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a
nondecreasing continuous function such that $\psi(0)=0$. If $D$ is bounded and
regular, it is well known that, for every nonnegative continuous function $f$
on $\partial D$, the Poisson-Dirichlet problem

$\left\\{\begin{array}[]{rcll}-Ls&=&g&\mbox{in }D,\\\ s&=&f&\mbox{on }\partial
D.\end{array}\right.$ (1.2)

has a unique solution $s=S_{D}(f,g)$. Observe that $H_{D}f=S_{D}(f,0)$ is the
$L-$harmonic extension of $f$ to $\overline{D}$ and $G_{D}g=S_{D}(0,g)$ is the
Green potential of $g$.

In this paper, it will first be shown that for every nonnegative continuous
function $f$ on $\partial D$, the semilinear boundary value problem

$\left\\{\begin{array}[]{rcll}-Lu+\xi\psi(u)&=&g&\mbox{in }D,\\\
u&=&f&\mbox{on }\partial D,\end{array}\right.$ (1.3)

admits one and only one solution which will be denoted by $U_{D}^{\xi}(f,g)$.
Besides, it is simple to see that

$U_{D}^{\xi}(f,g)\leq S_{D}(f,g)\quad\mbox{in }D.$ (1.4)

Our main goal is to establish a lower bound estimate of $U_{D}^{\xi}(f,g)$ as
a function of $S_{D}(f,g)$, which especially allows us to conclude lower bound
estimates of any nonnegative solution to inequality (1.1).

It should be noted that semilinear problems of type (1.3) play a vital role in
the modeling of many phenomena in the fields of physics, astrophysics,
logistics and differential geometry (see [18, 7]). We also notice that (1.3)
has been investigated in many papers, under various aspects, which deal with
existence and uniqueness as well as the estimation of its solution (see for
instance [8, 20, 2, 5, 4, 14, 9, 15] and references therein).

Taking $\xi$ a positive constant in $D$ ($\xi=2$), Ben Fredj and the first
author studied problem (1.3) in [2] when $L=\Delta$ is the Laplacian operator
and $g$ is the null function. They proved that for every nonnegative
continuous function $f$ on $\partial D$, (1.3) has a unique solution $u$ and
there exists a real constant $\kappa>0$ such that for all $x\in D$,

$\kappa H_{D}f(x)\leq u(x)\leq H_{D}f(x).$ (1.5)

The case of $\xi\leq 0$ is recently investigated by Grigor’yan and Verbitsky
in [12] where they gave a real positive function $\varrho$, depending only on
$\psi$, such that if $S_{D}(f,g)\geq 1$ in $D$ then every nonnegative solution
$u$ to problem (1.3) satisfies for all $x\in D$,

$u(x)\geq\varrho\left(G_{D}\xi\right(x)).$ (1.6)

In this context, Mâagli and Zribi [17] have already treated the special
setting where $L=\Delta$ and $f,g$ are identically zero.

For signed functions $\xi$, Grigor’yan and Verbitsky obtained in [11] lower
bounded estimates for nonnegative solutions to

$-Lu+\xi u^{\gamma}\geq g\mbox{ in }D,$ (1.7)

which are twice continuously differentiable in $D$.

In the current paper, we deal with nonnegative solutions to inequality (1.1)
in the distributional sense. We only consider functions $\xi$ that are
nonnegative everywhere, however $D$ can be any Greenian domain of
$\mathbb{R}^{d}$, not necessary bounded, and $\psi$ belongs to a large class
of nondecreasing functions which contains in addition to polynomial functions,
logarithmic functions as well as exponential functions.

We further assume that $\psi$ is continuously differentiable in $]0,\infty[$
and there exists $c>0$ such that

$\psi(rt)\leq c\psi(r)\psi(t)\quad\mbox{for all
}r\in[0,1],t\in\mathbb{R}_{+},$ (1.8)

and consider the function $\Theta:]0,1]\rightarrow[0,\ell[$ ($\ell$ may be
infinite) given, for every $t\in]0,1]$ by

$\Theta(t)=\int_{t}^{1}\frac{1}{c\,\psi(s)}\,ds\quad\mbox{where}\quad\ell=\int_{0}^{1}\frac{1}{c\,\psi(s)}\,ds.$

We obviously see that $\Theta$ admits an inverse function which will be
denoted by $\varphi$ and extended to $\mathbb{R}_{+}$ by setting
$\varphi(t)=0$ if $t\geq\ell$. Let $p=G_{D}g$ and assume that $0<p<\infty$ in
$D$. We will prove that every nonnegative solutions $u$ to (1.1) satisfies,
for all $x\in D$,

$u(x)\geq p(x)\varphi\left(\frac{G_{D}(\xi\psi(p))(x)}{p(x)}\right).$ (1.9)

To do this, we will go through lower bounded estimates of solution
$U_{D}^{\xi}(f,g)$ to the boundary value problem (1.3). In this regard, we had
to carry out a detailed study of $L$-harmonic functions in order to show, when
$D$ is bounded regular, that for any continuous function $f:\partial
D\to\mathbb{R}_{+}$ such that $s=S_{D}(f,g)$ is finite not identically zero
and $G_{D}(\xi\psi(s))$ is finite in $D$, the integral equation

$u+G_{D}\big{(}\xi\psi(u)\big{)}=s\quad\mbox{in }D,$ (1.10)

possesses a unique solution $u$ and that for every $x\in D$,

$s(x)\varphi\Big{(}\frac{G_{D}(\xi\psi(s))(x)}{s(x)}\Big{)}\leq u(x)\leq
s(x).$ (1.11)

Notice that in the setting where Green potentials $G_{D}\xi$ and $G_{D}g$ are
continuous in $\overline{D}$ vanishing on $\partial D$, $u$ is a solution to
(1.10) if an only if $u$ is a solution to problem (1.3).

For every $n\geq 1$, we define $v_{n}=U_{D_{n}}^{\xi}(0,g)$ where
$(D_{n})_{n\geq 1}$ is a sequence of bounded regular domains such that for
every $n\geq 1$,

$\overline{D_{n}}\subset D_{n+1}\quad\mbox{and}\quad\cup_{n\geq 1}D_{n}=D.$

Define $p_{n}=G_{D_{n}}g$ and choose a subsequence $(p_{n_{k}})_{k\geq 1}$ of
$(p_{n})_{n\geq 1}$ such that $p_{n_{k}}>0$ for all $k\geq 1$. We then apply
(1.11) to $v_{n_{k}}$ in $D_{n_{k}}$, to obtain that for all $k\geq 1$,

$v_{n_{k}}\geq
p_{n_{k}}\varphi\left(\frac{G_{D_{n_{k}}}(\xi\psi(p_{n_{k}}))}{p_{n_{k}}}\right).$
(1.12)

Given a solution $u$ to inequality (1.1), it follows from a comparison
principal that $u\geq v_{n_{k}}$ in $D_{n_{k}}$, which in turn allows us to
get estimate (1.9) in view of some convergence theorems and monotonicities of
functions $\psi$ and $\varphi$.

Estimates of type (1.9) encompass results given by Grigor’yan and Verbitsky
[11] for $\psi(t)=t^{\gamma}$ with $\gamma>0$. In fact, in this particular
case, simple computation of $\varphi$ yields that every nonnegative solution
$u$ to (1.1) satisfies, for all $x\in D$,

$u(x)\geq\left\\{\begin{array}[]{ll}p(x)\left(1+(\gamma-1)\frac{G_{D}\left(p^{\gamma}\xi\right)(x)}{p(x)}\right)_{+}^{1/(1-\gamma)}&\mbox{if
}\gamma<1,\\\ p(x)\exp\left({-\frac{G_{D}(\xi p)(x)}{p(x)}}\right)&\mbox{if
}\gamma=1,\\\
p(x)\left(1+(\gamma-1)\frac{G_{D}(p^{\gamma}\xi)(x)}{p(x)}\right)^{1/(1-\gamma)}&\mbox{if
}\gamma>1.\end{array}\right.$ (1.13)

Our approach is still valid for various functions $\psi$ such as

$\psi(t)=at+bt^{\gamma},\quad\psi(t)=\sinh
t\quad\mbox{and}\quad\psi(t)=a(1+bt)\log(1+bt)$

where $a,b,\gamma>0$. These examples are presented and discussed in detail in
the last section, where main results are stated and proved. In section 2, we
introduce some basic notions that are useful to study the Dirichlet problem
(1.2). After treating fundamentals properties of the Green potential in
section 3, we study in section 4 the semilinear problem (1.3) in regular
domains. We prove the existence and uniqueness of the nonnegative continuous
solution to problem (1.3) and investigate its regularity. In section 5, we are
concerned with estimates of the solution to (1.3) as well as estimates of
nonnegative solutions of the inequality (1.1).

##  2 $L-$harmonic functions

For every domain $D$ in $\mathbb{R}^{d}$, $d\geq 1$, let $\mathcal{B}(D)$ be
the set of all real-valued Borel measurable functions in $D$ and let
$\mathcal{B}_{b}(D)$ (resp. $\mathcal{B}^{+}(D)$) be the set of all functions
in $\mathcal{B}(D)$ which are bounded (resp. nonnegative) in $D$. More
generally, given any class of numerical functions $\mathcal{F}$, we will
denote by $\mathcal{F}_{b}$ (resp. $\mathcal{F}^{+}$) the set of all functions
in $\mathcal{F}$ which are bounded (resp. nonnegative). We define
$\mathcal{C}(D)$ to be the set of all real-valued continuous functions in $D$
and $\mathcal{C}_{c}^{\infty}(D)$ to be the set of all infinity differentiable
functions with compact support in $D$. For every $1\leq i,j\leq d$,
$\partial_{i}$ stands for the first partial derivative with respect to the
coordinate $x_{i}$ of $x=(x_{1},....,x_{d})\in\mathbb{R}^{d}$, and
$\partial_{ij}:=\partial_{i}\partial_{j}$. Let $\mathcal{C}^{1}(D)$ (resp.
$\mathcal{C}^{2}(D)$) be the set of all functions $u\in\mathcal{C}(D)$ such
that $\partial_{i}u\in\mathcal{C}(D)$ (resp.
$\partial_{i}u,\partial_{ij}u\in\mathcal{C}(D)$) for every $1\leq i,j\leq d$.
If $D$ is bounded, we denote by $\mathcal{C}_{0}(D)$ the set of all functions
$u\in\mathcal{C}(D)$ such that $\lim_{x\in D,x\rightarrow z}u(x)=0$ for all
$z\in\partial D$.

Let $D$ be a domain in ${\mathbb{R}}^{d}$ and let $0<\alpha\leq 1$. A real-
valued function $u$ is called $\alpha-$Hölder continuous in $\overline{D}$ if
there exists a real constant $C>0$ such that

$|u(x)-u(y)|\leq C|x-y|^{\alpha}\quad\mbox{for all }x,y\in\overline{D}.$

The set of all $\alpha-$Hölder continuous functions in $\overline{D}$ is
denoted by $\mathcal{C}^{\alpha}(\overline{D})$. We say that $u$ is locally
$\alpha-$Hölder continuous in the domain $D$ and we write
$u\in\mathcal{C}^{\alpha}(D)$ if $u\in\mathcal{C}^{\alpha}(\overline{V})$ for
all domains $V\Subset D$. Here and in the sequel, writing $V\Subset D$ means
that $\overline{V}$ is compact with closure $\overline{V}\subset D$. We also
define $\mathcal{C}^{1,\alpha}(D)$ (resp. $\mathcal{C}^{2,\alpha}(D)$) to be
the class of all functions $u\in\mathcal{C}^{\alpha}(D)$ such that
$\partial_{i}u\in\mathcal{C}^{\alpha}(D)$ (resp.
$\partial_{i}u,\partial_{ij}u\in\mathcal{C}^{\alpha}(D)$) for all $1\leq
i,j\leq d$.

We consider a second order differential operator of type

$Lu:=\sum\limits_{i,j=1}^{d}a_{ij}\partial_{ij}u+\sum\limits_{i=1}^{d}b_{i}\partial_{i}u$
(2.1)

where $a_{ij}\in\mathcal{C}^{2,\alpha}(\mathbb{R}^{d})$ and
$b_{i}\in\mathcal{C}^{1,\alpha}(\mathbb{R}^{d})$ for all $1\leq i,j\leq d$ and
we assume that $L$ is uniformly elliptic, that is, there exists a real
constant $\beta>0$ such that for all $x\in\mathbb{R}^{d}$ and for all
$\xi=(\xi_{1},\cdots,\xi_{d})\in\mathbb{R}^{d}$,

$\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\geq\beta\,|\xi|^{2}.$

Given a Radon measure $\mu$ in $D$ and a locally Lebesgue integrable function
$u$ in the open set $D$, equation

$Lu=\mu\mbox{ in }D$ (2.2)

should be understood in the distributional sense, that is, for all
$\varphi\in\mathcal{C}_{c}^{\infty}(D)$,

$\int_{D}uL^{*}\varphi(x)\,dx=\int_{D}\varphi(x)\,d\mu(x)$ (2.3)

where $L^{*}$ denotes the adjoin operator of $L$ given by

$L^{*}u:=\sum\limits_{i,j=1}^{d}\partial_{ij}(a_{ij}u)-\sum_{i=1}^{d}\partial_{i}(b_{i}u).$
(2.4)

We call $L-$harmonic in $D$ any continuous function $u:D\to\mathbb{R}$ such
that $Lu=0$ in $D$ and define $\mathcal{H}_{L}(D)$ to be the set of all
$L-$harmonic functions in $D$. We also assume that $L$ is hypoelliptic in the
sense that $\mathcal{H}_{L}(D)\subset\mathcal{C}^{2}(D)$.

An open bounded set $D\subset\mathbb{R}^{d}$ is called regular if for each
continuous function $~{}{f:\partial D\rightarrow\mathbb{R}}$, the Dirichlet
problem

$\left\\{\begin{array}[]{rl}Lh=0&\mbox{ in }D,\\\ h=f&\mbox{ on }\partial
D,\end{array}\right.$

has a unique solution which will be denoted by $H_{D}f.$ In this setting, for
every $x\in D$, the map $f\mapsto H_{D}f(x)$ defines a probability measure
$H_{D}(x,\cdot)$ on $\partial D$ which is called harmonic measure relative to
$x$ and $D$. For $x\in\mathbb{R}^{d}\backslash D$, it will be convenient to
put $H_{D}(x,\cdot):=\delta_{x}$ where $\delta_{x}$ denotes the Dirac measure
concentrated at the point $x$. It is well known (see [10, p.106]) that $D$ is
regular provided it satisfies the exterior sphere condition at every boundary
point, that is, for every $z\in\partial D$ there exists a ball $B$ of
$\mathbb{R}^{d}$ such that $B\cap D=\emptyset$ and $z\in\partial B$. This
immediately yields that every ball $B\subset\mathbb{R}^{d}$ is regular and
consequently a continuous function $u$ is $L-$harmonic in $D$ if and only if
$H_{B}u=u$ for all balls $B\Subset D$.

Due to J. Serrin [21], nonnegative $L-$harmonic functions in the domain $D$
satisfy the following Harnack’s inequality: For every compact set
$\Gamma\subset D$, there exists a real constant $C\geq 1$ (depending only on
$L$, $\Gamma$ and $D$) such that for all $u\in\mathcal{H}_{L}^{+}(D)$,

$\sup_{x\in\Gamma}u(x)\leq C\inf_{x\in\Gamma}u(x).$ (2.5)

######  Theorem 2.1.

Let $D\subset\mathbb{R}^{d}$ be a domain and let $(h_{n})_{n\geq 1}$ be a
nondecreasing (or nonincreasing) sequence of $L-$harmonic functions in $D$
such that $h:=\lim\limits_{n\rightarrow\infty}h_{n}$ is finite in some point
$x_{0}\in D$. Then $h$ is $L-$harmonic in $D$.

###### Proof.

Let $\Gamma$ be a compact subset of $D$ containing $x_{0}$ (otherwise, take
$\Gamma^{\prime}=\Gamma\cup\\{x_{0}\\}$) and let $\varepsilon>0$. It follows
from the Harnack’s inequality (2.5) and the fact that $h_{n}(x_{0})$ converges
to $h(x_{0})$ that, there exist $n_{0}\geq 1$ and a real constant $C>0$ such
that for every $x\in\Gamma$ and for every $m\geq n\geq n_{0}$, we have

$0\leq h_{m}(x)-h_{n}(x)\leq C\big{(}h_{m}(x_{0})-h_{n}(x_{0})\big{)}\leq
C\big{(}h(x_{0})-h_{n}(x_{0})\big{)}\leq C\varepsilon.$

Letting $m$ tend to $\infty$, we get that for every $x\in\Gamma$ and for every
$n\geq n_{0}$,

$0\leq h(x)-h_{n}(x)\leq C\varepsilon.$

We then conclude that the sequence $(h_{n})_{n\geq 1}$ converges to $h$
locally uniformly in $D$. In particular, $h$ is finite and continuous in $D$.
Now, for every $B\Subset D$ and every $x\in B$,

$H_{B}h_{n}(x)=\int_{\partial
B}h_{n}(y)H_{B}(x,dy)\underset{n\to\infty}{\longrightarrow}\int_{\partial
B}h(y)H_{B}(x,dy)=H_{B}h(x).$

Since $H_{B}h_{n}=h_{n}$ for every $n\geq 1$ and $h_{n}(x)$ converges to
$h(x)$, we deduce that $H_{B}h=h$ for every ball $B\Subset D$. This yields
that $h$ is $L-$harmonic in $D$. ∎

######  Proposition 2.2.

Let $D\subset\mathbb{R}^{d}$ be a regular set and let $f:\partial
D\to\overline{\mathbb{R}}_{+}$ be a Borel measurable function. Then, for every
connected component $E$ of $D$, $H_{D}f$ is either $L-$harmonic or identically
infinite in $E$.

###### Proof.

Without loss of generality, we assume that $D$ is connected. In virtue of the
previous theorem, we see that $\mathcal{A}:=\\{\Gamma\subset\partial
D:H_{D}(\cdot,\Gamma)\in\mathcal{H}_{L}(D)\\}$ is a monotone class on
$\partial D$. Furthermore, $\mathcal{A}$ contains the set $\mathcal{O}$ of all
open subsets of $\partial D$. In fact, let $O\in\mathcal{O}$ and consider

$f_{n}:x\mapsto f_{n}(x)=\inf(1,n\,d(x,O^{c}))$

where $d(x,O^{c})=\inf_{y\in O^{c}}|x-y|$ denotes the distance between $x$ and
$O^{c}:=\mathbb{R}^{d}\backslash O$. Then $(f_{n})_{n\geq 1}$ is a
nondecreasing sequence in $\mathcal{C}^{+}(\partial D)$ which converges to
$\textbf{1}_{O}$ and hence $(H_{D}f_{n})_{n\geq 1}$ is a nondecreasing
sequence of $L-$harmonic functions in $D$ which converges to $H_{D}(\cdot,O)$
in view of the monotone convergence theorem. We then conclude by Theorem 2.1,
that $H_{D}(\cdot,O)$ is $L-$harmonic and thereby
$\mathcal{O}\subset\mathcal{A}$. Therefore, it follows from the monotone class
lemma that $\mathcal{A}$ contains $\mathcal{B}(\partial D)$. This yields that
$H_{D}f\in\mathcal{H}_{L}(D)$ whenever

$f=\sum\limits_{i=1}^{k}\gamma_{i}\textbf{1}_{A_{i}}$ (2.6)

with $k\geq 1$, $\gamma_{1},...,\gamma_{k}\in\mathbb{R}_{+}$ and
$A_{1},...,A_{k}\in\mathcal{B}(\partial D)$. Finally, let $f:\partial
D\to\overline{\mathbb{R}}_{+}$ be a Borel measurable function such that
$H_{D}f\not\equiv\infty$ and choose a nondecreasing sequence $(f_{n})_{n\geq
1}$ of nonnegative functions of the form (2.6) which converges pointwise to
$f$. Then $(H_{D}f_{n})_{n\geq 1}$ is a nondecreasing sequence of $L-$harmonic
functions in $D$ pointwise converging to $H_{D}f$ by the monotone convergence
theorem. Thus, $H_{D}f\in\mathcal{H}_{L}(D)$ in virtue of Theorem 2.1. ∎

Let us point out that the previous result can be extended to any Borel
measurable function $f:\partial D\rightarrow\overline{\mathbb{R}}$ without
having to assume that $f$ is nonnegative in $\partial D$. More precisely, if
$D$ is connected and $f$ is $H_{D}(x_{0},\cdot)-$integrable on $\partial D$
for some $x_{0}\in D$ then $f$ is $H_{D}(x,\cdot)-$integrable on $\partial D$
for all $x\in D$ and the function $H_{D}f$ is $L-$harmonic in $D$.

We recall that, due to the hypoellipticity of $L$ and [10, Theorem 3.5], a non
constant function $h\in~{}\mathcal{H}_{L}(D)$ can not achieve its minimum (or
maximum) in the interior of the domain $D$. This yields that every
$L-$harmonic function $h$ in an open set $D$ is necessarily nonnegative
provided $\liminf_{x\in D,x\to z}h(x)~{}\geq~{}0$ for every $z\in\partial D$.

A lower semicontinuous function $v>-\infty$ is said to be $L-$superharmonic in
$D$ if for every regular set $U\Subset D$, $H_{U}v\in\mathcal{H}_{L}(U)$ and
$H_{U}v\leq v$ in $U$. The set of all $L-$superharmonic functions in $D$ will
be denoted by $\mathcal{S}_{L}(D)$. We will say that $v$ satisfies the
property $(\mathcal{M}_{*})$ if for every $x\in D$, there exists $r=r(x)>0$
such that $B(x,r)\subset D$ and for all $0<\rho<r$,

$\int_{\partial B(x,\rho)}v(z)H_{B(x,\rho)}(x,dz)\leq v(x).$ (2.7)

It is obvious that every $L-$superharmonic function satisfies property
$(\mathcal{M}_{*})$. The converse will be shown later.

######  Proposition 2.3.

Let $D\subset\mathbb{R}^{d}$ be an open set and let $v:D\to]-\infty,\infty]$
be a lower semicontinuous function which satisfies the property
$(\mathcal{M}_{*})$. The following holds true:

  1. a)

If $v$ attains its minimum in a connected component $E$ of $D$, then $v$ is
constant in $E$.

  2. b)

If $\liminf_{x\in D,\;x\to z}v(x)\geq 0$ for every $z\in\partial D$, then
$v\geq 0$ in $D$.

###### Proof.

Without loss of generality, we consider the case where $D$ is connected and
$v$ is not identically infinite in $D$. Assume that there exists $x_{0}\in D$
such that $\min_{x\in D}v(x)=v(x_{0})$. By $(\mathcal{M}_{*})$, there exists
$r_{0}>0$ such that $B(x_{0},r_{0})\subset D$ and for every ball
$B=B(x_{0},\rho)$ with $0<\rho<r_{0}$,

$H_{B}v(x_{0})=\int_{\partial B}v(z)H_{B}(x_{0},dz)\leq v(x_{0}).$

Furthermore, $H_{B}v\geq v(x_{0})$ in $B$ which yields that the $L-$harmonic
function $H_{B}v$ reaches its minimum at the point $x_{0}$ and thereby
$H_{B}v=v(x_{0})$ in $B$. On the other hand, since $v$ is lower semicontinuous
and bounded below by $v(x_{0})$, there exists (see for instance [1, Lemma
3.2.1]) a nondecreasing sequence $(f_{n})_{n\geq 1}$ of continuous functions
in $\partial B$ such that $\lim_{n\to\infty}f_{n}(z)=v(z)$ for all
$z\in\partial B$. Then $H_{B}f_{n}\leq H_{B}v=v(x_{0})$ in $B$ for every
$n\geq 1$. Hence, for a given point $z\in\partial B$,

$f_{n}(z)=\lim_{x\in B,\,x\rightarrow z}H_{B}f_{n}(x)\leq
v(x_{0})\quad\mbox{for all }n\geq 1.$ (2.8)

The equality in the formula above follows from the fact that $f_{n}$ is
continuous in $\partial B$ for all $n\geq 1$ and the ball $B$ is regular.
Letting $n$ tend to $\infty$, we obtain that $v(z)\leq v(x_{0})$ for every
$z\in\partial B$ and so $v=v(x_{0})$ in $\partial B(x_{0},\rho)$. Since
$0<\rho<r_{0}$ is arbitrary, we conclude that $v(x)=v(x_{0})$ for all $x\in
B(x_{0},r_{0})$. Therefore, the set

$U=\\{x\in D:v(x)=v(x_{0})\\}$

is open. By the lower semicontinuity of $v$, we also see that $U$ is closed
relative to $D$. Consequently $U=D$ and the proof of assertion $(a)$ is
complete. In order to show $(b)$, let us assume that $\liminf_{x\in D,x\to
z}v(x)\geq 0$ for every $z\in\partial D$ but there exists $x_{0}\in D$ such
that $v(x_{0})<0$. We extend $v$ to $\overline{D}$ by defining

$v(z)=\liminf_{x\in D,x\to z}v(x)\mbox{ for all }z\in\partial D.$

It is clear that $v$ is lower semicontinuous in the compact set
$\overline{D}$. Then, there exists $y_{0}\in\overline{D}$ such that
$\min_{x\in\overline{D}}v(x)=v(y_{0})$. Since $v\geq 0$ in $\partial D$ and
$v(x_{0})<0$, we conclude that $y_{0}\in D$ and consequently $v$ achieves its
minimum in the interior of $D$. Using $(a)$ we get that $v(x)=v(x_{0})<0$ for
every $x\in D$ which yields a contradiction. ∎

Throughout this paper, $\lambda_{d}$ will denote the Lebesgue measure in
$\mathbb{R}^{d}$.

######  Proposition 2.4.

Let $D\subset\mathbb{R}^{d}$ be an open set and let $v:D\to]-\infty,\infty]$
be a lower semicontinuous function which is finite $\lambda_{d}-$a.e in $D$.
The following assertions are equivalent:

  1. a)

$v$ is $L-$superharmonic in $D$.

  2. b)

$H_{B}v\leq v$ for every ball $B\Subset D$.

  3. c)

$v$ satisfies property $(\mathcal{M}_{*})$ in $D$.

###### Proof.

We obviously see that $(a)$ yields $(b)$ which in turn implies $(c)$. To show
that $(c)$ yields $(a)$, we take a regular set $U\Subset D$ and choose a
nondecreasing sequence $(f_{n})_{n\geq 1}$ of continuous functions on
$\partial U$ such that $\lim_{n\to\infty}f_{n}(z)=v(z)$ for all $z\in\partial
U$. Then, for every $n\geq 1$, $w_{n}:=v-H_{U}f_{n}$ is lower semicontinuous,
$w_{n}>-\infty$ in $U$ and for all $z\in\partial U$,

$\liminf_{x\in U,x\to z}w_{n}(x)\geq v(z)-f_{n}(z)\geq 0.$

It is also immediate that $w_{n}$ satisfies $(\mathcal{M}_{*})$ in $U$.
Thereby, according to assertion $(b)$ in the previous proposition, we conclude
that $w_{n}\geq 0$ in $U$ for all $n\geq 1$ and consequently

$H_{U}v=\lim_{n\to\infty}H_{U}f_{n}\leq v\quad\mbox{in }U.$ (2.9)

Furthermore, the $L-$harmonicity of $H_{U}v$ follows from Proposition 2.2 and
the fact that $v$ is finite $\lambda_{d}-$a.e. Thus, $v$ is $L-$superharmonic
in $D$ if $(\mathcal{M}_{*})$ is satisfied. ∎

We end this section with the following convergence property of a monotone
sequences of $L-$superharmonic functions which follows from Theorem 2.1.

######  Proposition 2.5.

Let $D\subset\mathbb{R}^{d}$ be an open set and let $(v_{n})_{n\geq 1}$ be a
nondecreasing sequence of nonnegative $L-$superharmonic functions in $D$ such
that $v:=\sup_{n\geq 1}v_{n}<\infty$ $\lambda_{d}-$a.e. Then $v$ is
$L-$superharmonic in $D$.

##  3 Green operator

We call Green function of $L$ in the regular set $D\subset\mathbb{R}^{d}$, the
Borel measurable function $G_{D}:D\times D\to]0,\infty]$ having the following
properties for any $y\in D$:

  1. G1)

$G_{D}(\cdot,y)$ is finite and continuous in $D\backslash\\{y\\}$ and
$\lim_{x\in D,x\to y}G_{D}(x,y)=G_{D}(y,y)$.

  2. G2)

$G_{D}(\cdot,y)$ is locally $\lambda_{d}-$integrable in $D$ and
$LG_{D}(\cdot,y)=-\delta_{y}$.

  3. G3)

$\lim_{x\in D,x\rightarrow z}G_{D}(x,y)=0$ for all $z\in\partial D$.

The fact that coefficients $a_{ij},b_{i}$ are sufficiently smooth guarantees
the existence of the Green function $G_{D}$ (see for instance [19]). We also
notice that a simple application of the maximum principle in the domain
$D\backslash\\{y\\}$ yields that

$G_{D}(y,y)=\sup_{x\in D}G_{D}(x,y).$ (3.1)

######  Proposition 3.1.

Let $D\subset\mathbb{R}^{d}$ be a regular set, $y\in D$ and
$0<\beta<G_{D}(y,y)$. Then, the function $w:=G_{D}(\cdot,y)\wedge\beta$ is
$L-$superharmonic in $D$. In particular, $G_{D}(\cdot,y)$ is $L-$superharmonic
in $D$.

###### Proof.

In virtue of Proposition 2.4, it is sufficient to show that $w$ satisfies
$(\mathcal{M}_{*})$. Since $LG_{D}(\cdot,y)=-\delta_{y}$, we see that $w$ is
$L-$harmonic in $\\{G_{D}(\cdot,y)<\beta\\}$ and consequently (2.7) is valid
for all $x\in D$ such that $G_{D}(x,y)<\beta$. If $G_{D}(x,y)\geq\beta$,
inequality (2.7) is immediate with $r(x)=d(x,D^{c})$. Thus,
$w=G_{D}(\cdot,y)\wedge\beta$ is $L-$superharmonic in $D$ for every
$0<\beta<G_{D}(y,y)$. To prove the last part of the proposition, we consider
$\beta_{n}=G_{D}(y,y)\wedge n$ and $w_{n}=G_{D}(\cdot,y)\wedge\beta_{n}$ for
every $n\geq 1$. Therefore, Proposition 2.5 and (3.1) yield that
$G_{D}(\cdot,y)=\lim_{n\to\infty}w_{n}$ is $L-$superharmonic in $D$. ∎

We then conclude that $G_{D}(\cdot,y)$ is a potential with support $\\{y\\}$
in the sense of R.M. Hervé [13]. Hence, the uniqueness of the Green function
$G_{D}$ follows from Theorem 16.5 in [13] and property (G2) above. In
particular, if $D$ and $\Omega$ are regular sets such that $D\subset\Omega$,
then

$G_{D}(\cdot,y)=G_{\Omega}(\cdot,y)-H_{D}G_{\Omega}(\cdot,y)\quad\mbox{in }D$
(3.2)

which yields that $G_{D}(x,y)\leq G_{\Omega}(x,y)$ for all $x,y\in D$.

Given an arbitrary open set $\Omega\subset\mathbb{R}^{d}$, we define for every
$x,y\in\Omega$,

$G_{\Omega}(x,y):=\sup_{D\in\mathcal{R}(\Omega)}G_{D}(x,y),$ (3.3)

where $\mathcal{R}(\Omega)$ denotes the family of all regular sets
$D\Subset\Omega$.

######  Definition 3.2.

The open set $\Omega$ of $\mathbb{R}^{d}$ is called Greenian if for every
$y\in\Omega$, $G_{\Omega}(\cdot,y)$ is not identically infinite in the
connected component of $\Omega$ which contains $y$. In this setting,
$G_{\Omega}$ is called Green function of $L$ in $\Omega$.

If $\Omega$ is regular then $\Omega\in\mathcal{R}(\Omega)$ and $G_{\Omega}$ is
just the Green function of $L$ in $\Omega$ given at the beginning of this
section. Let us also note that for any sequence $(D_{n})_{n\geq 1}$ of regular
sets such that $D_{n}\Subset D_{n+1}$ for all $n\geq 1$ and $\cup_{n\geq
1}D_{n}=\Omega$, we have

$G_{\Omega}(x,y)=\lim_{n\to\infty}G_{D_{n}}(x,y).$ (3.4)

Hence, it is not difficult to show that $G_{\Omega}(\cdot,y)$ satisfies
properties (G1) and (G2) for any Greenian set $\Omega$. However, instead of
(G3), we observe that $G_{\Omega}$ satisfies

  1. G3’)

For every $h\in\mathcal{H}_{L}^{+}(\Omega)$, if $h\leq G_{\Omega}(\cdot,y)$ in
$\Omega$ then $h\equiv 0$ in $\Omega$.

We define $\Gamma(x)=\overline{\Gamma}(|x|)$ for every $x\in\mathbb{R}^{d}$,
where $\overline{\Gamma}$ is given for every real $r>0$ by

$\overline{\Gamma}(r)=\left\\{\begin{array}[]{lr}r^{2-d}&\mbox{if }d\geq 3,\\\
1\vee(-\log r)&\mbox{if }d=2,\\\ 1&\mbox{if }d=1.\end{array}\right.$ (3.5)

Let $D\subset\mathbb{R}^{d}$ be a bounded open set. It is well known (see
[16]) that there exists a real constant $c>0$ such that

$G_{D}(x,y)\leq c\,\Gamma(x-y)\quad\mbox{for all }x,y\in D.$ (3.6)

Therefore, every bounded open subset of $\mathbb{R}^{d}$, $d\geq 1$, is
Greenian. Moreover, if $d\geq 3$ then $\mathbb{R}^{d}$ and thereby all open
subsets are Greenian. However, $\mathbb{R}$ and $\mathbb{R}^{2}$ are not
Greenian.

Given a Greenian set $D\subset\mathbb{R}^{d}$, the Green operator in $D$ is
defined by

$G_{D}p:=\int_{D}G_{D}(\cdot,y)p(y)dy$

for every function $p\in\mathcal{B}(D)$ for which the above integral exists.
The following proposition is well proved in [7].

######  Proposition 3.3.

Let $D\subset\mathbb{R}^{d}$ be a bounded domain and let
$p\in\mathcal{B}^{+}(D)$.

  1. a)

If $p$ is bounded, then $G_{D}p$ is bounded and $\alpha-$Hölder continuous in
$D$. If moreover $D$ is regular, then $\lim_{x\in D,\,x\rightarrow
z}G_{D}p(x)=0$ for all $z\in\partial D$.

  2. b)

If $p$ is bounded and $\beta-$Hölder continuous in $D$ for $0<\beta\leq 1$,
then $G_{D}p\in C^{2}(D)$.

We need the following topological lemma which should be well known. For the
completeness, we give here the proof.

######  Lemma 3.4.

Let $D$ be a connected open subset of $\mathbb{R}^{d}$ and let $a\in D$. Then
for every $x\in D$, there exist $n\geq 1$ and $x_{0},x_{1},\ldots,x_{n}\in D$
such that

$x_{0}=a,\;x_{n}=x\mbox{ and }[x_{k-1},x_{k}]\subset D\mbox{ for all
}k\in\\{1,\ldots,n\\}.$ (3.7)

###### Proof.

Let $W$ be the set of all points of $D$ having the property given in the
lemma. We claim that $W$ is open. Indeed, let $x\in W$ and let $n\geq 1$,
$x_{0},x_{1},\ldots,x_{n}\in D$ satisfying (3.7). Choose $r>0$ such that
$B(x,r)\subset D$ and let $y\in B(x,r)$. Then, setting $x_{n+1}=y$ we get
$x_{0},x_{1},...,x_{n},x_{n+1}\in D$ such that $x_{0}=a$, $x_{n+1}=y$ and
$[x_{k-1},x_{k}]\subset D$ for every $1\leq k\leq n+1$. This means that
$B(x,r)\subset W$ and the claim is proved. To show that $W$ is a closed subset
of $D$, we consider a sequence $(y_{p})_{p\geq 1}$ in $W$ which converges to a
point $y\in D$. Let $\varepsilon>0$ and $q\geq 1$ such that $y_{q}\in
B(y,\varepsilon)$. Since $y_{q}\in W$, there exist $m\geq 1$ and
$z_{0},z_{1},...,z_{m}\in D$ such that $z_{0}=a,z_{m}=y_{q}$ and
$[z_{k-1},z_{k}]\subset D$ for every $1\leq k\leq m$. To see that $y\in W$, it
suffices to take $z_{m+1}=y$. Finally, we conclude that $W$ is open and closed
in $D$ at the same time, which yields that $W=D$ since $W\not=\emptyset$ and
$D$ is connected. ∎

######  Theorem 3.5.

Let $D\subset\mathbb{R}^{d}$ be a bounded domain and let
$p\in\mathcal{B}^{+}(D)$ be a locally bounded function such that $u:=G_{D}p$
is finite at some point in $D$. Then $u$ is $\alpha-$Hölder continuous,
$L-$superharmonic in $D$ and $L-$harmonic in
$D\backslash\overline{\\{p>0\\}}$.

###### Proof.

The proof is divided in three steps. Given a set $V\subset D$, we define
$u_{V}=G_{D}(p\textbf{1}_{V})$.

Step 1. We claim that if $u(x_{0})<\infty$ for any $x_{0}\in D$ then $u$ is
finite and continuous in the ball $B_{r}:=B(x_{0},r)\Subset D$ where $r>0$.
Indeed, we decompose $u$ into the sum $u=v_{r}+w_{r}$ where $v_{r}=u_{B_{r}}$
and $w_{r}=u_{B_{r}^{c}}$. We obviously see that $u_{B_{r}}$ is finite and
continuous in $B_{r}$ because $p\textbf{1}_{B_{r}}$ is bounded in $D$. On the
other hand, given $0<\rho<r$ and $z\in B_{r}^{c}:=D\backslash B_{r}$, we
easily see that $G_{D}(\cdot,z)$ is $L-$harmonic in $D\backslash\\{z\\}$ and
hence Fubini-Tonelli’s theorem yields that for every $x\in B_{\rho}$,

$\displaystyle H_{B_{\rho}}w_{r}(x)$ $\displaystyle=$
$\displaystyle\int_{D\backslash B_{r}}\left(\int_{\partial
B_{\rho}}G_{D}(y,z)\,H_{B_{\rho}}(x,dy)\right)p(z)\,dz$ $\displaystyle=$
$\displaystyle\int_{D\backslash B_{r}}H_{B_{\rho}}G_{D}(\cdot,z)(x)\,p(z)\,dz$
$\displaystyle=$ $\displaystyle\int_{D\backslash B_{r}}G_{D}(x,z)p(z)\,dz$
$\displaystyle=$ $\displaystyle w_{r}(x).$

Since $w_{r}(x_{0})<\infty$, it follows that
$H_{B_{\rho}}w_{r}\not\equiv\infty$ in $B_{\rho}$. Then, in virtue of
Proposition 2.2, $w_{r}$ is $L-$harmonic in $B_{\rho}$ for every $0<\rho<r$.
Whence $w_{r}$ is $L-$harmonic in $B_{r}$, which yields in particular that
$w_{r}$ is finite and continuous in $B_{r}$. Consequently, $u$ is finite and
continuous in $B_{r}$.

Step 2. Assume that $u(x)<\infty$ for some point $x\in D$ and let $y\in D$
such that the segment $[x,y]\subset D$. We claim that $u(z)<\infty$ for all
$z\in[x,y]$. Indeed, it is enough to show that $u(y)<\infty$. To do this, put
$\eta:=d([x,y],D^{c})$ and let $0<r<\eta$. Choose an integer $n$ such that
$nr>|y-x|$ and define

$z_{k}=x+\frac{k}{n}(y-x)\quad\mbox{for }k=0,1,\ldots,n.$
$D$$r$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\;\cdots\;\cdots$$\eta$$x$$y$$z_{0}$$z_{1}$$z_{2}$$z_{n-1}$$z_{n}$

Seeing that $u(z_{0})=u(x)<\infty$, we obtain by Step 1 that $u<\infty$ in
${B(z_{0},r)}$ and then $u(z_{1})<\infty$ since $|z_{1}-z_{0}|<r$. Using same
arguments, we deduce that $u(z_{k})<\infty$ for every $k\in\\{1,\ldots,n\\}$.
In particular $u(y)=u(z_{n})<\infty$ and the claim is proved.

Step 3. From Steps 1 and 2, we first conclude that $u$ is finite and
continuous in the whole domain $D$. To see this, let $x_{0}\in D$ such that
$u(x_{0})<\infty$. According to Lemma 3.4 above, for each $x\in D$ we can find
$n\geq 1$ and $x_{1},\ldots,x_{n}\in D$ such that $x_{n}=x$ and
$[x_{k-1},x_{k}]\subset D$ for all $1\leq k\leq n$. Hence, by successive
applications of Step 2, we get that $u(x)=u(x_{n})<\infty$ and consequently
$u$ is finite in $D$. The fact that $u$ is continuous in $D$ follows from Step
1. Now, in order to prove that $u$ is $\alpha-$Hölder continuous in $D$,
consider a domain $V\Subset D$ and choose a domain $W$ such that $V\Subset
W\Subset D$. Since $p\textbf{1}_{W}$ is bounded in $D$, it is clear that
$u_{W}\in\mathcal{C}^{\alpha}(D)\subset\mathcal{C}^{\alpha}(\overline{V})$. On
the other hand, the function $u_{W^{c}}$ is continuous in $W$ and same
arguments as in Step 1 yield that $H_{B}u_{W^{c}}=u_{W^{c}}$ for every ball
$B\Subset W$. Therefore, $u_{W^{c}}$ is $L-$harmonic in $W$ and consequently
$\alpha-$Hölder continuous in $W$. Thus
$u=u_{W}+u_{W^{c}}\in\mathcal{C}^{\alpha}(\overline{V})$ for every domain
$V\Subset D$ which means that $u\in\mathcal{C}^{\alpha}(D)$. It remains to
show that $u$ is $L-$superharmonic in $D$ and $L-$harmonic in
$U:=D\backslash\overline{\\{p>0\\}}$. The first claim follows easily from
Fubini-Tonelli’s theorem by observing that $G_{D}(\cdot,y)$ is
$L-$superharmonic in $D$ for all $y\in D$. Finally, the function $u$ is
$L-$harmonic in $U$ because it is continuous and $H_{B}u=u$ for any ball
$B\Subset U$ by analogous calculus as in Step 1. ∎

We deduce that for every bounded domain $D$ of $\mathbb{R}^{d}$ and for every
locally bounded function $p\in\mathcal{B}^{+}(D)$ such that
$G_{D}p\not\equiv\infty$ in $D$, we have

$LG_{D}p=-p\mbox{ in }D.$ (3.8)

Indeed, the previous theorem guarantees that $G_{D}p$ is finite and continuous
in $D$ and for every function $\varphi\in\mathcal{C}^{\infty}_{c}(D)$,
Fubini’s theorem yields that

$\int_{D}G_{D}p(x)L^{*}\varphi(x)dx=\int_{D}p(y)\Big{(}\int_{D}G_{D}(x,y)L^{*}\varphi(x)dx\Big{)}dy=-\int_{D}p(y)\varphi(y)dy.$

######  Proposition 3.6.

Let $D\subset\mathbb{R}^{d}$ be a bounded domain. Then $G_{D}$ is a compact
operator on $\mathcal{B}_{b}(D)$ endowed with the uniform norm
$\|\cdot\|_{\infty}$.

###### Proof.

We have to show that ${\mathcal{F}}=\\{G_{D}u:u\in\mathcal{B}_{b}(D)\mbox{ and
}\|u\|_{\infty}\leq 1\\}$ is relatively compact in $\mathcal{B}_{b}(D)$
endowed with $\|\cdot\|_{\infty}$. We obviously see that
$\|G_{D}u\|_{\infty}\leq\|G_{D}1\|_{\infty}<\infty$ for all
$u\in\mathcal{B}_{b}(D)$ such that $\|u\|_{\infty}\leq 1$, which means that
${\mathcal{F}}$ is uniformly bounded. Thus, in view of Ascoli’s theorem, the
proof will be finished provided we show that the family ${\mathcal{F}}$ is
equicontinuous in $D$. In order to do this, let $A\subset D$ be Borel set,
$x\in D$ and $r>0$. Then, in virtue of (3.6), we have

$\int_{A}G_{D}(x,y)dy\leq c\int_{B(x,r)}\Gamma(x-y)\,dy+c\int_{A\backslash
B(x,r)}\Gamma(x-y)dy.$

By a simple computation, there exists a real constant $c_{1}>0$, depending
only on $d$, such that

$\int_{B(x,r)}\Gamma(x-y)\,dy\leq
c_{1}\int_{0}^{r}\overline{\Gamma}(t)t^{d-1}\,dt\quad\mbox{and}\quad\int_{A\backslash
B(x,r)}\Gamma(x-y)dy\leq\overline{\Gamma}(r)\lambda_{d}(A).$

Therefore, for every $r>0$ we get that

$\limsup_{\lambda_{d}(A)\to 0}\left(\sup_{x\in
D}\int_{A}G_{D}(x,y)dy\right)\leq
c\,c_{1}\int_{0}^{r}\overline{\Gamma}(t)t^{d-1}\,dt.$

Seeing that $\int_{0}^{r}\overline{\Gamma}(t)t^{d-1}\,dt\to 0$ if $r\to 0$, we
deduce that

$\sup_{x\in D}\int_{A}G_{D}(x,y)dy\to 0\quad\mbox{if}\quad\lambda_{d}(A)\to
0.$

Whence the family ${\mathcal{R}}=\\{G_{D}(x,\cdot):x\in D\\}$ is uniformly
$\lambda_{d}$-integrable in $D$. Consequently, by Vitali’s convergence
theorem, we conclude that for all $x,z\in D$,

$\sup_{\|u\|_{\infty}\leq
1}\left|G_{D}u(z)-G_{D}u(x)\right|\leq\int_{D}\left|G_{D}(z,y)-G_{D}(x,y)\right|dy\to
0\quad\mbox{if}\quad z\to x.$

Thus $\mathcal{F}$ is equicontinuous in the domain $D$. ∎

##  4 Semilinear Dirichlet problem

Throughout this work, we consider a regular domain $D$ in $\mathbb{R}^{d}$, a
Borel measurable locally bounded function $\xi:D\rightarrow\mathbb{R}_{+}$ and
a nondecreasing continuous function
$\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that $\psi(t)=0$ if and
only if $t=0$. We are concerned with existence and uniqueness of nonnegative
solution to the semilinear Dirichlet problem

$\left\\{\begin{array}[]{rcll}-Lu+\xi\psi(u)&=&g&\mbox{in }D,\\\
u&=&f&\mbox{on }\partial D,\end{array}\right.$ (4.1)

where $f:\partial D\rightarrow\mathbb{R}_{+}$ is continuous and
$g:D\rightarrow\mathbb{R}_{+}$ is Borel measurable locally bounded. It will be
convenient to extend the function $\psi$ to $\mathbb{R}$ by setting
$\psi(t)=0$ if $t\leq 0.$ By a nonnegative solution in $D$ to equation

$-Lu+\xi\psi(u)=g,$ (4.2)

we mean a continuous function $u:D\to\mathbb{R}_{+}$ satisfying (4.2) in the
distributional sense, i.e, for all $\varphi\in\mathcal{C}_{c}^{\infty}(D)$,

$-\int_{D}uL^{*}\varphi\,d\lambda_{d}+\int_{D}\xi\psi(u)\varphi\,d\lambda_{d}=\int_{D}g\varphi\,d\lambda_{d}.$
(4.3)

The boundary condition in problem (4.1) means that for all $z\in\partial D$,

$\lim_{x\in D,\,x\rightarrow z}u(x)=f(z).$ (4.4)

######  Proposition 4.1.

Let $\Psi:\mathbb{R}\to\mathbb{R}_{+}$ be nondecreasing and let
$u,v\in\mathcal{B}(D)$ be two locally bounded functions such that
$G_{D}(\xi\Psi(u))$ and $G_{D}(\xi\Psi(v)))$ are not identically infinite in
$D$. Assume that

$s:=u-v+G_{D}(\xi\Psi(u))-G_{D}(\xi\Psi(v))$

is a nonnegative $L-$superharmonic function in $D$. Then $u\geq v$ in $D$.

###### Proof.

We first observe that $G_{D}w$ is finite and continuous in $D$ by Theorem 3.5,
where $w:=\xi\Psi(u)-\xi\Psi(v)$. The fact that $\Psi$ is nondecreasing yields
that $\\{w^{+}>0\\}\subset\\{u>v\\}$ and consequently

$s+G_{D}w^{-}\geq G_{D}w^{+}\mbox{ in }\\{w^{+}>0\\}.$ (4.5)

We define $p_{n}:=n\wedge w^{+}\textbf{1}_{K_{n}}$ where $(K_{n})_{n\geq 1}$
is a sequence of compact sets satisfying

$K_{n}\subset K_{n+1}\subset\\{w^{+}>0\\}\mbox{ for all }n\geq
1\quad\mbox{and}\quad\lambda_{d}(\cup_{n\geq 1}K_{n})=\lambda_{d}(w^{+}>0).$

According to Theorem 3.5, for every $n\geq 1$, $G_{D}p_{n}$ is $L$-harmonic in
$U_{n}:=D\backslash K_{n}$ and consequently
$\theta_{n}:=s+G_{D}w^{-}-G_{D}p_{n}$ is $L-$superharmonic in $U_{n}$. On the
other hand, since $p_{n}$ is bounded and $D$ is regular, Proposition 3.3
yields that $\lim_{x\in D,x\to z}G_{D}p_{n}(x)=0$ for every $z\in\partial D$.
Then, for every $z\in\partial U_{n}\cap\partial D$,

$\liminf_{x\in U_{n},\,x\to z}\theta_{n}(x)=\liminf_{x\in U_{n},\,x\to
z}(s+G_{D}w^{-})(x)\geq 0$

Moreover, since $K_{n}\subset\\{w^{+}>0\\}$, it follows from (4.5) that
$\theta_{n}\geq 0$ in $K_{n}$. Hence, seeing that $\theta_{n}$ is lower
semicontinuous in $D$, we deduce that for every $z\in\partial U_{n}\cap
D\subset K_{n}$, $\liminf_{x\in U_{n},\,x\to z}\theta_{n}(x)\geq 0$. We then
conclude by Proposition 2.3 that for every $n\geq 1$,

$s+G_{D}w^{-}-G_{D}p_{n}\geq 0\mbox{ in }U_{n}.$

Letting $n$ tend to $\infty$, we obtain that $u-v=s+G_{D}w^{-}-G_{D}w^{+}\geq
0$ in $D\backslash\\{w^{+}>0\\}$ which yields that $u-v\geq 0$ in $D$. ∎

######  Theorem 4.2.

Let $s\in\mathcal{C}^{+}(D)$ be $L-$superharmonic in $D$ such that
$G_{D}(\xi\psi(s))\not\equiv\infty$. Then there exists one and only one
function $u\in\mathcal{C}^{+}(D)$ such that

$u+G_{D}(\xi\psi(u))=s\mbox{ in }D.$ (4.6)

###### Proof.

Due to the previous proposition, the integral equation (4.6) has at most one
solution. The existence of a solution will be established in two steps.

Step 1. Consider first an open set $V\Subset D$. Let
$a=\sup_{x\in\overline{V}}|s(x)|$ and define

$\Lambda(u)=s-G_{V}(\xi{\psi_{a}}(u))\quad\mbox{for all
}u\in\mathcal{B}_{b}(V),$

where ${\psi_{a}}(t)=\psi(a\wedge t)$ for every $t\in\mathbb{R}$. We claim
that $\Lambda$ is continuous in $\mathcal{B}_{b}(V)$ endowed with its uniform
norm. Indeed, let $(u_{n})_{n\geq 1}\subset\mathcal{B}_{b}(V)$ be a sequence
converging to a function $u\in\mathcal{B}_{b}(V)$. Then, due to the dominated
convergence theorem, $(G_{V}(\xi{\psi_{a}}(u_{n}))_{n\geq 1}$ is pointwise
convergent to $G_{V}(\xi\psi_{a}(u))$. Assume that the last convergence does
not hold uniformly. Thus, there exist $\varepsilon>0$ and a subsequence
$(u_{n_{k}})_{k\geq 1}$ of $(u_{n})_{n\geq 1}$ such that

$\sup_{x\in
V}\left|G_{V}(\xi{\psi_{a}}(u_{n_{k}}))(x)-G_{V}(\xi{\psi_{a}}(u))(x)\right|\geq\varepsilon\quad\mbox{for
all }k\geq 1.$ (4.7)

Since the operator $G_{V}$ is compact and the sequence
$(\xi{\psi_{a}}(u_{n_{k}}))_{k\geq 1}$ is bounded in $\mathcal{B}_{b}(V)$, we
can find a subsequence $(w_{k})_{k\geq 1}$ of $(u_{n_{k}})_{k\geq 1}$ such
that $(G_{V}(\xi{\psi_{a}}(w_{k})))_{k\geq 1}$ converges to
$G_{V}(\xi{\psi_{a}}(u))$ with respect to the uniform norm in
$\mathcal{B}_{b}(V)$. This yields a contradiction with (4.7) above. Hence
$\Lambda$ is continuous in $\mathcal{B}_{b}(V)$ and the claim is proved. Using
again the fact that the operator $G_{V}$ is compact, we check out that
$\Lambda(\mathcal{B}_{b}(V))$ is relatively compact in $\mathcal{B}_{b}(V)$.
Then, we conclude in virtue of the Schauder’s fixed point theorem, that there
exists $v\in\mathcal{B}_{b}(V)$ such that $\Lambda(v)=v$. On the other hand,
using Proposition 4.1, it is clear that $v\geq 0$ in $V$ and thereby $0\leq
v\leq s\leq a$ in $V$. This yields that ${\psi_{a}}(v)=\psi(v)$ and
consequently

$v+G_{V}(\xi\psi(v))=s\quad\mbox{in }V.$ (4.8)

Observe that $v$ is continuous in $V$ since $G_{V}(\xi\psi(v))$ is continuous
in $V$ by Proposition 3.3.(a).

Step 2. Choose a sequence $(D_{n})_{n\geq 1}$ of regular open sets such that

$D_{n}\Subset D_{n+1}\quad\mbox{for all }n\geq 1\quad\mbox{ and }\cup_{n\geq
1}D_{n}=D.$

Let, for every $n\geq 1$, $u_{n}$ be the solution of

$u_{n}+G_{D_{n}}(\xi\psi(u_{n}))=s\quad\mbox{in }D_{n}.$ (4.9)

Then, in virtue of formula (3.2) we observe that

$u_{n+1}+G_{D_{n}}(\xi\psi(u_{n+1}))=s-H_{D_{n}}G_{D_{n+1}}(\xi\psi(u_{n+1}))\quad\mbox{in
}D_{n}$

and hence
$u_{n}-u_{n+1}+G_{D_{n}}(\xi\psi(u_{n}))-G_{D_{n}}(\xi\psi(u_{n+1}))$ is a
nonnegative $L-$harmonic function in $D_{n}$. So, in virtue of Proposition
4.1, we have $0\leq u_{n+1}\leq u_{n}\leq s$ in $D_{n}$ for all $n\geq 1$. Let
$u:D\to{\mathbb{R}}_{+}$ be the function given for every $x\in D$ by

$u(x)=\lim_{n\to\infty}u_{n}(x)=\inf_{n\geq 1}u_{n}(x).$

Seeing that $G_{D}(\xi\psi(s))<\infty$ in $D$ and applying the dominated
convergence theorem, we get for every $x\in D$ that

$\lim_{n\rightarrow\infty}\int_{D_{n}}G_{D_{n}}(x,y)\xi(y)\psi(u_{n}(y))dy=\int_{D}G_{D}(x,y)\xi(y)\psi(u(y))dy.$

Thus, letting $n$ tend to infinity in (4.9) we conclude that $u$ satisfies
(4.6). The continuity of $u$ follows obviously from the fact that
$G_{D}(\xi\psi(u))$ is continuous in $D$. ∎

For every $f\in\mathcal{B}^{+}(\partial D)$ and every locally bounded function
$g\in\mathcal{B}^{+}(D)$ such that $H_{D}f$ and $G_{D}g$ are not identically
infinite in $D$, it follows from Proposition 2.2 and Theorem 3.5 that the
function

$S_{D}(f,g)=H_{D}f+G_{D}g$

is continuous and $L-$superharmonic in $D$. Then, assuming that
$G_{D}(\xi\psi(S_{D}(f,g)))\not\equiv\infty$ in $D$, the previous theorem
yields that the integral equation

$u+\int_{D}G_{D}(\cdot,y)\xi(y)\psi(u(y))\,dy=S_{D}(f,g)$ (4.10)

admits a unique solution $u\in\mathcal{C}^{+}(D)$. In all the following, we
use the notation

$u=U_{D}^{\xi}(f,g).$ (4.11)

For $i=1,2,$ let $f_{i}\in\mathcal{B}^{+}(\partial D)$ and
$g_{i},\xi_{i}\in\mathcal{B}^{+}(D)$ be locally bounded in $D$ such that
$s_{i}=S_{D}(f_{i},g_{i})<\infty$ and $G_{D}(\xi_{i}\psi(s_{i})))<\infty$ in
$D$. If $f_{1}\leq f_{2}$, $g_{1}\leq g_{2}$ and $\xi_{1}\geq\xi_{2}$, it
follows immediately from Proposition 4.1 that

$U_{D}^{\xi_{1}}(f_{1},g_{1})\leq U_{D}^{\xi_{2}}(f_{2},g_{2}).$ (4.12)

######  Proposition 4.3.

We consider a locally bounded function $g\in\mathcal{B}^{+}(D)$ and let
$f\in\mathcal{B}^{+}(\partial D)$ such that $s=S_{D}(f,g)<\infty$ and
$G_{D}(\xi\psi(s))<\infty$ in $D$. Let $(\xi_{n})_{n\geq 1},(g_{n})_{n\geq
1}\subset\mathcal{B}^{+}(D)$ and $(f_{n})_{n\geq
1}\subset\mathcal{B}^{+}(\partial D)$ be nondecreasing sequences which are
pointwise convergent respectively to $\xi,g$ and $f$. Then:

  1. a)

$U_{D}^{\xi_{n}}(f,g)$ decreases to $U_{D}^{\xi}(f,g)$.

  2. b)

$U_{D}^{\xi}(f_{n},g_{n})$ increases to $U_{D}^{\xi}(f,g)$.

###### Proof.

We define $u_{n}:=U_{D}^{\xi_{n}}(f,g)$ for every $n\geq 1$. Applying (4.12),
we get that $(u_{n})_{n\geq 1}$ is nonincreasing in $D$ and thereby it
converges to $u:=\inf_{n\geq 1}u_{n}$. Furthermore, applying the dominated
convergence theorem, we obtain

$\lim_{n\to\infty}\int_{D}G_{D}(\cdot,y)\xi_{n}(y)\psi(u_{n}(y))\,dy=\int_{D}G_{D}(\cdot,y)\xi(y)\psi(u(y))\,dy.$

Consequently, $u+G_{D}(\xi\psi(u))=s$ in $D$ which yields that
$u=U_{D}^{\xi}(f,g)$ and statement (a) is proved. Statement (b) is similar. ∎

Under appropriate conditions, the following theorem ensures that
$U_{D}^{\xi}(f,g)$ is none other than the solution to the semilinear boundary
value problem (4.1).

######  Theorem 4.4.

Assume that $G_{D}\xi,G_{D}g\in\mathcal{C}_{0}(D)$. Then, for every function
$f\in\mathcal{C}^{+}(\partial D)$, problem (4.1) admits one and only one
solution $u\in\mathcal{C}^{+}(D)$. More precisely $u=U_{D}^{\xi}(f,g)$.

###### Proof.

It is sufficient to prove that $u$ is a solution to (4.10) if and only if $u$
is the solution to problem (4.1). Let $a=\sup_{x\in D}|S_{D}(f,g)|$ and let
$u=U_{D}^{\xi}(f,g)$ be the solution to (4.10). It is easy to see $0\leq u\leq
a$ in $D$ which yields that $G_{D}(\xi\psi(u))\leq\psi(a)G_{D}\xi<\infty$ and
thereby $G_{D}(\xi\psi(u))\in\mathcal{C}(D)$ and according to formula (3.8),

$-LG_{D}(\xi\psi(u))=\xi\psi(u)\quad\mbox{in }D.$

We also have $-LG_{D}g=g$ in $D$ since $G_{D}g$ is continuous. Therefore,
$u\in\mathcal{C}^{+}(D)$ and

$-Lu+\xi\psi(u)=g\quad\mbox{in }D.$

On the other hand, for every $z\in\partial D$, $0\leq\lim_{x\to
z}G_{D}(\xi\psi(u))(x)\leq\psi(a)\lim_{x\to z}G_{D}\xi=0$ and consequently
$\lim_{x\rightarrow z}u(x)=f(z)$ by (4.10). Thus $u=U_{D}^{\xi}(f,g)$ is a
solution to the boundary problem (4.1). Conversely, let $u$ be a solution to
(4.1). Using the fact that $G_{D}\xi$ and $G_{D}g$ are in the class
$\mathcal{C}_{0}(D)$, we easily check out that

$h:=u+G_{D}(\xi\psi(u))-G_{D}g$

is $L-$harmonic in $D$ and $\lim_{x\to z}h(x)=f(z)$ for all $z\in\partial D$.
Thus $h=H_{D}f$ and $u$ satisfies (4.10). ∎

In virtue of the previous theorem, we observe that for any function
$u\in\mathcal{C}^{+}(D)$, the following three statements are equivalent:

  1. i)

The function $u$ is a solution to equation (4.2) in $D$.

  2. ii)

$U_{V}^{\xi}(u,g)=u$ in $V$ for every regular set $V\Subset D$.

  3. iii)

$U_{B}^{\xi}(u,g)=u$ in $B$ for every ball $B\Subset D$.

######  Proposition 4.5.

Assume that $\xi\in\mathcal{C}^{\alpha}(D)$,
$\psi\in\mathcal{C}^{\alpha}(\mathbb{R}_{+})$ and let
$f\in\mathcal{B}(\partial D)$, $g\in\mathcal{C}^{\alpha}(D)$ be nonnegative
functions such that $S_{D}(f,g)<\infty$ and
$G_{D}\big{(}\xi\psi(S_{D}(f,g))\big{)}<\infty$ in $D$. Then
$U_{D}^{\xi}(f,g)$ is twice continuously differentiable in $D$.

###### Proof.

Consider a ball $B\Subset D$. Notice that $u=U_{D}^{\xi}(f,g)$ is continuous
in $D$ and, in virtue of (4.10) and (3.2), we have

$u=H_{B}u+G_{B}g-G_{B}\big{(}\xi\psi(u)\big{)}\quad\mbox{in }B.$

We first observe that $H_{B}u\in\mathcal{C}^{2}(B)$ by the hypoellipticity
property of $L$ and $G_{B}g\in\mathcal{C}^{2}(B)$ in view of (b) in
Proposition 3.3. Next, since $p=\xi\psi(u)$ is bounded in $B$, statement (a)
in Proposition 3.3 yields that $G_{B}p\in\mathcal{C}^{\alpha}(B)$ and
consequently $u\in\mathcal{C}^{\alpha}(B)$. By simple computation, we check
out that $p\in\mathcal{C}^{\alpha^{2}}(B)$ and thereby
$G_{B}p\in\mathcal{C}^{2}(B)$ by Proposition 3.3(b). Whence
$u\in\mathcal{C}^{2}(B)$ for any arbitrary ball $B\Subset D$ which means that
$u\in\mathcal{C}^{2}(D)$. ∎

Let us note that all results obtained in this section can easily be extended
to a more general setting, where we consider instead of functions $\xi(x)$ and
$\psi(t)$, a function

$\Phi:(x,t)\in D\times\mathbb{R}_{+}\mapsto\Phi(x,t)\in\mathbb{R}$

satisfying some hypotheses analogous to those assumed on $\xi$ and $\psi$.
Besides, it should be specified that the fact that $\xi$ and $g$ are locally
bounded is not really necessary in our approach. In fact, it is enough to
assume that $\xi$ and $g$ belong to the local Kato class [6].

##  5 Estimates of solutions

This section is devoted to establish our main results which consist in lower
bound estimates of nonnegative solutions to equation (4.2) and more general to
inequality

$-Lu+\xi\psi(u)\geq g\mbox{ in }D.$ (5.1)

More precisely, under appropriate conditions on $\psi$, we will determine a
function $\varphi$ with values in $]0,1]$ such that for every
$f\in\mathcal{C}^{+}(\partial D)$, $g\in\mathcal{B}^{+}(D)$ and for all $x\in
D$,

$U_{D}^{\xi}(f,g)(x)\geq
s(x)\varphi\left(\frac{G_{D}(\xi\psi(s))(x)}{s(x)}\right).$ (5.2)

The function $g$ is assumed to be locally bounded in $D$ such that
$G_{D}g\in\mathcal{C}_{0}(D)$ and the $L-$superharmonic function
$s=S_{D}(f,g)>0$ in $D$.

We are only interested in locally bounded functions $\xi\in\mathcal{B}(D)$
which are nonnegative. Analogous problems with $\xi\leq 0$ are recently
investigated by Grigor’yan and Verbitsky in [12]. They gave a real positive
function $\varrho$, depending only on $\psi$, such that if $S_{D}(f,g)\geq 1$
in $D$, then the solution $u$ to problem (4.1) satisfies for all $x\in D$,

$u(x)\geq\varrho\left(G_{D}\xi\right(x)).$ (5.3)

The particular case where $L=\Delta$ is the Laplacian operator and $f,g$ are
identically zero, is already studied by Mâagli and Zribi [17]. For signed
function $\xi$, problem (4.1) is examined by Grigor’yan and Verbitsky [11] in
the special case where $\psi(t)=t^{\gamma}$ with $\gamma>0$. Assuming that
$p=G_{D}g>0$ in $D$, they showed that if $u\in\mathcal{C}^{2}(D)$ is a
nonnegative solution to

$-Lu+\xi u^{\gamma}\geq g\mbox{ in }D,$ (5.4)

then for every $x\in D$,

$u(x)\geq\left\\{\begin{array}[]{ll}p(x)\left(1+(\gamma-1)\frac{G_{D}\left(p^{\gamma}\xi\right)(x)}{p(x)}\right)_{+}^{1/(1-\gamma)}&\mbox{if
}\gamma<1,\\\ p(x)\exp\left({-\frac{G_{D}(\xi p)(x)}{p(x)}}\right)&\mbox{if
}\gamma=1,\\\
p(x)\left(1+(\gamma-1)\frac{G_{D}(p^{\gamma}\xi)(x)}{p(x)}\right)^{1/(1-\gamma)}&\mbox{if
}\gamma>1.\end{array}\right.$ (5.5)

In this paper, we establish similar estimates for a large class of functions
$\psi$ which contains polynomial functions, logarithmic functions as well as
exponential functions. More precisely, we consider a continuous function
$\psi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $\psi(0)=0$, $\psi(t)>0$ for
all $t>0$, $\psi$ is continuously differentiable in $]0,\infty[$ and

$\psi(rt)\leq c\psi(r)\psi(t)\quad\mbox{for all }r\in[0,1],t\in\mathbb{R}_{+}$
(5.6)

where $c$ is a positive constant not depending on either $s$ or $t$. Let
$0<\ell\leq\infty$ given by

$\ell c=\int_{0}^{1}\frac{1}{\psi(s)}\,ds$

and define the function $\Theta:]0,1]\rightarrow[0,\ell[$ for every
$t\in]0,1]$ by

$\Theta(t)=\int_{t}^{1}\frac{1}{c\psi(s)}ds.$

Then, obviously $\Theta$ admits an inverse function which will be denoted by
$\varphi$ and extended to $\mathbb{R}_{+}$ by setting $\varphi(t)=0$ if
$t\geq\ell$. Hence, $\varphi$ is continuous and nonincreasing in
$\mathbb{R}_{+}$, twice continuously differentiable in $]0,\ell[$ and for
every $r\in]0,\ell[$,

$\varphi^{\prime}(r)=-c\psi(\varphi(r))<0\quad\mbox{ and
}\quad\varphi^{\prime\prime}(r)=c^{2}\psi^{\prime}(\varphi(r))\psi(\varphi(r))>0.$

In order to prove Theorem 5.3 below, we first establish some technical lemmas.

######  Lemma 5.1.

Let $u:D\rightarrow\mathbb{R}_{+}$ and $v:D\rightarrow]0,\ell[$ be twice
continuously differentiable functions. If $u$ is $L-$superharmonic in $D$,
then

$L(uv)\geq\frac{Lu-L(u\varphi(v))}{c\psi(\varphi(v))}.$

###### Proof.

Obviously $w=\varphi(v)\in\mathcal{C}^{2}(D)$ and by elementary computation,
we get that

$Lw=\varphi^{\prime}(v)Lv+\varphi^{\prime\prime}(v)<A\nabla v,\nabla v>.$

Consequently

$\displaystyle L(uw)$ $\displaystyle=$ $\displaystyle uLw+wLu+<\nabla
u,A\nabla w>+<A\nabla u,\nabla w>$ $\displaystyle=$
$\displaystyle\varphi^{\prime}(v)uLv+\varphi^{\prime\prime}(v)u<A\nabla
v,\nabla v>+\varphi(v)Lu$ $\displaystyle\qquad+\varphi^{\prime}(v)<\nabla
u,A\nabla v>+\varphi^{\prime}(v)<A\nabla u,\nabla v>$ $\displaystyle=$
$\displaystyle\varphi(v)Lu-\varphi^{\prime}(v)vLu+\varphi^{\prime\prime}(v)u<A\nabla
v,\nabla v>$ $\displaystyle\qquad+\varphi^{\prime}(v)\left(uLv+vLu+<\nabla
u,A\nabla v>+<A\nabla u,\nabla v>\right)$ $\displaystyle=$
$\displaystyle\varphi(v)Lu-\varphi^{\prime}(v)vLu+\varphi^{\prime\prime}(v)u<A\nabla
v,\nabla v>+\varphi^{\prime}(v)L(uv)$

Since the matrix $A$ is positive-definite and the function $\varphi$ is
convex, it follows that

$\displaystyle
L(u\varphi(v))-\left(\varphi(v)-v\varphi^{\prime}(v)\right)Lu-\varphi^{\prime}(v)L(uv)$
$\displaystyle\qquad\qquad\qquad=\quad\varphi^{\prime\prime}(v)u<A\nabla
v,\nabla v>$ $\displaystyle\qquad\qquad\qquad\geq\quad 0.$

Hence

$\displaystyle-\varphi^{\prime}(v)L(uv)$ $\displaystyle\geq$
$\displaystyle-L(u\varphi(v))+\left(\varphi(v)-v\varphi^{\prime}(v)\right)Lu$
$\displaystyle=$
$\displaystyle-L(u\varphi(v))+\left(\varphi(v)-1-v\varphi^{\prime}(v)\right)Lu+Lu$

In virtue of the mean value theorem and the fact that $\varphi^{\prime}$ is
nondecreasing, we obtain that $\varphi(v)-1-v\varphi^{\prime}(v)\leq 0$ in
$D$. Assume that $u$ is $L-$superharmonic in $D$. Then we deduce from [10,
Theorem 3.5] that $Lu\leq 0$ in $D$ and consequently
$\left(\varphi(v)-1-v\varphi^{\prime}(v)\right)Lu\geq 0$ in $D$. Finally,
using the fact that $\varphi^{\prime}(v)=-c\psi(\varphi(v))\leq 0$, we get
that

$L(uv)\geq\frac{Lu-L(u\varphi(v))}{c\psi(\varphi(v))}.$

∎

######  Lemma 5.2.

Let $f\in\mathcal{B}_{b}^{+}(\partial D)$ and $g\in\mathcal{B}_{b}^{+}(D)$
such that $s=S_{D}(f,g)>0$ in $D$. Assume that $\xi$ is bounded and let
$(\xi_{n})_{n\geq 1}$, $(g_{n})_{n\geq 1}\subset\mathcal{B}^{+}(D)$ and
$(f_{n})_{n\geq 1}\in\mathcal{B}^{+}(\partial D)$ be bounded sequences which
are pointwise convergent respectively to $\xi$, $g$ and $f$. Assume that for
all $n\geq 1$, $s_{n}=S_{D}(f_{n},g_{n})>0$ and

$U_{D}^{\xi_{n}}(f_{n},g_{n})\geq
s_{n}\varphi\left(\frac{G_{D}(\xi_{n}\psi(s_{n}))}{s_{n}}\right)\quad\mbox{in
}D.$

Then (5.2) holds true.

###### Proof.

We have $u_{n}+G_{D}p_{n}=s_{n}$ where $u_{n}=U_{D}^{\xi_{n}}(f_{n},g_{n})$
and $p_{n}=\xi_{n}\psi(u_{n})$ for all $n\geq 1$. Since $(p_{n})_{n\geq 1}$ is
a bounded sequence, it follows from Proposition 3.6 that there exists a
subsequence $(p_{n_{k}})_{k\geq 1}$ such that $(G_{D}p_{n_{k}})_{k\geq 1}$ is
uniformly convergent in $D$. Seeing that $(s_{n_{k}})_{k\geq 1}$ is pointwise
convergent to $s$, we conclude that $(u_{n_{k}})_{k\geq 1}$ converges in $D$
to a function $u\in\mathcal{B}_{b}^{+}(D)$. Applying the dominated convergence
theorem, we obtain that for every $x\in D$,

$\lim_{k\to\infty}G_{D}p_{n_{k}}(x)=G_{D}(\xi\psi(u))(x).$

Therefore $u+G_{D}(\xi\psi(u))=s$ in $D$, which yields in view of Theorem 4.2
that $u=U_{D}^{\xi}(f,g)$. Again, since sequences $(\xi_{n})_{n\geq 1}$ and
$(s_{n})_{n\geq 1}$ are bounded, we see that
$(G_{D}(\xi_{n_{k}}\psi(s_{n_{k}})))_{k\geq 1}$ is pointwise convergent to
$G_{D}(\xi\psi(s))$. Hence, letting $k$ tend to infinity in the formula

$u_{n_{k}}\geq
s_{n_{k}}\varphi\left(\frac{G_{D}(\xi_{n_{k}}\psi(s_{n_{k}}))}{s_{n_{k}}}\right)\quad\mbox{in
}D,$

we obtain (5.2) in virtue of the continuity of $\varphi$. ∎

######  Theorem 5.3.

Let $g\in\mathcal{B}^{+}(D)$ be locally bounded and let
$f\in\mathcal{B}^{+}(\partial D)$ such that $0<s<\infty$ and
$G_{D}(\xi\psi(s))<\infty$ in $D$ where $s=S_{D}(f,g)$. Then

$s\,\varphi\left(\frac{G_{D}(\xi\psi(s))}{s}\right)\leq U_{D}^{\xi}(f,g)\leq
s\quad\mbox{ in }D.$

###### Proof.

Seeing that $u\leq s$ is immediate with $u=U_{D}^{\xi}(f,g)$, we only need to
prove that

$\frac{u}{s}\geq\varphi\left(\frac{G_{D}(\xi\psi(s))}{s}\right)\quad\mbox{in
}D.$ (5.7)

Step 1. We first assume that $\xi$, $g$ are bounded and $\alpha-$Hölder
continuous in $D$ and that $f$ is bounded in $\partial D$. Then
$s=H_{D}f+G_{D}g\in\mathcal{C}^{2}(D)$ since $H_{D}f\in\mathcal{C}^{2}(D)$ by
the hypoellipticity property of $L$ and $G_{D}g\in\mathcal{C}^{2}(D)$ in view
of Proposition 3.3.(b). Besides, $u\in\mathcal{C}^{2}(D)$ in virtue of
Proposition 4.5 and the fact that $\xi$ and $\psi$ are $\alpha-$Hölder
continuous. Therefore, for every $\varepsilon>0$,
$v_{\varepsilon}=\Theta\left(\frac{u+\varepsilon}{s+\varepsilon}\right)$ is
twice continuously differentiable in $D$ with values in $]0,\ell[$ and
consequently using Lemma 5.1 we obtain that

$\displaystyle
L((s+\varepsilon)v_{\varepsilon})\geq\frac{L(s+\varepsilon)-L(u+\varepsilon)}{c\psi(\frac{u+\varepsilon}{s+\varepsilon})}=-\xi\frac{\psi(u)}{c\psi(\frac{u+\varepsilon}{s+\varepsilon})}\geq-\xi\psi(s).$

The last inequality follows from (5.6) and the fact that $\psi$ is
nondecreasing in $\mathbb{R}_{+}$. We also see that
$G_{D}(\xi\psi(s))\in\mathcal{C}^{2}(D)$ by Proposition 3.3.(b) and therefore

$w_{\varepsilon}=(s+\varepsilon)v_{\varepsilon}-G_{D}(\xi\psi(s))\in\mathcal{C}^{2}(D)\quad\mbox{and}\quad
Lw_{\varepsilon}\geq 0\mbox{ in }D.$

On the other hand, for every $z\in\partial D$, $\lim_{x\to
z}G_{D}(\xi\psi(s))(x)=0$ by Proposition 3.3.(a). Moreover, seeing that

$1\leq\frac{s+\varepsilon}{u+\varepsilon}=\frac{u+\varepsilon+G_{D}(\xi\psi(u))}{u+\varepsilon}\leq
1+\frac{G_{D}(\xi\psi(s))}{\varepsilon}\quad\mbox{in }D,$

we also obtain that $\lim_{x\to z}v_{\varepsilon}(x)=\Theta(1)=0$ and thereby
$\lim_{x\to z}w_{\varepsilon}(x)=0$ for all $z\in\partial D$. We then conclude
in view of the maximum principle [10, Theorem 3.1] that $w_{\varepsilon}\leq
0$ in $D$ and consequently

$\frac{u+\varepsilon}{s+\varepsilon}\geq\varphi\left(\frac{G_{D}(\xi\psi(s))}{s+\varepsilon}\right)\quad\mbox{
in }D.$

Letting $\varepsilon$ tend to $0$, we conclude that (5.7) holds true whenever
$\xi$ and $g$ are bounded $\alpha-$Hölder continuous in $D$ and that $f$ is
bounded in $\partial D$.

Step 2. We consider at this step the case where $\xi$ and $g$ are only bounded
in $D$ and we still assume that $f$ is bounded in $\partial D$. In virtue of
Corollary 4.23 and Theorem 4.9 in [3], we choose two sequences
$(\xi_{n})_{n\geq 1},(g_{n})_{n\geq 1}\subset\mathcal{C}_{c}^{\infty}(D)$ such
that $0\leq\xi_{n}\leq\|\xi\|_{\infty}$, $0\leq g_{n}\leq\|g\|_{\infty}$ for
all $n\geq 1$ and

$\lim_{n\to\infty}\xi_{n}=\xi,\;\lim_{n\to\infty}g_{n}=g\;\mbox{a.e in }D.$

Seeing that $\lim_{n\to\infty}S_{D}(f,g_{n})(x)=s(x)$ and $s(x)>0$ for all
$x\in D$, we can impose on the $L$-superharmonic function
$s_{n}=S_{D}(f,g_{n})$ to be positive everywhere in $D$ for every $n\geq 1$.
It follows from Step 1 above that, for all $n\geq 1$,

$\frac{U_{D}^{\xi_{n}}(f,g_{n})}{s_{n}}\geq\varphi\left(\frac{G_{D}(\xi_{n}\psi(s_{n}))}{s_{n}}\right)\quad\mbox{in
}D.$

Hence, (5.7) holds true by Lemma 5.2.

Step 3. We return to the general setting where functions $\xi,g$ are
nonnegative locally bounded in $D$ and $f\in\mathcal{B}^{+}(\partial D)$. We
define for every $n\geq 1$, $\xi_{n}=\xi\wedge n$, $g_{n}=g\wedge n$,
$f_{n}=f\wedge n$ and let $s_{n}=S_{D}(f_{n},g_{n})>0$ in $D$. According to
Step 2, for every $n\geq 1$ and every $k\geq 1$,

$\frac{U_{D}^{\xi_{k}}(f_{n},g_{n})}{s_{n}}\geq\varphi\left(\frac{G_{D}(\xi_{k}\psi(s_{n}))}{s_{n}}\right)\quad\mbox{in
}D.$ (5.8)

On the other hand, for every $k\geq 1$,
$\lim_{n\to\infty}U_{D}^{\xi_{k}}(f_{n},g_{n})=U_{D}^{\xi_{k}}(f,g)$ by
Proposition 4.3.(b) and
$\lim_{n\to\infty}G_{D}(\xi_{k}\psi(s_{n}))=G_{D}(\xi_{k}\psi(s))$ by the
monotone convergence theorem. Hence, letting $n$ tend to infinity in (5.8), we
get that for all $k\geq 1$,

$\frac{U_{D}^{\xi_{k}}(f,g)}{s}\geq\varphi\left(\frac{G_{D}(\xi_{k}\psi(s))}{s}\right)\quad\mbox{in
}D.$

Letting $k$ tend to infinity, we conclude that (5.7) holds true in virtue of
the continuity of $\varphi$. Observe that
$\lim_{k\to\infty}U_{D}^{\xi_{k}}(f,g)=u$ by Proposition 4.3 and
$\lim_{k\to\infty}G_{D}(\xi_{k}\psi(s))=~{}G_{D}(\xi\psi(s))$ by the monotone
convergence theorem.

∎

Now, we use our previous results to establish a lower bound estimate of
nonnegative solutions to inequality

$-Lu+\xi\Psi(u)\geq g\quad\mbox{in }D,$ (5.9)

where $\Psi:\mathbb{R}_{+}\to\mathbb{R}$ is any locally bounded Borel
measurable function which is bounded above by some function $\psi$ satisfying
hypotheses given at the beginning of this section. Here, we do not need to
assume that $\Psi$ is monotone nor that it keeps a constant sign. We point out
that a solution $u$ to inequality (5.9) should be understood in the
distributional sense, that is, $u:D\to\mathbb{R}$ is continuous and for every
nonnegative function $\varphi\in\mathcal{C}_{c}^{\infty}(D)$,

$-\int_{D}uL^{*}\varphi\,d\lambda_{d}+\int_{D}\xi\Psi(u)\varphi\,d\lambda_{d}\geq\int_{D}g\varphi\,d\lambda_{d}.$

######  Theorem 5.4.

Let $\Psi$ and $\psi$ be as above, $D$ be any Greenian domain of
$\mathbb{R}^{d}$ (not necessarily bounded) and let $g\in\mathcal{B}^{+}(D)$ be
locally bounded such that $0<p<\infty$ in $D$ where $p=G_{D}g$. Then, for
every nonnegative solution $u$ to inequality (5.9) such that $Lu$ is a locally
bounded function in $D$,

$u\geq p\,\varphi\Big{(}\frac{G_{D}(\xi\psi(p))}{p}\Big{)}\quad\mbox{in }D.$
(5.10)

###### Proof.

Let $u$ be a solution to inequality (5.9), define $g_{0}:=-Lu+\xi\psi(u)$ and
choose a sequence $(D_{n})_{n\geq 1}$ of regular bounded domains such that
$D_{n}\Subset D_{n+1}$ for all $n\geq 1$ and $\cup_{n\geq 1}D_{n}=D$. For
every $n\geq 1$, define $v_{n}:=U_{D_{n}}^{\xi}(0,g)$, that is,

$\left\\{\begin{array}[]{rcll}-Lv_{n}+\xi\psi(v_{n})&=&g&\mbox{in }D_{n},\\\
v_{n}&=&0&\mbox{on }\partial D_{n}.\end{array}\right.$

Since $u\geq 0$ and since $g_{0}\geq-Lu+\xi\Psi(u)\geq g$ in $D$, it follows
immediately from (4.12) that $u=U_{D_{n}}^{\xi}(u,g_{0})\geq v_{n}$ in
$D_{n}$. On the other hand, we obviously see that the sequence $(p_{n})_{n\geq
1}$ of $L$-superharmonic functions in $D_{n}$ given by
$p_{n}=G_{D_{n}}g=S_{D_{n}}(0,g)$, is nondecreasing and pointwise convergent
to $p>0$ in $D$. Then, there exists a subsequence $(p_{n_{k}})_{k\geq 1}$ such
that $p_{n_{k}}>0$ in $D_{n_{k}}$ for every $k\geq 1$. We then apply Theorem
5.3 in the regular domain $D_{n_{k}}$ to obtain

$v_{n_{k}}\geq
p_{n_{k}}\varphi\left(\frac{G_{D_{n_{k}}}(\xi\psi(p_{n_{k}}))}{p_{n_{k}}}\right)\quad\mbox{in
}D_{n_{k}}.$

Hence, seeing that $p_{n_{k}}\leq p$ in $D_{n_{k}}$ and using monotonicities
of functions $\psi,\varphi$, we get that for all $k\geq 1$,

$u\geq
p_{n_{k}}\varphi\left(\frac{G_{D_{n_{k}}}(\xi\psi(p))}{p_{n_{k}}}\right)\quad\mbox{in
}D_{n_{k}}.$

Consequently, by letting $k$ tend to $\infty$ we obtain estimate (5.10) and
the proof is complete. ∎

In order to illustrate our results regarding estimates of type (5.10), we give
the following examples which deal with three different kinds of functions
$\psi$.

Example 1- Polynomial functions:  For $\psi(t)=t^{\gamma}$ we have
$\ell=1/(1-\gamma)$ if $0<\gamma<1$ and $\ell=\infty$ if $\gamma\geq 1$. By
elementary computation of $\varphi(t)$, Theorem 5.4 above yields that
estimates given by (5.5) hold true for every nonnegative solution to
inequality (5.4). As mentioned above, these estimates are already obtained by
Grigor’yan and Verbitsky [11] provided that the solution $u$ is twice
continuously differentiable in $D$. Notice that, in this work, we deal with
continuous solutions in the distributional sense. More generally, for
$\psi(t)=at+bt^{\gamma}$ where $a,b,\gamma>0$, we obtain

$\ell=\left\\{\begin{array}[]{cl}\displaystyle\frac{a+b}{a(1-\gamma)}\log\frac{a+b}{b}&\mbox{if
}\gamma<1,\\\ \vspace{0,3cm}\infty&\mbox{if }\gamma>1,\end{array}\right.$

and by simple computation of $\varphi(t)$, we deduce that every nonnegative
solution $u$ to

$-Lu+\xi(au+bu^{\gamma})\geq g\quad\mbox{in }D$ (5.11)

satisfies, for all $x\in D$,

$u(x)\geq\left\\{\begin{array}[]{ll}\displaystyle{p(x)\left(\frac{a+b}{a}\exp\Big{(}\frac{a}{a+b}(\gamma-1)\frac{G_{D}\xi(ap+bp^{\gamma})(x)}{p(x)}\Big{)}-\frac{b}{a}\right)_{+}^{1/(1-\gamma)}}&\mbox{if
}\gamma<1,\vspace{0,3cm}\\\
\displaystyle{p(x)\left(\frac{a+b}{a}\exp\Big{(}(\gamma-1)\frac{G_{D}\xi(ap+bp^{\gamma})(x)}{p(x)}\Big{)}-\frac{b}{a}\right)^{1/(1-\gamma)}}&\mbox{if
}\gamma>1.\end{array}\right.$

Example 2- Exponential functions: For $\psi(t)=\sinh t=(e^{t}-e^{-t})/2$, we
prove that $l=\infty$ and elementary computation yields that for every $t\geq
0$,

$\varphi(t)=2\,\mbox{artanh}\left(\alpha
e^{-t}\right)\quad\mbox{with}\quad\alpha=\tanh\frac{1}{2}.$

Hence, we conclude that for every nonnegative solution $u$ to inequality

$-Lu+\xi\sinh u\geq g\quad\mbox{in }D$ (5.12)

and for all $x\in D$, we have

$u(x)\geq
2\,p(x)\,\mbox{artanh}\left(\alpha\exp\Big{(}{-\frac{G_{D}(\xi\sinh(p))(x)}{p(x)}}\Big{)}\right).$

Example 3- Logarithmic functions: Consider $\psi(t)=a(1+bt)\log(1+bt)$ with
$a,b>0$. By simple computation, we get that $\ell=\infty$ and for every real
$t\geq 0$,

$\varphi(t)=\frac{(1+b)^{e^{-t}}-1}{b}.$

In particular, for $a=b=1$, it follows that every nonnegative solution $u$ to
inequality

$-Lu+\xi(1+u)\log(1+u)\geq g\quad\mbox{in }D,$ (5.13)

satisfies, for all $x\in D$,

$u(x)\geq
p(x)\left(2^{\displaystyle{\exp\left(-\frac{G_{D}(\xi(1+p)\log(1+p))(x)}{p(x)}\right)}}-1\right)$

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