#  Existence and regularity results for some Fully Non Linear singular or
degenerate equation

###### Abstract

In this article we prove existence, uniqueness and regularity for the singular
equation

$\displaystyle\begin{cases}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)+c(x)|u|^{\alpha}u+p(x)u^{-\gamma}=0\mbox{\ \ in }\ \Omega\\\ u>0\ \mbox{\ \
in}\ \Omega,\mbox{\ \ }u=0\mbox{\ \ \ \ on}\ \partial\Omega\end{cases}$

when $p$ is some continuous and positive function, $c$ and $h$ are continuous,
$\alpha>-1$ and $F$ is Fully non linear elliptic. Some conditions on the first
eigenvalue for the operator $-|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)-c(x)|u|^{\alpha}u$ are required. The results generalize the well known
results of Lazer and McKenna.

C. O. Ndaw

UMR 8088

CY Cergy Paris University

2 avenue Adolphe Chauvin, Cergy, France

Key words: Viscosity solutions, Second-order elliptic equations, Nonlinear
elliptic equations, Degenerate elliptic equations, Singular problem.
2020 Mathematical subject classification: 35D40, 35J15, 35J60, 35J70, 35J75.

##  1 Introduction and some useful tools

In this work we study the existence, uniqueness and regularity of solution for
the singular equation

$\displaystyle\begin{cases}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)+c(x)|u|^{\alpha}u+p(x)u^{-\gamma}=0\mbox{\ \ in }\ \Omega\\\ u>0\ \mbox{\ \
in}\ \Omega,\mbox{\ \ }u=0\mbox{\ \ \ \ on}\ \partial\Omega\end{cases}$ (1)

where $\Omega$ is a bounded $\mathcal{C}^{2}$ domain, $\alpha>-1$, $p>0$ is
continuous on $\bar{\Omega}$, $c$ and $h$ are continuous on $\bar{\Omega}$.
$F$ is Fully Non Linear elliptic, and the solutions are intented in the
viscosity sense, this will be precised below.

The results here enclosed generalize the pionneer work of Lazer and McKenna
[21], which consider the case where $h=c=\alpha=0$ and $F$ is the Laplacian.
In this simple framework, the solutions can be intended in the variational
sense, even if the presence of the singular zero order term $p(x)u^{-\gamma}$
lead the authors in [21] to use some tools usually employed in viscosity
framework : Existence of convenient sub- and super-solutions, comparison
Theorem, compactness of bounded sequences. In the present work, the
difficulties are linked both to the singularity/degeneracy of the term of
derivative of order $1$, and to the singularity of the zero order term.

When $\alpha=0$ there is a big amount of articles about the existence, the
uniqueness and the regularity of viscosity solutions for the equation
$F(D^{2}u)=f$, as the paper of Caffarelli [12], mainly devoted to the
${\mathcal{C}}^{1}$ and higher regularity, the paper of Ishii and Lions [19],
the user’s guide of Crandall, Ishii and Lions [15], the book of Caffarelli and
Cabré [13], and the famous paper of Ishii [18].

The case $\alpha\neq 0$ was introduced by Birindelli and Demengel in [4], [5],
which consider the equations

$|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)=f,$

with $f$ and $h$ continuous and bounded. They provide a definition of
viscosity solution which fits the case $\alpha<0$ ( note that in that case,
the equation is not well defined on a point where the gradient is zero). In
particular this definition can also be used for the $(\alpha+2)$-Laplacian
(when one works with viscosity solutions in place of classical solutions), or
to the infinity Laplacian. It can also be used for the case $\alpha>0$, note
that it is shown in [8] that the solutions are the same as the classical
viscosity solutions in that case.

More recently Attouchi and Ruosteenoja in [2] have refined this definition,
for equations of the form

$|\nabla u|^{\gamma}(\Delta u+(p-2)\Delta_{\infty}^{N}u)=f,$

with $\gamma>-1$, $p>1$.

Concerning the optimal regularity expected, the example of
$\varphi(r)=r^{2+\alpha\over 1+\alpha}$ which solves

$|\nabla\varphi|^{\alpha}\Delta\varphi=\dfrac{(2+\alpha)^{1+\alpha}}{(1+\alpha)^{2+\alpha}}((N-1)\alpha+N)$

in the ball $B(0,1)$, shows that , for $\alpha>0$, this regularity cannot be
better than ${\mathcal{C}}^{1,{1\over 1+\alpha}}$.

Coming back to the case $\alpha<0$, a first regularity result is proved in [7]
for solutions of the homogeneous Dirichlet problem. When the operator $F$ is
concave or convex, the ${\mathcal{C}}^{2}$ regularity holds. In the case
$\alpha>0$ and $h=0$, Imbert and Silvestre proved a ${\mathcal{C}}^{1,\beta}$
interior regularity result in [20]. This result is extended to a local result
”up to the boundary” and to the case where $h\neq 0$, [8]. The interior
regularity is precised in [1] in the case $\alpha>0$, which can be
${\mathcal{C}}^{1,{1\over 1+\alpha}}$ in the cases where for example $F$ is
convex or concave.

Concerning the equation (1) in the variational setting, the work of Crandall
Rabinowitz and Tartar [14] extend the results of Lazer McKenna when the
Laplacian is replaced by some linear uniformly elliptic operator, positively
homogeneous of degree $1$, while, for the Fully non linear setting, in [17]
the authors consider $F(x,u,Du,D^{2}u)$ which is homogeneous of degree $1$
with respect to all its arguments and Fully Non Linear Elliptic, with some
more general singularity than $p(x)u^{-\gamma}$. Always when $\alpha=0$ and
$F$ is replaced by some degenerate Pucci’s operators ${\cal P}_{k}^{\pm}$,
Birindelli and Galise [11] proved existence, uniqueness and some regularity
result, result extended to more general singular zero order term in [22].

We now precise the assumptions, and present the main result. We assume that
$F$ satisfies the assumptions:

There exist ellipticity constants $a$ and $A$, $0<a<A$ so that for any $M$ and
$N$ symmetric matrices on $\mathbb{R}^{N}$, $N\geq 0$,

$atr(N)\leq F(M+N)-F(M)\leq Atr(N).$ (2)

We will also assume that $F$ is positively homogenous, i.e

$F(tM)=tF(M)$

for all $t>0$.

A well known example of such operators are the Pucci’s operator ${\cal
M}^{\pm}_{a,A}$ ( ${\cal P}^{+}_{\lambda,\Lambda}$ for most of the authors),
defined as

${\cal
M}^{+}_{a,A}(S)=a\sum_{i,\lambda_{i}>0}\lambda_{i}(S)+A\sum_{i,\lambda_{i}<0}\lambda_{i}(S),\
{\cal M}^{-}_{a,A}(S)=-{\cal M}^{+}_{a,A}(-S)$

where the $\lambda_{i}(S)$ are the eigenvalue of the symmetric matrix $S$.

In the following Theorem $\lambda_{1}^{\bar{c}}$ denotes the first eigenvalue
for the operator $-|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)-\bar{c}(x)|u|^{\alpha}u$ with the definition precised later.

###### Theorem 1.1

Let $\Omega$ be a bounded ${\cal C}^{2}$ domain in $\mathbb{R}^{N}$. Suppose
that $F$ satisfies (2), is positively homogeneous of degree $1$, that $c$, $h$
, $p$ are continous and bounded, with $p>0$ on $\overline{\Omega}$. Let us
suppose that $\lambda_{1}^{c},\
\lambda_{1}^{\frac{(1+\alpha+\gamma)c}{2+\alpha}}>0,$ then there exists a
unique solution to (1). In addition $u\ \in\ \mathcal{C}^{1,\beta}(\Omega)$
for some $\beta>0$.

The plan of this paper is as follows : In Section 2 we remind the definition
of viscosity solutions adapted to the present context, and recall the maximum
and comparison principles, and the regularity results needed in the paper. In
Section 3 we prove the existence’s result and provide the convenient
comparison principle for such equations, which allows to prove the uniqueness
of solutions. In Section 4 we study the regularity of the solutions.

##  2 Background, definitions, and previous existence’s and regularity results
for singular and degenerate Elliptic equations $|\nabla
u|^{\alpha}F(D^{2}u)=f$.

We begin to recall the definition of viscosity solutions adapted to the
present context. We denote by $\mathcal{S}$ the space of symmetric matrices on
$\mathbb{R}^{N}$. Let us define for $f\in{\mathcal{C}}(\Omega)$

$\displaystyle G(x,u,q,X)=|q|^{\alpha}(F(X)+h(x)\cdot q)-f$ (3)

where $x\in\mathbb{R}^{N},\ q\in\mathbb{R}^{N},\ u\in\mathbb{R}\ \mbox{and \
}X\in\mathcal{S}$.

###### Definition 2.1

A function $u$, upper semicontinuous (USC for short) in $\Omega$ is a
viscosity sub-solution for (3) (or a solution of $G[u]\geq 0$, a sub-solution
of $G[u]=0$) if whenever $\varphi\in\mathcal{C}^{2}(\Omega)$ and $u-\varphi$
attains a local maximum at $\bar{x}\in\Omega$, then

1) Either $\nabla\varphi(\bar{x})\neq 0$ and

$G(\bar{x},u(\bar{x}),D\varphi(\bar{x}),D^{2}\varphi(\bar{x}))\geq 0.$

2) Or there exists a ball around $\bar{x}$ on which $u(x)=u(\bar{x})$, and

$-f(\bar{x})\geq 0.$

Similarly, $u$, lower semicontinous ( LSC for short) in $\Omega$ is a
viscosity super-solution for (3) (or a solution of $G[u]\leq 0$, a super-
solution of $G[u]=0$) if whenever $\varphi\in\mathcal{C}^{2}(\Omega)$ and
$u-\varphi$ attains a local minimum at $\bar{x}\in\Omega$, and
$\nabla\varphi(\bar{x})\neq 0$ then

$G(\bar{x},u(\bar{x}),D\varphi(\bar{x}),D^{2}\varphi(\bar{x}))\leq 0.$

If $u$ is locally constant around $\bar{x}$

$-f(\bar{x})\leq 0.$

Of course, $u\in\mathcal{C}(\Omega)$ is a viscosity solution of (3) (or a
solution of $G[u]=0$) if $u$ is both a viscosity sub-solution and a viscosity
super-solution.

Let $u\in\mathcal{C}(\Omega)$. We define the superjet $J^{2,+}u(x)$ and the
subjet $J^{2,-}u(x)$ of the second order.

###### Definition 2.2

$\displaystyle
J^{2,+}u(x)=\left\\{(p,X)\in\mathbb{R}^{N}\times\mathbb{S}\right.,{\rm\ so\
that}$ $\displaystyle u(x)\leq
u(\bar{x})+<p,x-\bar{x}>+\frac{1}{2}<X(x-\bar{x}),x-\bar{x}>$
$\displaystyle\left.+o(|x-\bar{x}|^{2})\right\\}.$ $\displaystyle
J^{2,-}u(x)=\left\\{(p,X)\in\mathbb{R}^{N}\times\mathbb{S}\right.{\rm,\ so\
that}$ $\displaystyle u(x)\geq
u(\bar{x})+<p,x-\bar{x}>+\frac{1}{2}<X(x-\bar{x}),x-\bar{x}>$
$\displaystyle\left.+o(|x-\bar{x}|^{2})\right\\}.$

More useful are the closed superjet and closed subjet,

$\overline{J^{2,+}u(\bar{x})}=\\{(\bar{p},\bar{X}),\ \exists x_{n},\
x_{n}\rightarrow\bar{x},\ {\rm and}\ (p_{n},X_{n})\in J^{2,+}u(x_{n}),\
(p_{n},X_{n})\rightarrow(p,X)\\}.$

$\overline{J^{2,-}u(\bar{x})}$ being defined in an obvious symmetric manner.

In the definition of viscosity solutions the test functions can be substituted
by the elements of the semi-jets in the sense that in the definition above one
can restrict to the functions $\varphi$ defined by

$\varphi(x)=u(\bar{x})+<p,x-\bar{x}>+\frac{1}{2}<X(x-\bar{x}),x-\bar{x}>$

with $(p,X)\in J^{2,-}u(\bar{x})$ when $u$ is a super-solution and $(p,X)\in
J^{2,+}u(\bar{x})$ when u is a sub-solution.
A key tool for the existence’s results is the comparison principle, [4].

###### Theorem 2.3

Let $\Omega$ be a bounded ${\mathcal{C}}^{2}$ domain in $\mathbb{R}^{N}$. Let
$k$ be continuous with respect to its variables. Suppose that $u$ is a USC
solution of

$\displaystyle|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)-k(x,u)\geq f$

and $v$ a LSC solution of

$\displaystyle|\nabla v|^{\alpha}(F(D^{2}v)+h(x)\cdot\nabla v)-k(x,v)\leq g$

with $f,g\in\mathcal{C}(\Omega)$, $f\geq g$ and $u\mapsto k(x,u)$ increasing,
or $f>g$ and $u\mapsto k(x,u)$ non decreasing.
Then if $u\leq v$ on $\partial\Omega$, $u\leq v$ in $\Omega$.

This comparison theorem, and the construction of sub- and super-solutions
which are zero on the boundary, together with a mere adaptation to our context
of Perron’s method, permit to prove the existence of solutions of the
Dirichlet problem, for

$\left\\{\begin{array}[]{lc}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)-k(x,u)=g&{\rm in}\ \Omega\\\ u=0&{\rm on}\
\partial\Omega.\end{array}\right.$

Since we are dealing with positive solutions, we will many times use the
strong maximum principle :

###### Theorem 2.4

Let $\Omega$ be a bounded ${\cal C}^{2}$ domain in $\mathbb{R}^{N}$. Suppose
that $h$ and $c$ are continuous and let $u\geq 0$ satisfies

$\displaystyle|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)+c(x)u^{1+\alpha}\leq 0\mbox{\ \ \ \ \ \ }in\ \Omega.$

Then one has

$\displaystyle u>0\mbox{\ \ \ \ \ or\ \ \ \ \ }u\equiv 0.$

For some of the results enclosed we will need ”Hopf boundary principle”, say
the fact that near the boundary, the gradient of positive solutions cannot be
zero in a neighborhood of the boundary :

###### Theorem 2.5

Let $\Omega$ be a bounded ${\cal C}^{2}$ domain in $\mathbb{R}^{N}$. Let
$x_{0}\in\partial\Omega$ and $\partial\Omega$ satisfy the interior sphere
condition at $x_{o}$. Let $\overrightarrow{n}$ denotes the inner normal to
$\partial\Omega$ at $x_{0}$. If $u>0$ in $\Omega$ and

$\displaystyle|\nabla u|^{\alpha}F(D^{2}u)+h(x)\cdot\nabla u|\nabla
u|^{\alpha}+c(x)u^{1+\alpha}\leq 0\mbox{\ \ \ \ \ \ }in\ \Omega\mbox{\ \ and\
\ }u(x_{0})=0$

then

$\displaystyle\dfrac{\partial u}{\partial\overrightarrow{n}}(x_{0})>0.$

in the sense

$\displaystyle\underline{\lim}_{h\rightarrow
0,h>0}{\dfrac{u(x_{0}+h\overrightarrow{n})-u(x_{0})}{h}>0}.$

###### Remark 1

As an easy consequence, if $u$ is a solution of the equation which is
$\mathcal{C}^{1}$ up to the boundary, we have for some $\kappa>0$ , for any
$x_{o}\in\partial\Omega$ such that $u(x_{0})=0$,

$\displaystyle|\dfrac{\partial
u}{\partial\overrightarrow{n}}(x_{0})|\geq\kappa\ \mbox{\rm\ \ \ and then\ \ \
\ \ }|\nabla u|\geq\kappa\mbox{\rm\ \ \ in a neighborhood of \ \ }x_{o}.$

We now recall the Lipschitz estimates between sub-and super-solutions (See
[19] for example in the case $\alpha=0$).

###### Theorem 2.6

1) Let $u$ be USC such that

$\displaystyle|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)\geq f\ \mbox{\
\ \ \ \ }{\rm in}\ B(0,1)$

and $v$ is LSC and satisfies

$\displaystyle|\nabla v|^{\alpha}(F(D^{2}v)+h(x)\cdot\nabla v)\leq g\ \mbox{\
\ \ \ \ }{\rm in}\ B(0,1)$

with $f$ and $g$ continuous and bounded.

Then $\forall\ r\in(0,1)$, there exists $\ L_{r}>0$ such that $\forall\
(x,y)\in({B}(0,r))^{2}$,

$\displaystyle u(x)-v(y)\leq\sup(u-v)+L_{r}|x-y|.$

Here $L_{r}$ depends on $r,$ $\sup u-\inf v$, $|f|_{\infty}$ and
$|g|_{\infty}$.

In particular any solution of $|\nabla v|^{\alpha}(F(D^{2}v)+h(x)\cdot\nabla
v)=f$, with $f$ continuous and bounded, is locally Lipschitz continuous.

2) Suppose that $u$ is a solution of the Dirichlet problem

$\left\\{\begin{array}[]{lc}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)=f,\mbox{\ \ \ \ \ }&f\ {\rm in}\ B(0,1),\\\ u=0&{\rm on}\
\partial(B(0,1)).\end{array}\right.$

Then $u$ is Lipschitz continuous, with some Lipschitz constant depending on
$|f|_{\infty}$ and $|u|_{\infty}$.

As a corollary one easily has by using the definition of viscosity sub- and
super-solutions

###### Theorem 2.7

1) Let $\Omega$ be a bounded $\mathcal{C}^{2}$ domain in $\mathbb{R}^{N}$. Let
$u_{n}$ be a viscosity solution of

$|\nabla u_{n}|^{\alpha}(F(D^{2}u_{n})+h(x)\cdot\nabla u_{n})=f_{n}$

in $\Omega$ and suppose that $f_{n}$ converges locally uniformly to $f$, and
$u_{n}$ is locally uniformly bounded, then one can extract from $(u_{n})_{n}$
a subsequence such that this subsequence converges locally uniformly towards a
solution of

$|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)=f.$

2) If moreover $u_{n}=0$ on the boundary and $f_{n}$ converges uniformly to
$f$, $u_{n}$ converges uniformly up to a subsequence towards $u$ on
$\overline{\Omega}$.

Let us now make precise the ${\mathcal{C}}^{1,\beta}$ regularity results for
the solutions of

$|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)=f,$

when $f$ is continuous in $\Omega$. The first result obtained in [7] is

###### Theorem 2.8

Let $\Omega$ be a bounded ${\cal C}^{2}$ domain in $\mathbb{R}^{N}$. Let
$\alpha\in]-1,0]$. Let $f$ and $h$ be continuous on $\overline{\Omega}$. There
exists $\beta$ so that for any $u$ solution of

$\left\\{\begin{array}[]{lc}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)=f&{\rm in}\ \Omega\\\ u=0&{\rm on}\ \partial\Omega,\end{array}\right.$

$u\in{\mathcal{C}}^{1,\beta}(\overline{\Omega})$.

The case $\alpha>0$ is treated in [20], for $h=0$, the precise result is an
interior estimate:

###### Theorem 2.9

Suppose that $\alpha\geq 0$. There exists $\beta\in]0,1[$, such that for all
$r\in]0,1[$, there exists $C_{r}$ so that for any $f\in{\mathcal{C}}(B(0,1))$,
and for any $u$ solution of

$|\nabla u|^{\alpha}F(D^{2}u)=f$

in $B(0,1)$, one has that $u$ is ${\mathcal{C}}^{1,\beta}(B(0,r))$, with

$|u|_{{\mathcal{C}}^{1,\beta}(B(0,r))}\leq
C_{r}(|u|_{\infty}+|f|_{\infty}^{1\over 1+\alpha}).$

To complete this interior estimate, a
${\mathcal{C}}^{1,\beta}(\overline{\Omega})$ regularity result is obtained in
[8] :

###### Theorem 2.10

Suppose that $\Omega$ is a bounded ${\mathcal{C}}^{2}$ domain in
$\mathbb{R}^{N}$, that $f$ and $h$ are continuous on $\overline{\Omega}$. Let
$\varphi\in{\mathcal{C}}^{1,\beta_{o}}(\partial\Omega)$. There exists
$\beta\leq\inf(\beta_{o},{1\over 1+\alpha})$, and some constant $C$ so that
for any $u$ solution of

$\left\\{\begin{array}[]{lc}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)=f&{\rm in}\ \Omega\\\ u=\varphi&{\rm on}\
\partial\Omega,\end{array}\right.$

one has that $u\in{\mathcal{C}}^{1,\beta}$ on $\overline{\Omega}$ and

$|u|_{{\mathcal{C}}^{1,\beta}(\overline{\Omega})}\leq
C(|\varphi|_{{\mathcal{C}}^{1,\beta_{o}}(\partial\Omega)}+|f|_{\infty}^{1\over
1+\alpha}).$

Other authors provide precises bounds on $\beta$, the interested reader can
see [1].

We now precise the definition of the first demi eigenvalues and
eigenfunctions, [5]. They are defined on the model of [3] :

$\lambda_{1}^{+,c}=\sup\\{\lambda,\exists\varphi>0,|\nabla\varphi|^{\alpha}(F(D^{2}\varphi)+h(x)\cdot\nabla\varphi)+(c(x)+\lambda)|\varphi|^{\alpha}\varphi\leq
0\\}.$

It is clear that $\lambda_{1}^{+,c}$ exists and
$\lambda_{1}^{+,c}\geq-|c|_{\infty}$.

Some precise estimates depending on the larger ball contained in $\Omega$, and
the smallest ball containing it, can be found in [5]. Even if this article
will only need $\lambda_{1}^{+,c}$, we give the definition of the other demi
eigenvalue, say:

$\lambda_{1}^{-,c}=\sup\\{\lambda,\exists\varphi<0,|\nabla\varphi|^{\alpha}(F(D^{2}\varphi)+h(x)\cdot\nabla\varphi)+(c(x)+\lambda)|\varphi|^{\alpha}\varphi\geq
0\\}.$

Note that in the case where $F$ is odd, the two eigenvalues coincide.

We have the following maximum and minimum principle ”under the first
eigenvalues”

###### Theorem 2.11

Let $\Omega$ be a bounded ${\cal C}^{2}$ domain in $\mathbb{R}^{N}$. Under the
previous assumptions on $F$, $h,c$, $\alpha$, suppose that
$\tau<\lambda_{1}^{+,c}$ and that $u$, USC is a sub-solution of

$|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)+(c(x)+\tau)|u|^{\alpha}u\geq
0\ {\rm in}\ \Omega$

and $u\leq 0$ on $\partial\Omega$. Then $u\leq 0$ in $\Omega$.

Suppose that $\tau<\lambda_{1}^{-,c}$ and that $u$, LSC is a super-solution of

$|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)+(c(x)+\tau)|u|^{\alpha}u\leq
0\ {\rm in}\ \Omega$

and $u\geq 0$ on $\partial\Omega$. Then $u\geq 0$ in $\Omega$.

This Theorem allows to prove the existence of a positive eigenfunction for
$\lambda_{1}^{+,c}$ and a negative one for $\lambda_{1}^{-,c}$:

###### Theorem 2.12

Let $\Omega$ be a bounded ${\cal C}^{2}$ domain in $\mathbb{R}^{N}$.Under the
previous assumptions on $F$, $h,c$, $\alpha$, there exists a positive
eigenfunction associated to $\lambda_{1}^{+,c}$, more precisely

$\left\\{\begin{array}[]{lc}|\nabla\varphi_{1}^{+}|^{\alpha}(F(D^{2}\varphi_{1}^{+})+h(x)\cdot\nabla\varphi_{1}^{+})+(c(x)+\lambda_{1}^{+,c})(\varphi_{1}^{+})^{1+\alpha}=0&{\rm
in}\ \Omega\\\ \varphi_{1}^{+}=0&{\rm on}\ \partial\Omega.\end{array}\right.$

There exists a negative eigenfunction associated to $\lambda_{1}^{+,c}$, more
precisely

$\left\\{\begin{array}[]{lc}|\nabla\varphi_{1}^{-}|^{\alpha}(F(D^{2}\varphi_{1}^{-})+h(x)\cdot\nabla\varphi_{1}^{-})+(c(x)+\lambda_{1}^{-,c})|\varphi_{1}^{-}|^{\alpha}\varphi_{1}^{-}=0&{\rm
in}\ \Omega\\\ \varphi_{1}^{-}=0&{\rm on}\ \partial\Omega.\end{array}\right.$

In the following we will drop the exponent $+$ since we will only use
$\lambda_{1}^{+,c}$.

##  3 Existence and uniqueness of viscosity solutions

###  3.1 Existence of viscosity sub- and super-solutions : Proof of Theorem
1.1

Let $\beta_{c,\alpha,\gamma}=\dfrac{(1+\alpha+\gamma)c}{2+\alpha}$. We begin
with some remark about the validity of assumption $\lambda_{1}^{c}>0$, assumed
in the existence’s Theorem:

###### Remark 2

If we take for example $|c|_{\infty}<\lambda_{1}^{0}$, which is known to be
$>0$ one has $\lambda_{1}^{c}>0$. The same is true for
$|\beta_{c,\alpha,\gamma}|_{\infty}<\lambda_{1}^{0}$.

We begin to exhibit a sub- and a super-solution. Let $\phi_{1}$ be an
eigenfunction for $\lambda_{1}^{\beta_{c,\alpha,\gamma}}$. We first treat the
case $\gamma>1$ :

###### Proposition 1

Let assume $\gamma>1$. Let $t=\dfrac{2+\alpha}{1+\alpha+\gamma}$. There exist
$b_{i},\ i=1,\ 2$ so that $\psi_{i}=b_{i}\phi_{1}^{t}$ are respectively sub-
and super-solutions.

Proof. We do the sub-solution case. Let $\psi=b\phi_{1}^{t}$. Then
$\nabla\psi=bt\phi_{1}^{t-1}\nabla\phi_{1}$.

$\displaystyle D^{2}\psi$ $\displaystyle=$ $\displaystyle
bt(t-1)\phi_{1}^{t-2}\nabla\phi_{1}\otimes\nabla\phi_{1}+bt\phi_{1}^{t-1}D^{2}\phi_{1}.$

Then

$\displaystyle|\nabla\psi|^{\alpha}(F(D^{2}\psi)$ $\displaystyle+$
$\displaystyle h(x)\cdot\nabla\psi)+c(x)\psi^{1+\alpha}$ $\displaystyle\geq$
$\displaystyle
Ab^{1+\alpha}t^{1+\alpha}(t-1)\phi_{1}^{t-2+(t-1)\alpha}|\nabla\phi_{1}|^{2+\alpha}$
$\displaystyle+$ $\displaystyle
b^{1+\alpha}t^{1+\alpha}\phi_{1}^{(t-1)(1+\alpha)}\left(|\nabla\phi_{1}|^{\alpha}(F(D^{2}\phi_{1})+h(x)\cdot\nabla\phi_{1})+\dfrac{c}{t^{1+\alpha}}\phi_{1}^{1+\alpha}\right)$
$\displaystyle=$
$\displaystyle(bt)^{1+\alpha}\phi_{1}^{(t-1)(1+\alpha)}\left(|\nabla\phi_{1}|^{\alpha}(F(D^{2}\phi_{1})+h(x)\cdot\nabla\phi_{1})+({c\over
t^{1+\alpha}}+\lambda_{1}^{\beta_{c,\alpha,\gamma}})\phi_{1}^{1+\alpha}\right)$
$\displaystyle-$ $\displaystyle
b^{1+\alpha}t^{1+\alpha}\phi_{1}^{(t-1)\alpha+t-2}(A(1-t)|\nabla\phi_{1}|^{2+\alpha}+\lambda_{1}^{\beta_{c,\alpha,\gamma}}\phi_{1}^{2+\alpha})$
$\displaystyle=$
$\displaystyle-b^{1+\alpha}t^{1+\alpha}\phi_{1}^{t-2+(t-1)\alpha}\left((1-t)A|\nabla\phi_{1}|^{2+\alpha}+\lambda_{1}^{\beta_{c,\alpha,\gamma}}\phi_{1}^{2+\alpha}\right)$

and then if we denote

$\displaystyle
q^{A}(x,b)=b^{1+\alpha}t^{1+\alpha}\phi_{1}^{t-2+(t-1)\alpha}\left((1-t)A|\nabla\phi_{1}|^{2+\alpha}+\lambda_{1}^{\beta_{c,\alpha,\gamma}}\phi_{1}^{2+\alpha}\right),$

one has

$\displaystyle|\nabla\psi|^{\alpha}(F(D^{2}\psi)+h(x)\cdot\nabla\psi)+c(x)\psi^{1+\alpha}+q^{A}(x,b)\geq
0.$

Note that using analogous computations, one has

$\displaystyle|\nabla\psi|^{\alpha}(F(D^{2}\psi)+h(x)\cdot\nabla\psi)+c(x)\psi^{1+\alpha}+q^{a}(x,b)\leq
0$

where $q^{a}$ is defined by replacing $A$ by $a$ in the definition of $q^{A}$.

Claim: one has the existence of positive constants $d_{i},\ i=1,2$ so that

$\displaystyle d_{2}\leq
A(1-t)|\nabla\phi_{1}|^{2+\alpha}+\lambda_{1}^{\beta_{c,\alpha,\gamma}}\phi_{1}^{2+\alpha}\leq
d_{1}.$ (4)

Let us admit for a while the claim, and let us take

$\displaystyle
b_{1}=\left(\dfrac{\min{p}}{d_{1}t^{1+\alpha}}\right)^{\frac{1}{1+\alpha+\gamma}},$

then $q^{A}(x,b_{1})\leq p(x)\psi_{1}^{-\gamma}$ and

$\displaystyle|\nabla(b_{1}\phi)^{t})|^{\alpha}(F(D^{2}(b_{1}\phi^{t}))$
$\displaystyle+$ $\displaystyle h(x)\cdot\nabla(b_{1}\phi^{t}))$
$\displaystyle+$ $\displaystyle
c(x)(b_{1}\phi_{1}^{t})^{1+\alpha}+p(x)(b_{1}\phi_{1}^{t})^{-\gamma}$
$\displaystyle\geq$
$\displaystyle|\nabla(b_{1}\phi_{1})^{t})|^{\alpha}(F(D^{2}(b_{1}\phi_{1})^{t})+h(x)\cdot\nabla(b_{1}\phi_{1})^{t}))$
$\displaystyle+c(x)(b_{1}\phi_{1}^{t})^{1+\alpha}+q^{A}(x,b_{1}).$

Then with that choice of $b_{1}$, $\psi_{1}=b_{1}\phi_{1}^{t}$ is a sub-
solution of (1).
In the same manner using the left hand side inequality of (4) and taking

$b_{2}=\left(\dfrac{\max{p}}{d_{2}t^{1+\alpha}}\right)^{\frac{1}{1+\alpha+\gamma}},$

$\psi_{2}=b_{2}\phi_{1}^{t}$ is a super-solution of (1). Note that
$b_{1}<b_{2}$ and then $\psi_{1}\leq\psi_{2}$.

We now prove claim (4): Since $\phi_{1}$ is in
$\mathcal{C}^{1}(\overline{\Omega})$ the inequality on the right side of (4)
is obvious.

To prove the left hand side inequality, near the boundary, using Hopf
principle and the fact that $\phi_{1}$ is of class $\mathcal{C}^{1}$,
$\exists\ \delta,\ d(x,\partial\Omega)<\delta\ $ $\Rightarrow|\
\nabla\phi_{1}|>m$ (anywhere else one will use $|\nabla\phi_{1}|\geq 0$).
Now, using $\phi_{1}>0$, $\exists\ \tilde{m}$ such that
$\phi_{1}\geq\tilde{m}$ on $d(x,\partial\Omega)\geq\delta$. From all this we
derive that

$\displaystyle
c_{1}|\nabla\phi_{1}|^{2+\alpha}+c_{2}\phi_{1}^{2+\alpha}\geq\min{(c_{1},c_{2})}\min{(m^{2+\alpha},\tilde{m}^{2+\alpha})}.$

Taking

$\displaystyle
d_{2}=\min{(c_{1},c_{2})}\min{(m^{2+\alpha},\tilde{m}^{2+\alpha})}$

we have the left-side inequality of (4). $\hfill\square$

We now treat the case $\gamma<1$. We begin to introduce a regularized problem
depending on some parameter $\delta>0$,

$\left\\{\begin{array}[]{lc}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)+c(x)u^{1+\alpha}+p(x)(u+\delta)^{-\gamma}=0&{\rm in}\ \Omega,\\\ u=0&{\rm
on}\ \partial\Omega.\end{array}\right.$ (5)

###### Lemma 3.1

Let $\gamma>0$, and $c$ be so that $\lambda_{1}^{c}>0$, let $\psi_{1}$ be some
positive eigenfunction for $\lambda_{1}^{c}$. Then, there exist
$\varepsilon_{0}$ and $\delta_{0}$ so that for $\varepsilon<\varepsilon_{0}$
and $\delta\in[0,\delta_{0}],$ then $u_{\star}=\varepsilon\psi_{1}$ is a sub-
solution of(5).

Proof.

Let $u_{\star}=\varepsilon\psi_{1}$. Take

$\displaystyle\delta_{0}=\dfrac{1}{2^{\frac{\gamma}{1+\alpha+\gamma}}}\left(\frac{\min{p}}{\lambda_{1}^{c}}\right)^{\dfrac{1}{1+\alpha+\gamma}}\mbox{\
\ and\ \
}\varepsilon_{0}=\dfrac{1}{2^{\frac{\gamma}{1+\alpha+\gamma}}|\psi_{1}|_{\infty}}\left(\frac{\min{p}}{\lambda_{1}^{c}}\right)^{\dfrac{1}{1+\alpha+\gamma}}.$

Then, by an easy computation, one has for all $\varepsilon<\varepsilon_{0}$
and $\delta<\delta_{0}$,

$|\nabla u_{\star}|^{\alpha}(F(D^{2}u_{\star})+h(x)\cdot\nabla
u_{\star})+c(x)u_{\star}^{1+\alpha}+p(x)(u_{\star}+\delta)^{-\gamma}\geq 0.$

$\hfill\square$

###### Proposition 2

Suppose that $\gamma<1$, and $s<1$ sufficiently close to $1$ in order that
$\lambda_{1}^{cs^{-(1+\alpha)}}>0$.
If $\psi_{2}$ is an eigenfunction corresponding to
$\lambda_{1}^{cs^{-(1+\alpha)}}$, then there exists $d$ great enough in order
that $u^{\star}=d\psi_{2}^{s}$ is a super-solution of (1).

###### Remark 3

The fact that $\lambda_{1}^{cs^{-(1+\alpha)}}>0$ when $s$ is sufficiently
close to $1$ is justified by the continuity result in the Proposition 3 below.

Proof. of Proposition 2 Let $u^{\star}=d\psi_{2}^{s}$. We have

$\displaystyle|\nabla u^{\star}|^{\alpha}(F(D^{2}u^{\star})$ $\displaystyle+$
$\displaystyle h(x)\cdot\nabla
u^{\star})+c(u^{\star})^{1+\alpha}+p(x)(u^{\star})^{-\gamma}$
$\displaystyle\leq$ $\displaystyle
d^{1+\alpha}s^{1+\alpha}\psi_{2}^{s-2+(s-1)\alpha}\left((s-1)|\nabla\psi_{2}|^{2+\alpha}a\right.$
$\displaystyle\left.+|\nabla\psi_{2}|^{\alpha}(F(D^{2}\psi_{2})+h(x)\cdot\nabla\psi_{2})\right)+c(x)(d\psi_{2}^{s})^{1+\alpha}+p(x)d^{-\gamma}\psi_{2}^{-\gamma
s}$ $\displaystyle=$
$\displaystyle(ds)^{1+\alpha}\psi_{2}^{(s-1)(1+\alpha)}\left(|\nabla\psi_{2}|^{\alpha}F(D^{2}\psi_{2})+h(x)\cdot\nabla\psi_{2}+({c\over
s^{1+\alpha}}+\lambda_{1}^{cs^{-(1+\alpha)}})\psi_{2}^{1+\alpha}\right)$
$\displaystyle-$ $\displaystyle
d^{1+\alpha}s^{1+\alpha}\psi_{2}^{(s-1)\alpha+t-2}(a(1-s)|\nabla\psi_{2}|^{2+\alpha}+\lambda_{1}^{cs^{-(1+\alpha)}}\psi_{2}^{2+\alpha})+p(x)d^{-\gamma}\psi_{2}^{-\gamma
s}$ $\displaystyle=$
$\displaystyle-d^{1+\alpha}s^{1+\alpha}\psi_{2}^{s-2+(s-1)\alpha}\left((1-s)|\nabla\psi_{2}|^{2+\alpha}a+\lambda_{1}^{\frac{c}{s^{1+\alpha}}}\psi_{2}^{2+\alpha}\right)+p(x)d^{-\gamma}\psi_{2}^{-\gamma
s}.$

Since $\gamma<1$ one has $\dfrac{2+\alpha}{1+\alpha+\gamma}>1$, and then if
$s<1$, one has $-s(\gamma+1)+(1-s)\alpha+2>0$.
Then denoting

$\kappa=\min_{x\in\overline{\Omega}}\left((1-s)|\nabla\psi_{2}|^{2+\alpha}a+\lambda_{1}^{\frac{c}{s^{1+\alpha}}}\psi_{2}^{2+\alpha}\right)$

(the existence of $\kappa$ can be proved in the same manner as the existence
of $d_{2}$ in (4)) and assuming $d$ large enough in order that

$\displaystyle
d\geq\left(\dfrac{|p|_{\infty}(\max\psi_{2}(x))^{-s(\gamma+1)+(1-s)\alpha+2}}{\kappa
s^{1+\alpha}}\right)^{1\over{\alpha+\gamma+1}}.$

We have

$\displaystyle|\nabla u^{\star}|^{\alpha}(F(D^{2}u^{\star})+h(x)\cdot\nabla
u^{\star})+c(u^{\star})^{1+\alpha}+p(x)(u^{\star})^{-\gamma}\leq 0.$

So $u^{\star}$ is a super-solution of equation (1). $\hfill\square$
Furthermore for $\varepsilon$ chosen small and and $d$ large we have that
$u_{\star}<u^{\star}$. Indeed let $\psi_{1}$ be a positive eigenfunction for
$\lambda_{1}^{c}$ and $\psi_{2}$ a positive eigenfunction for
$\lambda_{1}^{cs^{-(1+\alpha)}}$, by the results in [4], there exist positive
constants $c_{i}$ $i=1,\cdots,4$ so that

$c_{1}d(x,\partial\Omega)\leq\psi_{1}\leq c_{2}d(x,\partial\Omega),\
c_{3}d(x,\partial\Omega)\leq\psi_{2}\leq c_{4}d(x,\partial\Omega)$

and then taking $\varepsilon$ small enough and $d$ large enough so that

$\varepsilon c_{2}({\rm diam}\ \Omega)^{1-s}<dc_{3}^{s}$

one gets that

$\varepsilon\psi_{1}\leq\varepsilon
c_{2}d(x,\partial\Omega)<dc_{3}^{s}d(x,\partial\Omega)^{s}\leq d\psi_{2}^{s}.$

###### Proposition 3

The map $s\mapsto\lambda_{1}^{cs^{-(1+\alpha)}}$ is continuous. More
generally, if $c_{n}$ converge uniformly to $c$ we have
$\lim\lambda_{1}^{c_{n}}=\lambda_{1}^{c}$.

Proof. On one hand we have $\limsup\lambda_{1}^{c_{n}}\leq\lambda_{1}^{c}$.
Indeed, let $\lambda=\limsup\lambda_{1}^{c_{n}}$, there exists $\varphi_{n}$
so that $\varphi_{n}>0$, $|\varphi_{n}|_{\infty}=1$ and

$|\nabla\varphi_{n}|^{\alpha}(F(D^{2}\varphi_{n})+h(x)\cdot\nabla\varphi_{n})+(c_{n}+\lambda_{1}^{c_{n}})\varphi_{n}=0.$

Using the uniform Lipschitz estimate in Theorem 2.6, $(\varphi_{n})$ is
uniformly Lipschitz continuous, then one can extract from $(\varphi_{n})_{n}$
a subsequence which converges uniformly on $\overline{\Omega}$ toward some
function $\varphi$, in particular $(c_{n}+\lambda_{1}^{c_{n}})\varphi_{n}$
converges uniformly toward $(c+\lambda)\varphi$, and using Theorem 2.7, one
gets that $\varphi$ satisfies

$|\nabla\varphi|^{\alpha}(F(D^{2}\varphi)+h(x)\cdot\nabla\varphi)+(c+\lambda)\varphi=0.$

By the strong maximum principle, since $|\varphi|_{\infty}=1$, $\varphi>0$ in
$\Omega$ and then by the definition of $\lambda_{1}^{c}$,

$\lambda_{1}^{c}\geq\lambda.$

On the other hand let $\lambda<\lambda_{1}^{c}$, by the existence’s result in
[4] there exists $\psi$ so that $\psi>0$ and

$|\nabla\psi|^{\alpha}(F(D^{2}\psi)+h(x)\cdot\nabla\psi)+(\lambda+c)\psi=-1.$

Let then $N$ so that for $n>N$ $|c_{n}-c||\psi|_{\infty}<{1\over 2}$. Then for
such $n$

$|\nabla\psi|^{\alpha}(F(D^{2}\psi)+h(x)\cdot\nabla\psi)+(\lambda+c_{n})\psi\leq{-1\over
2}$

and then for this range of values of $n$, $\lambda_{1}^{c_{n}}\geq\lambda$.
Since $\lambda$ is arbitrary less than $\lambda_{1}^{c}$ one gets
$\liminf\lambda_{1}^{c_{n}}\geq\lambda_{1}^{c}$. $\hfill\square$

###### Remark 4

The function $u^{\star}$ is also a super-solution of the regularized problem
(5).

###  3.2 Proof of the existence’s Theorem

Let us now prove the existence result in Theorem 1.1. We will proceed in two
steps.

#### Step1:

Let $u_{\star}$ and $u^{\star}$ be respectively the sub- and super-solutions
(both for the regularized problem) constructed above, with $u_{\star}\leq
u^{\star}$. Let $k$ be defined by

$\displaystyle
k>\max\left\\{\dfrac{\gamma}{1+\alpha}\dfrac{|p|_{\infty}}{\delta^{\alpha+\gamma+1}};|c|_{\infty}\right\\}.$

Then the functions

$\displaystyle
f_{\delta}(x,u)=-k(u+\delta)^{1+\alpha}-p(x)(u+\delta)^{-\gamma}$

and

$\displaystyle
g_{\delta}(x,u)=c(x)(u+\delta(\alpha))^{1+\alpha}-k(u+\delta)^{1+\alpha}$

where

$\delta(\alpha)=\left\\{\begin{array}[]{c}\delta\ {\rm if}\ \alpha<0\\\ 0\
{\rm if\ not},\end{array}\right.$

are decreasing with respect to $u>0$.

In the sequel will suppose $\alpha\geq 0$, the case $\alpha<0$ is left to the
reader.

Let us consider the sequence $\\{w_{n}\\}$, defined in a recursive way by

$\left\\{\begin{array}[]{lc}|\nabla
w_{n}|^{\alpha}(F(D^{2}w_{n})+h(x)\cdot\nabla
w_{n})+c(x)|w_{n}|^{\alpha}w_{n}-k(w_{n}+\delta)^{1+\alpha}=f_{\delta}(x,w_{n-1})&{\rm
in}\ \Omega\\\ w_{n}=0\ \mbox{on }\partial\Omega&\end{array}\right.,$

with $w_{0}=u_{\star}$.
We will prove that for all $n\in\mathbb{N}$, $u_{\star}\leq w_{n}\leq
w_{n+1}\leq u^{\star}$. Let us show by induction that $\\{w_{n}\\}$ is non
decreasing.
To prove that $w_{1}\geq w_{0}$ note that :

$\displaystyle|\nabla w_{0}|^{\alpha}(F(D^{2}w_{0})+h(x)\cdot\nabla
w_{0})+c(x)|w_{0}|^{\alpha}w_{0}-k(w_{0}+\delta)^{1+\alpha}\geq
f_{\delta}(x,w_{0})$

and

$\displaystyle|\nabla w_{1}|^{\alpha}(F(D^{2}w_{1})+h(x)\cdot\nabla
w_{1})+c(x)|w_{1}|^{\alpha}w_{1}-k(w_{1}+\delta)^{1+\alpha}=f_{\delta}(x,w_{0}).$

Using the comparison Theorem 2.3 (with $w_{0}=w_{1}$ on $\partial\Omega$) we
have that $w_{1}\geq w_{0}$.
Suppose that $w_{n}\geq w_{n-1}$ and let us show that $w_{n+1}\geq w_{n}$:
Since $f_{\delta}$ is decreasing, we have that

$\displaystyle|\nabla w_{n}|^{\alpha}(F(D^{2}w_{n})$ $\displaystyle+$
$\displaystyle h(x)\cdot\nabla
w_{n})+c(x)|w_{n}|^{\alpha}w_{n}-k(w_{n}+\delta)^{1+\alpha}$ $\displaystyle=$
$\displaystyle f_{\delta}(x,w_{n-1})\geq f_{\delta}(x,w_{n}).$

This implies that

$\displaystyle|\nabla w_{n}|^{\alpha}(F(D^{2}w_{n})$ $\displaystyle+$
$\displaystyle h(x)\cdot\nabla
w_{n})+c(x)|w_{n}|^{\alpha}w_{n}-k(w_{n}+\delta)^{1+\alpha}$
$\displaystyle\geq$ $\displaystyle|\nabla
w_{n+1}|^{\alpha}(F(D^{2}w_{n+1})+h(x)\cdot\nabla w_{n+1})$ $\displaystyle+$
$\displaystyle c(x)|w_{n+1}|^{\alpha}w_{n+1}-k(w_{n+1}+\delta)^{1+\alpha},$

and using the comparison Theorem 2.3 (with $w_{n}=w_{n+1}$ on
$\partial\Omega$) one gets $w_{n+1}\geq w_{n}$.

We have shown that $\\{w_{n}\\}$ is non decreasing and since $w_{0}>0$ in
$\Omega$ one gets $w_{n}>0$ in $\Omega$ for all $n\geq 0$.

Using the fact that $u^{\star}$ satisfies $|\nabla
u^{\star}|^{\alpha}(F(D^{2}u^{\star})+h(x)\cdot\nabla
u^{\star})+c(x)|u^{\star}|^{\alpha}u^{\star}-k(u^{\star}+\delta)^{1+\alpha}\leq
f_{\delta}(x,u^{\star})\leq f_{\delta}(x,w_{n})$ we get at each step that
$w_{n+1}\leq u^{\star}$, once we have assumed that $w_{n}\leq u^{\star}$.

Since the sequence $\\{w_{n}\\}$ satisfies the Lipschitz estimates recalled in
Theorem 2.6 it converges uniformly to a function $Z_{\delta}$ which satisfies

$|\nabla Z_{\delta}|^{\alpha}(F(D^{2}Z_{\delta})+h(x)\cdot\nabla
Z_{\delta})+c(x)|Z_{\delta}|^{\alpha}Z_{\delta}+p(x)(Z_{\delta}+\delta)^{-\gamma}=0.$

Furthermore for any $\delta$ one has $u_{\star}\leq Z_{\delta}\leq u^{\star}$.

#### Step2: $\delta$ tends to $0$

Let $\delta$ and $Z_{\delta}$ defined by the first step. We note that since
$u_{\star}\leq Z_{\delta}\leq u^{\star}$, the term
$p(x)(Z_{\delta}+\delta)^{-\gamma}$ is uniformly locally bounded independently
on $\delta$, and then $Z_{\delta}$ is uniformly locally Lipschitz. It follows,
using the uniform Lipschitz estimates in Theorem 2.6, that one can extract
from $Z_{\delta}$ a sequence which converges locally uniformly to some
function $Z$, such that $u_{\star}\leq Z\leq u^{\star}$. Passing to the limit
with Theorem 2.7, one gets that, since $p(Z_{\delta}+\delta)^{-\gamma}$
converges locally uniformly (for a subsequence) towards $pZ^{-\gamma}$, $Z$ is
a solution of equation (1) .

###  3.3 Comparison principle and Uniqueness result

We begin to prove some Lipschitz estimate between sub- and super-solutions of
equation (1)

###### Theorem 3.2

Suppose that $u$ is a positive, bounded by above, solution of

$\left\\{\begin{array}[]{lc}|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)+p(x)u^{-\gamma}\geq f&\ {\rm in}\ \Omega\\\ u=0&\ {\rm on}\
\partial\Omega\end{array}\right.$

and $v$ is a positive solution of

$\left\\{\begin{array}[]{lc}|\nabla v|^{\alpha}(F(D^{2}v)+h(x)\cdot\nabla
v)+p(x)v^{-\gamma}\leq g&\ {\rm in}\ \Omega\\\ v=0&\ {\rm on}\
\partial\Omega\end{array}\right.$

with $f$, $g$ and $h$ continuous and bounded, and $p>0$ is Holder continuous
of exponent $\tau_{p}$. Then :

  * •

If $\sup_{\overline{\Omega}}(u-v)>0$, there exists $c$ depending on
$\Omega,|u|_{\infty},|f|_{\infty},|g|_{\infty},|h|_{\infty},|p|_{\infty}$ so
that for all $(x,y)$ in $\overline{\Omega}^{2}$

$u(x)-v(y)\leq\sup(u-v)+C|x-y|$

  * •

If $\sup_{\overline{\Omega}}(u-v)=0$, and if there exist
$\tau_{1}\leq\inf(1,{2\over 1+\gamma})$, and $C_{1}>0$ so that

$u(x)\leq C_{1}d(x,\partial\Omega)^{\tau_{1}},$

then, there exists some constant $C$ depending on
$\Omega,|u|_{\infty},|f|_{\infty},|g|_{\infty},|h|_{\infty},\tau_{p}$, and
$C_{1}$ so that for all $(x,y)$ in $\overline{\Omega}^{2}$

$u(x)-v(y)\leq C|x-y|^{\tau},$

with $\tau=\inf(\tau_{1},{2+\alpha+\tau_{p}\over 1+\alpha+\gamma})$.

###### Remark 5

In the sequel, we will apply this result with $f=-cu^{1+\alpha}$ and
$g=-cv^{1+\alpha},$ mainly to prove the uniqueness. On the other hand, we get
Hölder regularity of this solution, by the second part of the Theorem above,
by recalling that from the first sections, one has an exponent $\tau_{1}$
which can be taken arbitrarily close to $1$ in the case $\gamma<1$ and is
equal to ${2\over 1+\gamma}$ if $\gamma>1$.

Proof

Let $\omega$ be defined on $\mathbb{R}^{+}$ by

$\omega(s)=s-{s^{1+\varepsilon}\over 2(1+\varepsilon)}$

where $\varepsilon\in]0,1[$. Let us introduce

$\psi(x,y)=u(x)-v(y)-\sup(u-v)-M\omega(|x-y|)$

where $M$ will be chosen large enough later.

It is clear that it is sufficient to prove that for $|x-y|<{1\over 2}$,

$\psi(x,y)\leq 0$

Indeed, if $|x-y|>{1\over 2}$ and if we assume that ${M\over 2}>\sup u-\inf v$
the required result holds.

We argue by contradiction and suppose that
$\sup_{(x,y)\in\overline{\Omega}^{2}}\psi(x,y)>0$. Then by the upper-
semicontinuity of $\psi$, it is achieved on some pair
$(\bar{x},\bar{y})\in\overline{\Omega}^{2}$. In the following $\delta>0$ is a
positive parameter, take

$M={2(\sup u-\inf v)\over\delta},$

then from the definition of $\bar{x}$ and $\bar{y}$,
$|\bar{x}-\bar{y}|\leq\delta$. So saying that $M$ is large is equivalent to
say that $\delta$ is small.

We first remark that neither $\bar{x}$ nor $\bar{y}$ belongs to the boundary.
Indeed, $\bar{x}$ cannot be on the boundary by the positivity of $v$, and if
$\bar{y}\in\partial\Omega$ then we would have

$u(\bar{x})\geq\sup(u-v)+{M\over 2}|\bar{x}-\bar{y}|.$ (6)

This is contradicted by the continuity of $u$, since there exists $\delta_{1}$
so that for $d(x,\partial\Omega)<\delta_{1}$ one has

$u(x)\leq{\sup(u-v)\over 2}$

while if $d(x,\partial\Omega)>\delta$

${M\over 2}d(\bar{x},\partial\Omega)+\sup(u-v)\geq{M\delta\over 2}>\sup u$

which also contradicts (6).

We have obtained that $(\bar{x},\bar{y})\in\Omega^{2}$. Furthermore
$\bar{x}\neq\bar{y}$.

By Ishii’s lemma (Lemma 9) in [18] (see also [6]) for all $\zeta>0$ there
exist $X_{\zeta}$ and $Y_{\zeta}$ so that

$(q,X_{\zeta})\in\overline{J}^{2,+}u(\bar{x}),\
(q,-Y_{\zeta})\in\overline{J}^{2,-}v(\bar{y})$

with $q=M\omega^{\prime}(|\bar{x}-\bar{y}|)$, and

$\left(\begin{array}[]{cc}X_{\zeta}&0\\\ 0&Y_{\zeta}\end{array}\right)\leq
M\left(\begin{array}[]{cc}B&-B\\\ -B&B\end{array}\right)+\zeta
M^{2}\left(\begin{array}[]{cc}B^{2}&-B^{2}\\\ -B^{2}&B^{2}\end{array}\right)$

with $B=D^{2}(\omega(|\cdot|)(\bar{x}-\bar{y})$. One has

$B=\left(\omega^{\prime\prime}-{\omega^{\prime}\over r}\right){x\otimes
x\over|x|^{2}}+{\omega^{\prime}\over r}I$

and then

$B^{2}=\left((\omega^{\prime\prime})^{2}-{(\omega^{\prime})^{2}(r)\over
r^{2}}\right){x\otimes x\over|x|^{2}}+{(\omega^{\prime})^{2}(r)\over r^{2}}I.$

So taking $\zeta={1\over
M(1+2|\omega^{\prime\prime}(r)|+{\omega^{\prime}(r)\over r})}$, $B+\zeta
MB^{2}$ has the eigenvalues $\omega^{\prime\prime}+\zeta
M(\omega^{\prime\prime})^{2}\leq{\omega^{\prime\prime}\over 2}$ and
${\omega^{\prime}(r)\over r}+\zeta{(\omega^{\prime})^{2}\over r^{2}}\leq
2{\omega^{\prime}\over r}$. With that choice of $\zeta,$ dropping the index
$\zeta$ for $X_{\zeta}$ and $Y_{\zeta}$, one has $X+Y\leq 0$ and for any $x$

${}^{t}(x,-x)\left(\begin{array}[]{cc}X&0\\\
0&Y\end{array}\right)\left(\begin{array}[]{c}x\\\ -x\end{array}\right)\leq
4M^{t}x(B+\varepsilon B^{2})x.$

In particular there exist at least one eigenvalue which is less or equal to
$4{\omega^{\prime\prime}\over 2}=2\omega^{\prime\prime}$. Using $X+Y\leq 0$,
one has $tr(X+Y)\leq 2M\omega^{\prime\prime}$.

We have then

$tr(X+Y)\leq-CM|\bar{x}-\bar{y}|^{\varepsilon-1}$

while always by Ishii’s lemma

$|X|+|Y|\leq CM|B|_{\infty}\leq CM|\bar{x}-\bar{y}|.$

On the other hand

${M\over 2}\leq|q|\leq M.$

Then, using in the following lines $F(X)-F(-Y)\leq atr(X+Y)$,

$\displaystyle-|p|_{\infty}(\sup(u-v))^{-\gamma}+f(\bar{x})$
$\displaystyle\leq$ $\displaystyle-p(\bar{x})(u(\bar{x})^{-\gamma}+f(\bar{x})$
$\displaystyle\leq$ $\displaystyle|q|^{\alpha}F(X)+h(\bar{x})\cdot
q|q|^{\alpha}$ $\displaystyle\leq$
$\displaystyle|q|^{\alpha}F(-Y)+h(\bar{y})\cdot q|q|^{\alpha}+2|h|_{\infty}\
M^{1+\alpha}$ $\displaystyle+$ $\displaystyle M^{1+\alpha}({1\over
2})^{|\alpha|}atr(X+Y)$ $\displaystyle\leq$
$\displaystyle-p(\bar{y})v(\bar{y})^{-\gamma}+g(\bar{y})+2|h|_{\infty}\
M^{1+\alpha}$ $\displaystyle-$ $\displaystyle
CM^{1+\alpha}|\bar{x}-\bar{y}|^{\varepsilon-1}$ $\displaystyle\leq$
$\displaystyle 2|h|_{\infty}\
M^{1+\alpha}-CM^{1+\alpha}|\bar{x}-\bar{y}|^{\varepsilon-1}+g(\bar{y})$

from this we get for some positive constant $C$, and for $\delta$ small ( so
that $|h|_{\infty}\ \delta^{1-\varepsilon}<{C\over 2}$),

${C\over
2}M^{1+\alpha}|\bar{x}-\bar{y}|^{\varepsilon-1}\leq|p|_{\infty}(\sup(u-v))^{-\gamma}+|f|_{\infty}+|g|_{\infty}+2|h|_{\infty}M^{1+\alpha},$

clearly a contradiction as soon as $M$ is large enough, since $\varepsilon<1$.
We have then obtained that $\psi(x,y)\leq 0$ for all
$x,y\in\overline{\Omega}^{2}$.

We now do the case where $\sup(u-v)=0$. We take the function

$\psi(x,y)=u(x)-v(y)-M|x-y|^{\tau}$

where $M={2(\sup u-\inf v)\over\delta^{\tau}}$ , this forces $\bar{x}$ and
$\bar{y}$ to satisfy $|\bar{x}-\bar{y}|\leq\delta$. We take
$\tau=\inf(\tau_{1},{2+\alpha+\tau_{p}\over 1+\alpha+\gamma})$. We argue by
contradiction and suppose that the supremum of $\psi$ is $>0$. Then it is
achieved on some pair $(\bar{x},\bar{y})\in\overline{\Omega}^{2}$. By taking
$M$ larger than $C$ where $C$ is so that

$u(\bar{x})\leq Cd(\bar{x},\partial\Omega)^{\tau_{1}}$

one obtains that $\bar{y}$ cannot belong to the boundary. On the other hand,
the positivity of $v$ implies that $\bar{x}$ cannot be on the boundary.

Note for further purpose that

$-p(\bar{x})u(\bar{x})^{-\gamma}\geq-p(\bar{y})u(\bar{x})^{-\gamma}-C_{p}|\bar{x}-\bar{y}|^{\tau_{p}}(M|\bar{x}-\bar{y}|^{\tau})^{-\gamma}\geq-p(\bar{y})v(\bar{y})^{-\gamma}-CM^{-\gamma}|\bar{x}-\bar{y}|^{\tau_{p}-\tau\gamma}.$

We have then by Ishii’s lemma, for all $\zeta$ the existence of $X_{\zeta}$
and $Y_{\zeta}$ in ${\cal S}$ so that with $q=M\tau|\bar{x}-\bar{y}|^{\tau-1}$

$(q,X_{\zeta})\in\overline{J}^{2,+}u(\bar{x}),\
(q,-Y_{\zeta})\in\overline{J}^{2,-}v(\bar{y})$
$\left(\begin{array}[]{cc}X_{\zeta}&0\\\ 0&Y_{\zeta}\end{array}\right)\leq
M\left(\begin{array}[]{cc}B+\zeta MB^{2}&-B-\zeta MB^{2}\\\ -B+\zeta
MB^{2}&B+\zeta MB^{2}\end{array}\right)$

with $B=D^{2}(|\cdot|^{\tau})(\bar{x}-\bar{y})$. By the choice of $\zeta$ as
the first part of the proof, and dropping the index $\zeta$, one has $X+Y\leq
0$, and $tr(X+Y)\leq-CM|\bar{x}-\bar{y}|^{\tau-2}$.
We have then by using the fact that $u$ and $v$ are respectively sub-and
super-solutions

$\displaystyle-p(\bar{y})v(\bar{y})^{-\gamma}$ $\displaystyle-$ $\displaystyle
CM^{-\gamma}|\bar{x}-\bar{y}|^{\tau_{p}-\tau\gamma}+f(\bar{x})$
$\displaystyle\leq$
$\displaystyle|q|^{\alpha}F(X)+M^{1+\alpha}(h(\bar{x})\cdot\bar{x}-\bar{y})|\bar{x}-\bar{y}|^{(\tau-1)\alpha-1}$
$\displaystyle\leq$
$\displaystyle|q|^{\alpha}F(-Y)+M^{1+\alpha}|h|_{\infty}|\bar{x}-\bar{y}|^{(\tau-1)\alpha}-CM^{1+\alpha}|\bar{x}-\bar{y}|^{(\tau-1)\alpha+\tau-2}$
$\displaystyle\leq$
$\displaystyle-p(\bar{y})(v(\bar{y})^{-\gamma}+g(\bar{y})+M^{1+\alpha}|h|_{\infty}|\bar{x}-\bar{y}|^{(\tau-1)\alpha}-CM^{1+\alpha}|\bar{x}-\bar{y}|^{(\tau-1)\alpha+\tau-2}.$

From this one derives that for some constants

$CM^{1+\alpha}|\bar{x}-\bar{y}|^{(\tau-1)\alpha+\tau-2}\leq
CM^{1+\alpha}|h|_{\infty}|\bar{x}-\bar{y}|^{(\tau-1)\alpha}+CM^{-\gamma}|\bar{x}-\bar{y}|^{\tau_{p}-\tau\gamma}+|f|_{\infty}+|g|_{\infty}$

which is a contradiction as soon as $\delta$ is small enough, by the
assumption on $\tau$. We have obtained that for all $x,y$ in
$\overline{\Omega}$

$u(x)-v(y)\leq C|x-y|^{\tau}.$

$\hfill\square$

###### Corollary 1

The solutions constructed in the proof of the previous section are Hölder
continuous up to the boundary, with an exponent $\tau$ arbitrary close to $1$
when $\gamma<1$, and $\tau\leq{\tau_{p}+\alpha+2\over 1+\alpha+\gamma}$, when
$\gamma>1$. They are in both cases Lipschitz continuous inside $\Omega$.

Proof The Lipschitz interior continuity is immediate by using the results of
[5], remarking that on a compact set of $\Omega$, by the strong maximum
principle and the continuity of $u$, $pu^{-\gamma}$ is bounded.
$\hfill\square$
Using Theorem 3.2 we have the following comparison result between sub- and
super solutions.

###### Theorem 3.3

Suppose that $u,v>0$ are respectively sub- and super-solutions of

$|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla
u)+c(x)u^{1+\alpha}+p(x)u^{-\gamma}=0\mbox{ in }\Omega$

and are zero on the boundary. Then $u\leq v$ in $\Omega$.

Proof. Note that for $y<\varepsilon:=\left({\gamma\min
p\over(1+\alpha)|c|_{\infty}}\right)^{1\over 1+\alpha+\gamma}$ the function
$y\mapsto c(x)y^{1+\alpha}+p(x)y^{-\gamma}$ is decreasing. By upper-
semicontinuity, since $u$ is zero on the boundary, for all $\varepsilon$ there
exists $\delta$ so that for $d<\delta$ one has $u(x)\leq\varepsilon$ .

Suppose by contradiction that $u>v$ somewhere. We suppose first that the
supremum of $u-v$ is achieved inside $d(x,\partial\Omega)\leq\delta$. Let
$\bar{x}$ be some point in this set where the supremum is achieved. Then one
has $0<v(\bar{x})<u(\bar{x})\leq\varepsilon$. In particular
$c(\bar{x})u(\bar{x})^{1+\alpha}+p(\bar{x})(u(\bar{x}))^{-\gamma}<c(\bar{x})v(\bar{x})^{1+\alpha}+p(\bar{x})(v(\bar{x}))^{-\gamma}$.

Using the usual doubling of variables, say defining in the case $\alpha>0$

$\psi_{j}(x,y)=u(x)-v(y)-{j\over 2}|x_{j}-y_{j}|^{2}$

(the case $\alpha<0$ requires the changes provided at the end of the proof),
there exist $x_{j}$ and $y_{j}$ in $\\{x\in\Omega:\
d(x,\partial\Omega)\leq\delta\\}$ and $(X_{j},Y_{j})$ in ${\mathcal{S}}^{2}$
so that

$(j(x_{j}-y_{j}),X_{j})\in\overline{J^{2,+}}u(x_{j}),\
(j(x_{j}-y_{j}),-Y_{j})\in\overline{J^{2,-}}v(y_{j})$

and

$\left(\begin{array}[]{cc}X_{j}&0\\\ 0&Y_{j}\end{array}\right)\leq
2j\left(\begin{array}[]{cc}I&-I\\\ -I&I\end{array}\right).$

Furthermore by the boundary conditions, neither $x_{j}$, nor $y_{j}$ belong to
$\partial\Omega$.

From the Lipshitz estimates between sub and super-solutions in Theorem 3.2,
one has

$j|x_{j}-y_{j}|^{2}+\sup(u-v)\leq
u(x_{j})-v(y_{j})\leq\sup(u-v)+|x_{j}-y_{j}|.$

From what we derive that $j|x_{j}-y_{j}|$ is bounded. In particular, using $h$
continuous, one has $|h(x_{j})-h(y_{j})|(j|x_{j}-y_{j}|)^{1+\alpha}=o(1)$.

Using $X_{j}+Y_{j}\leq 0$, and, since $F$ satisfies (2), one can write

$\displaystyle-c(x_{j})u(x_{j})^{1+\alpha}-p(x_{j}){u(x_{j})}^{-\gamma}$
$\displaystyle\leq$
$\displaystyle|j(x_{j}-y_{j})|^{\alpha}(F(X_{j})+h(x_{j})\cdot
j(x_{j}-y_{j}))$ $\displaystyle\leq$
$\displaystyle|j(x_{j}-y_{j})|^{\alpha}(F(-Y_{j})+h(y_{j})\cdot
j(x_{j}-y_{j}))+o(1)$ $\displaystyle\leq$
$\displaystyle-c(y_{j})v(y_{j})^{1+\alpha}-p(y_{j})(v(y_{j}))^{-\gamma}+o(1),$

and using the continuity of $c$, $h$ and $p$ and passing to the limit when $j$
goes to infinity, one gets a contradiction.

Suppose now that $\bar{x}$ is not in $\\{x\in\Omega:\
d(x,\partial\Omega)\leq\delta\\}$. Then there exists $0<\kappa<M$ so that for
all $x\in\Omega$ such that $d(x)<\delta$, $u(x)\leq
v(x)+\sup(u-v)-\kappa:=v(x)+M-\kappa$. We then use the change of function
$U=\log u$ and $V=\log(v+M-\kappa)$ in the set
$\Omega_{\delta}=\\{x\in\Omega,d(x,\partial\Omega)>\delta\\}$ One has $U\leq
V$ on the boundary, and $U$ and $V$ are respectively sub-and super-solutions
of the equations

$|\nabla U|^{\alpha}(F(D^{2}U+\nabla U\otimes\nabla U)+h(x)\cdot\nabla
U)+c(x)+e^{-U(1+\alpha+\gamma)}p(x)\geq 0$

and

$|\nabla V|^{\alpha}(F(D^{2}V+\nabla V\otimes\nabla V)+h(x)\cdot\nabla
V)+c(x)+e^{-V(1+\alpha+\gamma)}p(x)\leq 0.$

Using the comparison principle for these type of equations, (see for example
[10], Theorem 5.1), remarking that $y\mapsto e^{-y(1+\alpha+\gamma)}p(x)$ is
decreasing one gets that $U\leq V$ everywhere in $\Omega_{\delta}$. This
implies that $u\leq v+M-\kappa$ everywhere in $\Omega$, a contradiction with
the definition of the supremum.

We breafly give the changes to bring in the case $\alpha<0$. In that case in
the first step, the function $\psi_{j}$ must be replaced by

$\psi_{j}(x,y)=u(x)-v(y)-{j\over q}|x_{j}-y_{j}|^{q}$

where $q>{\alpha+2\over 1+\alpha}$. We next follow the lines in the comparison
Theorem 3.3 in [9], (A key point consists in observing that $x_{j}\neq
y_{j}$).

$\hfill\square$

###### Corollary 2

There is uniqueness of solution for the equation (1).

##  4 Regularity of the unique viscosity solution of (1).

###  4.1 Interior regularity of the viscosity solution

Interior regularity of the unique viscosity solution of the problem (1) is
easily obtained by the results about regularity of viscosity solutions of the
following equation

$\displaystyle|\nabla u|^{\alpha}F(D^{2}u)+h(x)\cdot\nabla u|\nabla
u|^{\alpha}=f$ (7)

which are recalled in the introduction, ( [20], [9], [8]). Indeed, let $u$ be
a solution of equation (1), let $f=-c(x)u^{1+\alpha}-p(x)u^{-\gamma}$, and let
us consider the equation

$|\nabla v|^{\alpha}(F(D^{2}v)+h(x)\cdot\nabla
v)=-c(x)u^{1+\alpha}-p(x)u^{-\gamma}\ {\rm in}\ \Omega.$

It is immediate to check that $u$ is a viscosity solution of this equation,
hence, using the classical regularity results recalled below,
$u\in{\mathcal{C}}^{1,\beta}(\Omega)$.

###  4.2 ”Regularity” up to the boundary

We begin to remark that when $\gamma>1$, even the Lipschitz regularity up to
the boundary does not hold. Next we will see some cases in which we can ensure
the ${\mathcal{C}}^{1}$ regularity for $\gamma<1$. We note that if $x_{0}\
\in\ \partial\Omega$, $\overrightarrow{n}$ denotes the inner normal to
$\partial\Omega$ at $x_{0}$, and $\phi_{1}$ is some eigenfunction for
$\lambda_{1}^{\beta_{c,\alpha,\gamma}}$, then by Hopf boundary principle

$\lim_{s\rightarrow
0^{+}}{\dfrac{\phi_{1}(x_{0}+s\overrightarrow{n})}{s}}=\lim_{s\rightarrow
0^{+}}{\dfrac{\phi_{1}(x_{0}+s\overrightarrow{n})-\phi_{1}(x_{0})}{s}}=\nabla\phi_{1}(x_{0})\cdot\overrightarrow{n}.$

If $\gamma>1$, let us recall that for some convenient $b_{1}>0$ and for
$t=\dfrac{2+\alpha}{1+\alpha+\gamma}<1$, $b_{1}\phi_{1}(x)^{t}$ is a sub-
solution, and then by the comparison principle, $u(x)\geq\
b_{1}\phi_{1}(x)^{t}$. It follows that, for $s>0$,

$\dfrac{u(x_{0}+s\overrightarrow{n})-u(x_{0})}{s}\geq\
b_{1}\phi_{1}(x_{0}+s\overrightarrow{n})^{t-1}\dfrac{\phi_{1}(x_{0}+s\overrightarrow{n})}{s}.$

Therefore since $t<1$, $\phi_{1}=0$ on the boundary, using the existence of a
positive constant $C$ such that
$\dfrac{\phi_{1}(x_{0}+s\overrightarrow{n})}{s}>C$ given by Hopf principle,
then

$\lim_{s\rightarrow
0^{+}}{\dfrac{u(x_{0}+s\overrightarrow{n})-u(x_{0})}{s}}=+\infty,$

so $u$ cannot be Lipchitz continuous on $\overline{\Omega}$.

###  4.3 Regularity up to the boundary when $N=1$

Suppose that $N=1$, $h=c=0$ and $p\equiv 1$. We prove below the
${\mathcal{C}}^{1}$ regularity up to the boundary in the case $\gamma<1$,
while in the case $\gamma>1$ the solution cannot be Lipschitz continuous up to
the boundary. We can suppose without loss of generality that $\Omega=]0,1[$.
Let us consider the equation

$|u^{\prime}|^{\alpha}u^{\prime\prime}+u^{-\gamma}=0,$ $u(0)=u(1)=0,$

multiplying by $u^{\prime}$ and integrating one gets

${|u^{\prime}|^{2+\alpha}\over 2+\alpha}+{u^{1-\gamma}\over 1-\gamma}=C,$

where $C$ is a constant, and then when $\gamma>1$,
$\lim_{x\rightarrow(0,1)}{|u^{\prime}|^{2+\alpha}\over 2+\alpha}=+\infty$.

For the special case where $\gamma=1$, multiplying by $u^{\prime}$ and
integrating one gets the equation

${|u^{\prime}|^{2+\alpha}\over 2+\alpha}+\log{u}=C,$

Note that the equation is invariant by the change $x\mapsto 1-x$, $u$ is
concave and then ${1\over 2}$ is a maximum point for $u$, so
$u^{\prime}({1\over 2})=0$. Then, for some positive constant $C$ defined by
$u({1\over 2})=e^{C}$,

$u^{\prime}(x)=\left\\{\begin{array}[]{lc}\left((2+\alpha)(C-\log{u})\right)^{1\over
2+\alpha}&{\rm if}\ x<{1\over 2}\\\
-\left((2+\alpha)(C-\log{u})\right)^{1\over 2+\alpha}&{\rm if}\ x>{1\over
2}.\end{array}\right.$ (8)

Consequently,

$\lim_{x\rightarrow 0}u^{\prime}=\infty.$

If $\gamma<1$ the solutions are given , for some positive constant $C$ defined
by $u({1\over 2})=(C(1-\gamma))^{1\over 1-\gamma}$

$u^{\prime}(x)=\left\\{\begin{array}[]{lc}\left((2+\alpha)(C-{u^{1-\gamma}\over
1-\gamma})\right)^{1\over 2+\alpha}&{\rm if}\ x<{1\over 2}\\\
-\left((2+\alpha)(C-{u^{1-\gamma}\over 1-\gamma})\right)^{1\over
2+\alpha}&{\rm if}\ x>{1\over 2}.\end{array}\right.$ (9)

From equation (9), $u^{\prime}$ is continuous up to the boundary.

###  4.4 Existence of radial solution when $N>1$ and regularity up to the
boundary when $\gamma<1$ and $\Omega$ is a ball.

In all this sub-section we still suppose that $h=c=0$, $p\equiv 1$ and that
$N\geq 2$.
We begin to prove the existence of a radial solution in the particular case
where $F=tr$. We will do the general case later. We begin to construct in a
neighborhood of $0$ a solution by using a fixed point argument. First of all
suppose $v(0)=1,v^{\prime}(0)=0$ (necessary in the radial case). Note that we
follow the method employed in particular in [16]. Let us consider the map
$v\mapsto T(v)$ where

$T(v)(r)=1-\int_{0}^{r}\left({1\over
s^{(1+\alpha)(N-1)}}\int_{0}^{s}\lambda^{(1+\alpha)(N-1)}(1+\alpha)v^{-\gamma}(\lambda)d\lambda\right)^{1\over
1+\alpha}ds.$

We prove that for $r_{o}$ small enough, $T$ possesses a fixed point defined in
$[0,r_{o}[$. Let us define

$r_{o}=\left({{((N-1)(1+\alpha)+1)^{1\over 1+\alpha}}(2+\alpha)\over
2^{1+{|\alpha|+1\over 1+\alpha}\gamma}(\max(\gamma,1+\alpha))^{1\over
1+\alpha}(1+\alpha)}\right)^{1+\alpha\over 2+\alpha},$

and let us consider the ball

$B=\left\\{v\in{\mathcal{C}}(B(0,r_{o})),|v(x)-1|_{\infty}<{1\over
2}\right\\}.$

Then for $r<r_{o}$, $T$ maps $B$ into itself. Indeed :

$\displaystyle|T(v)-1|_{\infty}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{r}\left({1\over
s^{(1+\alpha)(N-1)}}\int_{0}^{s}\lambda^{(1+\alpha)(N-1)}(1+\alpha)2^{\gamma}d\lambda\right)^{1\over
1+\alpha}ds$ $\displaystyle=$ $\displaystyle r^{2+\alpha\over
1+\alpha}{(2^{\gamma}(1+\alpha))^{1\over
1+\alpha}(1+\alpha)\over{((N-1)(1+\alpha)+1)^{1\over 1+\alpha}}(2+\alpha)}$
$\displaystyle\leq$ $\displaystyle{1\over 2}.$

In order to check that $T$ is a contracting mapping on $B$, we denote for
$v,w$ in $B$

$X(s)={1\over
s^{(1+\alpha)(N-1)}}\int_{0}^{s}\lambda^{(1+\alpha)(N-1)}(1+\alpha)v^{-\gamma}d\lambda$

and

$Y(s)={1\over
s^{(1+\alpha)(N-1)}}\int_{0}^{s}\lambda^{(1+\alpha)(N-1)}(1+\alpha)w^{-\gamma}d\lambda.$

Note that

$\displaystyle\rm{i)}$ $\displaystyle|v^{-\gamma}-w^{-\gamma}|\leq
2^{\gamma+1}\gamma|v-w|,$ $\displaystyle\rm{ii)}$
$\displaystyle{{s2^{-\gamma}(1+\alpha)\over(1+\alpha)(N-1)+1}\leq
X(s)\leq{s2^{\gamma}(1+\alpha)\over(1+\alpha)(N-1)+1}}$ $\displaystyle\ \ \ \
\rm{\ and\ }{s2^{-\gamma}1+\alpha)\over(1+\alpha)(N-1)+1}\leq
Y(s)\leq{s2^{\gamma}(1+\alpha)\over(1+\alpha)(N-1)+1},$
$\displaystyle\rm{iii)}$
$\displaystyle|X(s)-Y(s)|\leq{1+\alpha\over(1+\alpha)(N-1)+1}|v^{-\gamma}-w^{-\gamma}|s.$

Then using i), ii), iii) and the mean value’s Theorem, for some
$\theta\in]0,1[$

$\displaystyle|X(s)^{1\over 1+\alpha}-Y(s)^{1\over 1+\alpha}|$
$\displaystyle\leq$ $\displaystyle{1\over
1+\alpha}|X(s)-Y(s)||X(s)+\theta(Y(s)-X(s))|^{-\alpha\over 1+\alpha}$
$\displaystyle\leq$ $\displaystyle 2^{1+{|\alpha|+1\over
1+\alpha}\gamma}\gamma^{1\over
1+\alpha}\left({1+\alpha\over(1+\alpha)(N-1)+1}\right)^{1\over
1+\alpha}s^{1\over 1+\alpha}|v-w|.$

As a consequence

$\displaystyle|T(v)-T(w)|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{r}|X(s)^{1\over 1+\alpha}-Y(s)^{1\over 1+\alpha}|ds$
$\displaystyle\leq$ $\displaystyle 2^{1+{1+|\alpha|\over
1+\alpha}\gamma}\gamma^{1\over
1+\alpha}\left({1+\alpha\over(1+\alpha)(N-1)+1}\right)^{1\over
1+\alpha}\left({1+\alpha\over 2+\alpha}\right)r^{2+\alpha\over
1+\alpha}|v-w|,$

and then $T$ is a contraction mapping.

This gives the local existence and uniqueness of a fixed point, denoted $u$,
around $0$. We can suppose, up to replace $r_{o}$ by some smaller number, that
$u^{\prime}<0$ and $u>0$ in the whole interval $]0,r_{o}[$. Now if $r_{1}>0$
is so that $u^{\prime}(r_{1})<0$, Cauchy Lipschitz Theorem gives the local
existence and uniqueness of a solution. For that it is sufficient to consider
the ordinary differential equation :

$\left(\begin{array}[]{c}v^{\prime}\\\
w^{\prime}\end{array}\right):=\varphi(v,w)=\left(\begin{array}[]{c}|w|^{-{\alpha\over\alpha+1}}w\\\
(-v^{-\gamma}-{N-1\over r}w)\end{array}\right),$

with $v(r_{1})=u(r_{1}),\ w(r_{1})=u^{\prime}(r_{1})$, and to observe that
$\varphi$ is a Lipschitz function of $(v,w)$ as long as neither $w$, nor $v$
takes the value $0$. We denote by $u$ the fixed point for $T$ in $[0,r_{1}]$
extended by the unique local solution of the previous ODE, as long as it is
defined.

Note that $u$ satisfies on $[r_{1},r[$

$u(r)=u(r_{1})-\int_{r_{1}}^{r}\left({1\over
s^{(1+\alpha)(N-1)}}\int_{0}^{s}\lambda^{(1+\alpha)(N-1)}(1+\alpha)u^{-\gamma}(\lambda)d\lambda\right)^{1\over
1+\alpha}ds.$

Then as long as $u>0$ one has

$u(r)=1-\int_{0}^{r}\left({1\over
s^{(1+\alpha)(N-1)}}\int_{0}^{s}\lambda^{(1+\alpha)(N-1)}(1+\alpha)u^{-\gamma}(\lambda)d\lambda\right)^{1\over
1+\alpha}ds.$

Since $u$ has values in $]0,1[$, and $u$ is not identically equal to $1$

$\int_{0}^{r}\left({1\over
s^{(1+\alpha)(N-1)}}\int_{0}^{s}\lambda^{(1+\alpha)(N-1)}(1+\alpha)u^{-\gamma}(\lambda)d\lambda\right)^{1\over
1+\alpha}ds\geq r^{2+\alpha\over 1+\alpha}{(1+\alpha)^{2+\alpha\over
1+\alpha}\over{((N-1)(1+\alpha)+1)^{1\over 1+\alpha}}(2+\alpha)}$

and then taking $R$ so that $R^{2+\alpha\over
1+\alpha}{(1+\alpha)^{2+\alpha\over 1+\alpha}\over{((N-1)(1+\alpha)+1)^{1\over
1+\alpha}}(2+\alpha)}>1$, one obtains that there exists $\bar{r}<R$, so that
$u(\bar{r})=0$.

We then consider $\tilde{u}$ defined as :

$\tilde{u}(r)=Cu(\bar{r}\ r),\rm{\ with}\
C:=\bar{r}^{-{2+\alpha\over\gamma+\alpha+1}}.$

Then $\tilde{u}$ solves the equation

$|\tilde{u}^{\prime}|^{\alpha}(\tilde{u}^{\prime\prime}+{N-1\over
r}\tilde{u}^{\prime})+\tilde{u}^{-\gamma}=0$

and $\tilde{u}(1)=0$.

The computations above can easily be generalized to the case where $p$ is a
radial function which satisfies the assumptions of the article.

We now observe that in the radial case, when $\gamma<1$ the solution
$\tilde{u}$ above is ${\mathcal{C}}^{1}$. Indeed, multiplying

$|\tilde{u}^{\prime}|^{\alpha}\tilde{u}^{\prime\prime}+{N-1\over
r}|\tilde{u}^{\prime}|^{\alpha}\tilde{u}^{\prime}+\tilde{u}^{-\gamma}=0$

by $\tilde{u}^{\prime}r^{(N-1)(2+\alpha)}$ and integrating, one has

${d\over
dr}\left(\frac{|\tilde{u}^{\prime}|^{2+\alpha}}{2+\alpha}r^{(N-1)(2+\alpha)}+{\tilde{u}^{1-\gamma}\over
1-\gamma}r^{(N-1)(2+\alpha)}\right)=(N-1)(2+\alpha)r^{(N-1)(2+\alpha)-1}{\tilde{u}^{1-\gamma}\over
1-\gamma},$

and then, integrating between ${1\over 2}$ and $r$ one gets that

$\frac{|\tilde{u}^{\prime}|^{2+\alpha}}{2+\alpha}r^{(N-1)(2+\alpha)}+{\tilde{u}^{1-\gamma}\over
1-\gamma}r^{(N-1)(2+\alpha)}-C={1\over 1-\gamma}\int_{1\over
2}^{r}(N-1)(2+\alpha)s^{(N-1)(2+\alpha)-1}\tilde{u}^{1-\gamma}(s)ds,$

where
$C=\left(\frac{|\tilde{u}^{\prime}|^{2+\alpha}}{2+\alpha}r^{(N-1)(2+\alpha)}+{\tilde{u}^{1-\gamma}\over
1-\gamma}r^{(N-1)(2+\alpha)}\right)({1\over 2})$, which proves that
$|\tilde{u}^{\prime}|^{2+\alpha}$ has a finite limit when $r$ goes to $1$.

We now do the general case . We begin with the case of one of the Pucci’s
operator. We will deduce the general case by using the fact that the operator
$F$ is sandwiched between the two Pucci’s operators, ${\mathcal{M}}^{+}_{a,A}$
and ${\mathcal{M}}^{-}_{a,A}$.

Suppose that $F={\mathcal{M}}^{+}_{a,A}$. We begin to prove local existence
and uniqueness of a solution near $0$. For that aim we argue as in the case of
the Laplacian , say we observe that if $r_{o}$ is replaced by $r_{o}a^{1\over
2+\alpha}$ and $B$ is defined as in the Laplacian case, we have a fixed point,
denoted $u_{o}$, defined on $[0,r_{o}[$ for the operator $T$ in $B$ :

$T(v)=1-\int_{0}^{r}\left({1\over
s^{(N-1)(1+\alpha)}}\int_{0}^{s}\lambda^{(N-1)(1+\alpha)}{(1+\alpha)\over
a}v^{-\gamma}d\lambda\right)^{1\over 1+\alpha}ds.$

Up to replace $r_{o}$ by some smaller number, one can assume that for
$r<r_{o}$, ${u^{-\gamma}\over a}+{(N-1)u^{\prime}\over r}<0$. Let $r_{1}$ be
so that $r_{1}\in]0,r_{o}[$. We consider for $r>r_{o}$, the ordinary
differential equation

$\left(\begin{array}[]{c}v^{\prime}\\\
w^{\prime}\end{array}\right):=\varphi(v,w)=\left(\begin{array}[]{c}|w|^{-{\alpha\over\alpha+1}}w\\\
f_{a,A}(-v^{-\gamma}-{N-1\over r}aw)\end{array}\right)$

where

$f_{a,A}(x)={x^{+}\over A}-{x^{-}\over a},$

with $v(r_{o})=u(r_{o})\neq 0$, $w(r_{o})=u^{\prime}(r_{o})\neq 0$. The
function $\varphi$ is Lipschitz continuous as long as $v\neq 0$ and $w\neq 0$.
Then Cauchy Lipschitz Theorem ensures local existence and uniqueness of
solution for $r>r_{o}$. Let $r_{1}>r_{o}$ and $u_{1}$ defined on
$[r_{o},r_{1}[$ which solves this ordinary differential equation. Let

$u=\left\\{\begin{array}[]{cc}u_{o}&{\rm if}\ r<r_{o}\\\ u_{1}&{\rm if}\
r\in[r_{o},r_{1}[\end{array}\right.$

We observe that $u^{\prime}<0$ as long as $u(r)>0$. Indeed, suppose that
$r_{2}>r_{o}$ is so that $u^{\prime}(r)\leq 0$, $u(r)>0$ for $r<r_{2}$, and
$u^{\prime}(r_{2})=0$. One would have, since
$(|u^{\prime}|^{\alpha}u^{\prime})$ is continuous,
$(|u^{\prime}|^{\alpha}u^{\prime})^{\prime}(r_{2})\geq 0$, while by the
equation this quantity is $<0$ on $r_{2}$, a contradiction. As a consequence,
$u$ is a solution of the equation related to ${\cal M}^{+}_{a,A}$ on $[0,R[$
where $R\leq\infty$ is so that $u(r)>0$, $u^{\prime}(r)<0$ for $r<R$ and
$\lim_{r\rightarrow R}\inf(u,u^{\prime})(r)=0$. As a conclusion we have
obtained a solution $u$ on some intervall $[0,R[$, with $R$ so that $u(r)>0$,
$u^{\prime}(r)<0$ for $r<R$, and $\lim_{r\rightarrow R}u(r)=0$. In the
following lines we prove that $R<\infty$.

Since $u^{\prime}\neq 0$ on $]0,R[$, $u^{\prime\prime}$ is continuous and then
the sets $\\{r>0,\ u^{\prime\prime}(r)<0\\}$ and $\\{r>0,\
u^{\prime\prime}(r)>0\\}$ are open. Then each of these sets is a countable
union of intervals.

So there exist a numerable set of $r_{i}$ so that on $]0,r_{1}[$, and on
$]r_{2i},r_{2i+1}[$, $u^{\prime\prime}<0$, on $]r_{2i+1},r_{2i+2}[$,
$u^{\prime\prime}>0$. Then on $]r_{2i},r_{2i+1}[$, $u$ satisfies

$\displaystyle-(-u^{\prime})^{1+\alpha}(s)$ $\displaystyle=$
$\displaystyle-(-u^{\prime})^{1+\alpha}(r_{2i})\left({r_{2i}\over
s}\right)^{(N-1)(1+\alpha)}$ $\displaystyle-$ $\displaystyle{1\over
s^{(N-1)(1+\alpha)}}\int_{r_{2i}}^{s}{(1+\alpha)\over
a}u^{-\gamma}(\lambda)\lambda^{(N-1)(1+\alpha)}d\lambda$ $\displaystyle:=$
$\displaystyle h_{2i}(s)(\leq 0),$

and then

$u(r)=u(r_{2i})-\int_{r_{2i}}^{r}(-h_{2i}(s))^{1\over 1+\alpha}ds.$

On $]r_{2i+1},r_{2i+2}[$ we have the analogous formula with

$\displaystyle h_{2i+1}(s)$
$\displaystyle=-(-u^{\prime})^{1+\alpha}(r_{2i+1})\left({r_{2i+1}\over
s}\right)^{(N-1)(1+\alpha)a\over A}$ $\displaystyle\ \ \ \ -{1\over
s^{(N-1)(1+\alpha)a\over A}}\int_{r_{2i+1}}^{s}{(1+\alpha)\over
A}u^{-\gamma}(\lambda)\lambda^{(N-1)(1+\alpha)a\over A}d\lambda.$

We observe next that there exists $\bar{r}$ so that $u(\bar{r})=0$. For that
aim we write for $r\in]r_{2k-1},r_{2k}[$

$u(r)=u(0)+(u(r_{0})-u(0))+\sum_{0}^{k-1}\left(u(r_{2i+1})-u(r_{2i})\right)+u(r)-u(r_{2k-1}).$

Note that $u(r_{2i+1})-u(r_{2i})<0$, and since $u$ is bounded by $u(0)$,

$\displaystyle u(r)-u(r_{2k-1})$ $\displaystyle\leq$
$\displaystyle-(u(0))^{-\gamma\over 1+\alpha}\int_{r_{2k-1}}^{r}\left({1\over
s^{(N-1)(1+\alpha)a\over A}}\int_{r_{2i+1}}^{s}{(1+\alpha)\over
A}\lambda^{(N-1)(1+\alpha)a\over A}d\lambda\right)^{1\over 1+\alpha}ds$
$\displaystyle\leq$ $\displaystyle-C(u(0))^{-\gamma\over
1+\alpha}(r^{2+\alpha\over 1+\alpha}-r_{2k-1}^{2+\alpha\over 1+\alpha}).$

where $C$ is a positive constant. An analogous formula holds for
$r\in]r_{2k},r_{2k+1}[$.

Suppose that $r_{2i}>1$, then for all $k\geq i$, on $[r_{2k},r_{2k+1}[$,
$-\lambda^{(N-1)(1+\alpha)}\leq-\lambda^{(N-1)(1+\alpha)a\over A}$ so in the
previous inequality we do ”as if ” the same formula holds for the case
$u^{\prime\prime}\geq 0$ or $\leq 0$ as soon as $r_{2i}$ is greater than $1$,
and then for $r\in[r_{2i},r_{2i+1}]$

$u(r)\leq u(r_{2i})-\int_{r_{2i}}^{r}\left({1\over
s^{(N-1)(1+\alpha)}}\int_{r_{2i}}^{s}{(1+\alpha)\over
a}u^{-\gamma}(\lambda)\lambda^{(N-1)(1+\alpha)a\over A}d\lambda\right)^{1\over
1+\alpha}ds.$

Finally one has for any $r$, $r>1$ , denoting by $i_{o}$ the first $i$ so that
$r_{2(i_{o}+1)}>1$, if it exists,

$\displaystyle u(r)$ $\displaystyle\leq$ $\displaystyle
u(0)+(u(r_{0})-u(0))+\sum_{i,i\leq
i_{o}-1}(u(r_{2i+1})-u(r_{2i}))+u(r)-u(r_{2i_{o}})$ $\displaystyle\leq$
$\displaystyle u(0)-\int_{r_{2i_{o}}}^{r}\left({1\over
s^{(N-1)(1+\alpha)}}\int_{r_{2i}}^{s}{(1+\alpha)\over
a}u^{-\gamma}(\lambda)\lambda^{(N-1)(1+\alpha)a\over A}d\lambda\right)^{1\over
1+\alpha}ds$ $\displaystyle\leq$ $\displaystyle u(0)-Cu(0)^{-\gamma\over
1+\alpha}(r^{2+\alpha\over 1+\alpha}-r_{2i_{o}}^{2+\alpha\over 1+\alpha})$
$\displaystyle\leq$ $\displaystyle u(0)+Cu(0)^{-\gamma\over
1+\alpha}-Cu(0)^{-\gamma\over 1+\alpha}r^{2+\alpha\over 1+\alpha}.$

And then for $r$ large enough this quantity becomes negative. If for all
$i,r_{2i}\leq 1$, since $r_{2i}$ is increasing let $l$ its limit, then one can
write for $r>l$

$u(r)\leq u(0)-Cu(0)^{-\gamma\over 1+\alpha}(r^{2+\alpha\over
1+\alpha}-l^{2+\alpha\over 1+\alpha})\leq u(0)-Cu(0)^{-\gamma\over
1+\alpha}(r^{2+\alpha\over 1+\alpha}-1)$

and the same conclusion follows. Let then $\bar{r}$ be so that $u(\bar{r})=0$.
We end the proof as in the case where ${\mathcal{M}}^{+}_{a,A}$ is replaced by
the Laplacian, and we have obtained a radial solution of the equation related
to ${\cal M}^{+}_{a,A}$ in the ball $B(0,1)$.

For the general case we observe that by the previous computations, there
exists $\bar{u}$ a radial solution for

$|\bar{u}^{\prime}|^{\alpha}{\mathcal{M}}^{+}_{a,A}(D^{2}\bar{u})=-\bar{u}^{-\gamma},\bar{u}(1)=0.$

Then it provides a super-solution for the equation, while, by obvious changes
in the analysis above, there exists $\underline{u}$ a radial solution for

$|\underline{u}^{\prime}|^{\alpha}{\mathcal{M}}^{-}_{a,A}(D^{2}\underline{u})=-{\underline{u}}^{\gamma},\underline{u}(1)=0$

which provides a sub-solution. By the comparison principle ( in the uniqueness
part), $\underline{u}\leq\overline{u}$. Using Perron’s method adapted to the
present context (see [4]), we obtain the existence of a radial solution of (1)
which lies between $\bar{u}$ and $\underline{u}$.

Acknowledgment : The author wishes to thank the anonymous referee for his
remarks which permit to improve considerably this paper.

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