#  $C^{\infty}$ regularity in semilinear free boundary problems

Daniel Restrepo  Department of Mathematics, Johns Hopkins University, 3400 N.
Charles Street, Baltimore, MD 21218<EMAIL_ADDRESS>and  Xavier Ros-Oton
ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain & Universitat de
Barcelona, Departament de Matemàtiques i Informàtica, Gran Via de les Corts
Catalanes 585, 08007 Barcelona, Spain & Centre de Recerca Matemàtica,
Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract.

We study the higher regularity of solutions and free boundaries in the Alt-
Phillips problem $\Delta u=u^{\gamma-1}$, with $\gamma\in(0,1)$. Our main
results imply that, once free boundaries are $C^{1,\alpha}$, then they are
$C^{\infty}$. In addition $u/d^{\frac{2}{2-\gamma}}$ and
$u^{\frac{2-\gamma}{2}}$ are $C^{\infty}$ too.

In order to achieve this, we need to establish fine regularity estimates for
solutions of linear equations with boundary-singular Hardy potentials $-\Delta
v=\kappa v/d^{2}$ in $\Omega$, where $d$ is the distance to the boundary and
$\kappa\leq\frac{1}{4}$. Interestingly, we need to include even the critical
constant $\kappa=\frac{1}{4}$, which corresponds to $\gamma=\frac{2}{3}$.

###### Key words and phrases:

Alt-Phillips, free boundary, higher regularity, Hardy potential, boundary
regularity

###### 2010 Mathematics Subject Classification:

35R35, 35B65, 35J61, 35J25

##  1\. Introduction

A classical problem in the Calculus of Variations concerns the study of
minimizers of functionals of the type

$J(u):=\int_{D}{\textstyle\frac{1}{2}}|\nabla u|^{2}+F(u).$

with boundary data $u=g\geq 0$ on $\partial D$. The potential $F$ is usually
asked to be nondecreasing, and with $F(0)=0$. Under these conditions,
minimizers of $J$ are non-negative and satisfy (in an appropriate sense) the
Euler-Lagrange equation

$\Delta u=F^{\prime}(u)\quad\mbox{in}\quad D.$ (1.1)

Two of the most famous and important cases are $F(u)=u_{+}$ (the obstacle
problem), and $F(u)=\chi_{\\{u>0\\}}$ (the Alt–Caffarelli or Bernoulli free
boundary problem). In general, when $F\notin C^{1,1}$, the solution will
develop free boundaries: there will be a region of $D$ in which $u=0$, and a
region in which $u>0$. The so-called free boundary is the interface
$\partial\\{u>0\\}$.

In this paper we are interested in the more general family of problems

$F(u)={\textstyle\frac{1}{\gamma}}u^{\gamma}\qquad\mbox{with}\qquad\gamma\in(0,2),$

which gives rise to the so-called Alt–Phillips problem, introduced in [4].
This family of problems obviously encompasses both the obstacle problem
($\gamma=1$), as well as the Alt–Caffarelli problem ($\gamma\to 0$).

The regularity theory for both the obstacle and the Alt–Caffarelli problems is
quite well understood, and in particular the free boundary is known to be
$C^{\infty}$ outside a closed set of singular points [30, 34, 17, 10].
Moreover, fine properties of the singular set have been established in [20,
25, 16, 22].

The regularity theory for minimizers in the Alt–Phillips problem has been
developed in the works [4, 6, 19, 36, 15], and the main known results may be
summarized as follows:

  1. (1)

Solutions $u$ are $C^{\frac{2}{2-\gamma}}$, and this is optimal.111Throughout
the paper, we denote $C^{\beta}=C^{k,\alpha}$ with $k+\alpha=\beta$ and
$\alpha\in(0,1]$, $k\in\mathbb{N}$.

  2. (2)

Free boundary points can be split into two sets: regular points, and singular
ponts.

  3. (3)

The regular set is an open subset of the free boundary, and it is
$C^{1,\alpha}$.

  4. (4)

The singular set is closed, and:

\- When $\gamma\in(0,1)$ its Hausdorff dimension is at most $n-3$.

\- When $\gamma\in[1,2)$ it is $(n-1)$-rectifiable.

Our main goal in this paper is to prove that the set of regular points is
actually $C^{\infty}$, and that solutions enjoy a $C^{\infty}$ regularity
estimate near regular points, namely $u/d^{\frac{2}{2-\gamma}}\in C^{\infty}$.

This has been only proved for $\gamma\in(1,2)$ by Fotouhi–Koch [21] and very
recently for $\gamma\in(\frac{2}{3},2)$ by Allen-Kriventsov-Shahgholian [2],
and has been open for a very long time in the case $\gamma\in(0,\frac{2}{3}]$.
Our proof is completely different from that in [21] and [2], and covers the
whole range of exponents $\gamma\in(0,2)$.

As we will see, quite interestingly, we find that in terms of higher order
regularity the problem behaves like the obstacle problem for
$\gamma\geq\frac{2}{3}$, while it is more similar to the Bernoulli problem for
$\gamma<\frac{2}{3}$. This is quite new and surprising, since in almost all
previous regularity results the only differences were in cases
$\gamma\in(1,2)$ vs $\gamma\in(0,1)$.

###  1.1. Main result

Our main result can be stated as follows. Recall that regular free boundary
points are those at which the free boundary $\partial\\{u>0\\}$ is
$C^{1,\alpha}$ in a neighborhood.

######  Theorem 1.1.

Let $\gamma\in(0,2)$ and $0\leq u\in C^{\frac{2}{2-\gamma}}$ be any viscosity
solution of

$\Delta u=u^{\gamma-1}\quad\textrm{in}\quad\\{u>0\\}\cap B_{1}.$ (1.2)

Assume $x_{0}\in\partial\\{u>0\\}\cap B_{1}$ is a regular free boundary point.
Then,

  * •

The free boundary $\partial\\{u>0\\}$ is $C^{\infty}$ in a neighborhood of
$x_{0}$.

  * •

The functions $u/d^{\frac{2}{2-\gamma}}$ and $u^{\frac{2-\gamma}{2}}$ are
$C^{\infty}$ in a neighborhood of $x_{0}$.

In addition to its own intrinsic interest, this result is important for
example in the study of stable solutions to the Alt–Phillips problem (in which
one needs at least second order expansions for solutions near
$\partial\Omega$), especially when $\gamma<\frac{2}{3}$; see [26].

Our approach to prove Theorem 1.1 is to find a PDE for the derivatives $u_{i}$
(and for their quotients $u_{i}/u_{j}$), to then use these equations to deduce
that solutions and free boundaries are smooth. This approach was introduced by
De Silva and Savin in the context of the obstacle problem [13], and also in
the Alt–Caffarelli problem; see [14].

In the obstacle problem, they proved that if $v_{2}>0$ in $\Omega\cap B_{1}$,
then

$\left.\begin{array}[]{r}\Delta v_{1}=\Delta
v_{2}=0\quad\textrm{in}\quad\Omega\cap B_{1}\\\
v_{1}=v_{2}=0\quad\textrm{on}\quad\partial\Omega\cap B_{1}\\\
\partial\Omega\quad\textrm{is}\quad C^{k,\alpha}\\\
\end{array}\right\\}\qquad\Longrightarrow\qquad\frac{v_{1}}{v_{2}}\in
C^{k,\alpha}(\overline{\Omega}\cap B_{1/2}),$

for any $k\geq 1$ and $\alpha\in(0,1)$. This, applied to the derivatives
$u_{i}$, and using that the normal vector $\nu$ to $\partial\Omega$ can be
written as

$\nu^{i}=\frac{u_{i}}{|\nabla
u|}=\frac{u_{i}/u_{n}}{\sqrt{\sum_{j}(u_{j}/u_{n})^{2}}}$

yields that

$\partial\Omega\in C^{k,\alpha}\quad\Longrightarrow\quad\frac{u_{i}}{u_{n}}\in
C^{k,\alpha}\quad\Longrightarrow\quad\nu\in
C^{k,\alpha}\quad\Longrightarrow\quad\partial\Omega\in C^{k+1,\alpha},$ (1.3)

thus yielding the $C^{\infty}$ regularity by a bootstrap argument.

In case of the Alt-Caffarelli free boundary problem, the argument goes in the
same spirit, but the PDE is completely different. It turns out that the
quotient $w:=u_{i}/u_{n}$ solves

$\begin{split}{\rm div}(u_{n}^{2}\nabla w)=0&\quad\textrm{in}\quad\Omega\cap
B_{1}\\\ \partial_{\nu}w=0&\quad\textrm{on}\quad\partial\Omega\cap
B_{1},\end{split}$

and thus by Schauder estimates we get

$\partial\Omega\in C^{k,\alpha}\quad\Longrightarrow\quad u_{n}\in
C^{k-1,\alpha}\quad\Longrightarrow\quad w\in
C^{k,\alpha}\quad\Longrightarrow\quad\nu\in
C^{k,\alpha}\quad\Longrightarrow\quad\partial\Omega\in C^{k+1,\alpha}.$ (1.4)

This yields that $\partial\Omega$ is $C^{\infty}$, and thus applying now
Schauder estimates (with Dirichlet conditions) to the solution $u$, we find
that $u\in C^{\infty}$.

###  1.2. The linearized equation

In our setting, derivatives $u_{i}$ are not harmonic but solve the linearized
equation

$\Delta u_{i}=(\gamma-1)u^{\gamma-2}u_{i}\quad\textrm{in}\quad\Omega\cap
B_{1}.$

At regular points, we know that

$u\sim c_{\gamma}d^{\frac{2}{2-\gamma}}$

where $d$ is the distance to the (free) boundary and222Intuitively, this is
because blow-ups of solutions at regular points converge to the function
$c_{\gamma}(x\cdot\nu)_{+}^{\frac{2}{2-\gamma}}$, the one dimensional solution
of the Alt-Phillips equation.

$c_{\gamma}:=\left(\frac{(2-\gamma)^{2}}{2\gamma}\right)^{\frac{1}{2-\gamma}};$
(1.5)

see [4]. Thus, since near regular points we also know that
$u/d^{\frac{2}{2-\gamma}}\in C^{\alpha}$, then the linearized equation becomes

$\Delta
u_{i}=\kappa_{\gamma}\frac{u_{i}}{d^{2}}+fd^{\frac{\gamma}{2-\gamma}-2}\quad\textrm{in}\quad\Omega\cap
B_{1},$ (1.6)

where $f|_{\partial\Omega}=0$, $f\in C^{\alpha}(\overline{\Omega}\cap B_{1})$,
and

$\kappa_{\gamma}:=(\gamma-1)c_{\gamma}^{\gamma-2}=\Big{(}\frac{\gamma}{2-\gamma}\Big{)}\Big{(}\frac{\gamma}{2-\gamma}-1\Big{)}.$
(1.7)

Notice that

$\kappa_{\gamma}\geq-{\textstyle\frac{1}{4}},$

with the minimum being attained at $\gamma=\frac{2}{3}$. Moreover, we also
have the symmetry property

$\kappa_{\gamma}=\kappa_{\frac{4-4\gamma}{4-3\gamma}}.$

Thus, we define the linearized operator

$L_{\kappa}(v):=-\Delta v+\kappa\,\frac{v}{d^{2}},$ (1.8)

Quite interestingly, this operator is the appearing in the Euler-Lagrange
equation of the Hardy-type inequality

$\int_{\Omega}|\nabla\xi|^{2}+\kappa\int_{\Omega}\frac{\xi^{2}}{d^{2}}\geq 0,$

which holds in bounded smooth domains for $\kappa>-\frac{1}{4}$, and does not
hold for $\kappa<-\frac{1}{4}$. In case $\kappa=-\frac{1}{4}$ the inequality
is known to hold in convex domains. While we will not use such variational
characterization, this shows that the range $\kappa\geq-\frac{1}{4}$ is very
natural, and the case $\kappa=-\frac{1}{4}$ is critical.

Our analysis will be based on non-variational arguments, and in particular we
will establish a comparison principle for $L_{\kappa}$ as well as suitable
barriers.

As we will see, there are two characteristic powers for the operator in
dimension one:

$L_{\kappa}(x^{\theta})=0\quad\textrm{for}\quad
x>0\qquad\Longleftrightarrow\qquad\theta=\frac{1\pm\sqrt{1+4\kappa}}{2}.$

We call the largest root $\theta_{+}$, whereas we will denote by $\theta_{-}$
the smallest root. The first one will behave somehow as the exponent of
solutions with Dirichlet conditions, while the second one somehow corresponds
to the exponent of solutions with Neumann conditions. The case
$\kappa=-\frac{1}{4}$ is slightly different, as there is only one exponent
$x^{1/2}$, but then there is a second 1D solution, namely $x^{1/2}\log x$.

Our main result for the linearized operator $L_{\kappa}$, which we believe is
of independent interest, reads as follows.

######  Theorem 1.2.

Let $\kappa\in[-\frac{1}{4},\infty)$, and let $\Omega\subset\mathbb{R}^{n}$ be
a $C^{k+1,\alpha}$ domain, $k\geq 0$ and $\alpha\in(0,1)$. Let $f\in
C^{k,\alpha}(\overline{\Omega}\cap B_{1})$ be such that
$f|_{\partial\Omega}\equiv 0$, and $d$ be the regularized distance to
$\partial\Omega$, given by Lemma 2.1.

Let $v\in C(\Omega\cap B_{1})$ be any solution to

$-\Delta
v+\kappa\,\frac{v}{d^{2}}=fd^{\theta_{+}-2}\quad\text{in}\quad\Omega\cap
B_{1},$ (1.9)

satisfying

$\begin{array}[]{rl}\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{\theta_{-}}}=0&\quad\textrm{if}\quad\kappa>-{\textstyle\frac{1}{4}\vspace{1mm}}\\\
\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{1/2}|\log
d|}=0&\quad\textrm{if}\quad\kappa=-{\textstyle\frac{1}{4}}.\end{array}$ (1.10)

Then, $v\in C^{\theta_{+}}(B_{1/2}\cap\overline{\Omega})$ and

$\big{\|}v/{d^{\theta_{+}}}\big{\|}_{C^{k,\alpha}(\overline{\Omega}\cap
B_{1/2})}\leq C\big{(}\|f\|_{C^{k,\alpha}(\overline{\Omega}\cap
B_{1})}+\|v\|_{L^{\infty}(\Omega\cap B_{1})}\big{)},$

where $C>0$ depends only on $\Omega$, $k$, $\alpha$, $n$, and $\kappa$.

Here, the assumption (1.10) acts as a Dirichlet boundary condition on
$\partial\Omega$ ruling out the existence of solutions behaving like the other
possible power. This is interesting because $v$ is not assumed to be bounded
and since $\theta_{-}<0$ when $\kappa>0$, (1.10) imposes a growth bound nearby
the boundary which, a posteriori, implies the boundedness of $v$.

We will also prove a modified version of the result allowing for general
pointwise expansions for $v$ of the form
$Q_{1}d^{\theta_{+}}+Q_{2}d^{\theta_{-}+1}$ nearby $\partial\Omega$ with
$Q_{1}$ and $Q_{2}$ polynomials of a suitable degree; see Theorem 4.2.

###  1.3. Strategy of the proofs

Recall that solutions $u$ to the Alt–Phillips problem behave like
$d^{\frac{2}{2-\gamma}}$, and thus their derivatives $u_{i}$ do so like
$d^{\frac{\gamma}{2-\gamma}}$. This means that we have two completely
different cases: for $\gamma\geq\frac{2}{3}$ the exponent
$\frac{\gamma}{2-\gamma}$ corresponds to $\theta_{+}$ of the linearized
operator $L_{\kappa}$

$\gamma\in[{\textstyle\frac{2}{3}},2)\qquad\Longrightarrow\qquad u_{i}\asymp
d^{\theta_{+}},$

while for $\gamma<\frac{2}{3}$ the exponent $\frac{\gamma}{2-\gamma}$
corresponds to $\theta_{-}$ of $L_{\kappa}$

$\gamma\in(0,{\textstyle\frac{2}{3}})\qquad\Longrightarrow\qquad u_{i}\asymp
d^{\theta_{-}}.$

In a sense, this means that _when $\gamma\geq\frac{2}{3}$ derivatives $u_{i}$
have zero Dirichlet boundary data_, while for $\gamma<\frac{2}{3}$ this is not
true anymore.

In both cases, the idea is to apply Theorem 1.2 inductively to show that
$u_{i}/d^{\frac{\gamma}{2-\gamma}}$, $u/d^{\frac{2}{2-\gamma}}$, and
$u_{i}/u_{n}$ are $C^{\infty}$, namely,

$\partial\Omega\in
C^{k+1,\alpha}\quad\overset{(a)}{\Longrightarrow}\quad\frac{u_{i}}{d^{\frac{\gamma}{2-\gamma}}},\frac{u}{d^{\frac{2}{2-\gamma}}}\in
C^{k,\alpha}\quad\overset{(b)}{\Longrightarrow}\quad\frac{u_{i}}{u_{n}}\in
C^{k+1,\alpha}\quad\Longrightarrow\quad\partial\Omega\in C^{k+2,\alpha}.$
(1.11)

Let us notice that since nearby regular points $u/d^{\frac{2}{2-\gamma}}\in
C^{\alpha}$ (see Lemma 2.3), from the scheme (1.11), (a) only has to be
considered when $k\geq 1$. However, it is not immediate to apply Theorem 1.2
to the derivatives $u_{i}$ in order to get (a). Indeed, by induction
hypothesis we know that $u/d^{\frac{2}{2-\gamma}}\in C^{k-1,\alpha}$ and
$u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{k-1,\alpha}$, so that the right hand
side $f$ in the equation (1.6) is then $C^{k-1,\alpha}$. Unfortunately, using
Theorem 1.2 (when $\gamma\geq\frac{2}{3}$) we then deduce that
$u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{k-1,\alpha}$, gaining no new
information.

To improve the smoothness of $u/d^{\frac{\gamma}{2-\gamma}}$ and
$u_{i}/d^{\frac{\gamma}{2-\gamma}}$, a key idea is to consider the function

$v:=u-c_{\gamma}d^{\frac{2}{2-\gamma}},$

and use the identity

$p^{\gamma-1}-q^{\gamma-1}=(\gamma-1)(p-q)q^{\gamma-2}+(p/q-1)^{2}q^{\gamma-1}\int_{0}^{1}(\gamma-1)(\gamma-2)\big{(}1+tp/q\big{)}^{\gamma-3}(1-t)dt$

for $p,q>0$, to show that

$\begin{split}L_{\kappa}v&=u^{\gamma-1}-\big{(}c_{\gamma}d^{\frac{2}{2-\gamma}}\big{)}^{\gamma-1}+gd^{\frac{2}{2-\gamma}-2}\\\
&=\kappa\frac{v}{d^{2}}+fd^{\frac{2}{2-\gamma}-2}+gd^{\frac{2}{2-\gamma}-2},\end{split}$
(1.12)

where $g\in C^{k,\alpha}$ and

$f:=\big{(}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{)}^{2}\int_{0}^{1}\bar{c}_{\gamma}\big{(}c_{\gamma}+tu/d^{\frac{2}{2-\gamma}}\big{)}^{\gamma-3}(1-t)dt.$

The crucial point here is that, even though by inductive hypothesis we only
know $u/d^{\frac{2}{2-\gamma}}$ is $C^{k-1,\alpha}$, we will show that $f$ is
actually _more_ regular, namely $f\in C^{k,\alpha}$ if $k\geq 2$.

The reason for this is that, roughly speaking, if a function $w$ is pointwise
$C^{m,\alpha}$ at $x=0$, and $w(0)=0$, then $w^{2}$ is actually pointwise
$C^{m+1,\alpha}$ at $x=0$ if 333When $m=0$, we have instead that $w^{2}$ is
$C^{2\alpha}$ at $0$. $m\geq 1$. And $w^{2}\eta$ will still be pointwise
$C^{m+1,\alpha}$ as long as $\eta$ is $C^{m,\alpha}$. This, yields that $f$ is
pointwise $C^{k,\alpha}$ at all boundary points, and combined with interior
estimates for $u$ allows us to prove that $f$ is indeed $C^{k,\alpha}$ in
$\overline{\Omega}$. Combining this with the fact that $v$ satisfies
simultaneously (1.10) and (4.47), it is possible to prove a modified version
of Theorem 1.2, i.e., Theorem 5.1, to deduce $v/d^{\frac{2}{2-\gamma}}\in
C^{k,\alpha}$.

Recalling the definition of $v$, and using again interior estimates for $u$,
we then prove the desired estimate $u_{i}/d^{\frac{\gamma}{2-\gamma}}\in
C^{k,\alpha}$ in (a).

Once we have (a) in (1.11), to prove (b) we notice that the quotient
$w:=u_{i}/u_{n}$ satisfies

$\begin{split}{\rm div}(u_{n}^{2}\nabla w)=0&\quad\textrm{in}\quad\Omega\cap
B_{1}\\\ \partial_{\nu}w=0&\quad\textrm{on}\quad\partial\Omega\cap
B_{1},\end{split}$

which is a degenerate elliptic equation since $u_{n}\to 0$ on
$\partial\Omega$. Still, thanks to (a), we have $u_{n}^{2}=d^{s}a(x)$, with
$s=\frac{2\gamma}{2-\gamma}$ and $a(x)\in C^{k,\alpha}$. Using this, we can
apply444Here, one has to be careful with the case $k=1$, in which the boundary
condition $\partial_{\nu}w$ must be understood in an appropriate sense.
Schauder estimates for degenerate elliptic equations with Neumann boundary
conditions due to Terraccini-Tortone-Vita [32], to show that $w\in
C^{k+1,\alpha}$.

Although a key point in showing (a) –for all values of $\gamma\in(0,2)$– is
Theorem 5.1 instead of Theorem 1.2, the former is actually a consequence of
the latter. Also, Theorem 1.2 has direct applications to the study of the
boundary behavior of solutions to equations involving operators with Hardy-
type potentials (see [24]).

To prove Theorem 1.2, we first need a comparison principle for the operator
$L_{\kappa}$ with boundary conditions (1.10). We prove it by constructing
explicit barriers, and reads as follows:

######  Lemma 1.3.

Let $\kappa\in[-\frac{1}{4},\infty)$, $\alpha\in(0,1)$,
$\Omega\subset\mathbb{R}^{n}$ be such that $\Omega\cap B_{1}$ is a
$C^{1,\alpha}$ domain, and $d$ be as in Theorem 1.2.

Let $v\in C(\Omega\cap B_{1})$ be any viscosity solution of

$-\Delta v+\kappa\,\frac{v}{d^{2}}\leq 0\quad\text{in}\quad\Omega\cap B_{1}$

satisfying

$\begin{array}[]{rl}\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{\theta_{-}}}\leq
0&\quad\textrm{if}\quad\kappa>-{\textstyle\frac{1}{4}\vspace{1mm}}\\\
\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{1/2}|\log d|}\leq
0&\quad\textrm{if}\quad\kappa=-{\textstyle\frac{1}{4}}.\end{array}$ (1.13)

Then, there exists $\rho\in(0,1)$ only depending on the $C^{1,\alpha}$ norm of
$\partial\Omega$ such that for any $r\in(0,\rho]$ if $v\leq 0$ on
$\partial\Omega\cap B_{r}$, then $v\leq 0$ in $\Omega\cap B_{r}$.

Using this, we prove uniform $C^{\theta_{+}}$ estimates for solutions to the
linearized equation $L_{\kappa}v=0$ with zero Dirichlet boundary conditions.
This gives us compactness on the family of solutions $v$ we are studying, and
thus we can run a contradiction and blow-up argument to get the higher order
expansion, which is developed in Section 4. This, combined with a
classification of global solutions in a half-space (see Proposition 3.5), will
yield Theorem 1.2, as well as its generalized version Theorem 4.2.

###  1.4. Acknowledgements

X.R. was supported by the European Research Council (ERC) under the Grant
Agreement No 101123223 (SSNSD), by the AEI project PID2021-125021NAI00
(Spain), AGAUR Grant 2021 SGR 00087 (Catalunya), AEI Grant RED2022-134784-T
funded by MCIN/AEI/10.13039/501100011033 (Spain), and the AEI Maria de Maeztu
Program for Centers and Units of Excellence in R&D CEX2020-001084-M.

###  1.5. Organization of the paper

In Section 2 we give some preliminary lemmas. Then, in Sections 3 and 4 we
develop the boundary regularity for the linearized operator $L_{\kappa}$,
proving in particular Theorem 1.2. Finally, in Section 5 we prove Theorem 1.1,
while in Section 6 we give some technical lemmas that are used throughout the
proofs.

###  1.6. Notations and conventions

We denote $\mathbf{P}_{k}$ the space of polynomials of order $k$ in $n$
variables. Polynomials will be written in their coefficient expansion
following convention

$Q\in\mathbf{P}_{k}\qquad\implies\qquad
Q(x)=\sum_{\alpha\in\mathbb{N}^{n}\,,\,|\alpha|\leq
k}q^{(\alpha)}x^{\alpha},\quad x\in\mathbb{R}^{n}.$

In the study of boundary regularity in a domain $\Omega$, we will always
assume that $0\in\partial\Omega$ and, whenever $\Omega$ is regular at $0$, we
will orient it so that the canonical vector $-e_{n}$ corresponds to its outer
unit normal at zero, unless otherwise stated. Note that these two
simplifications are possible thanks to the translation and rotation
invariances of (1.2). In concordance with this assumption, we will adopt the
notation $x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}$, calling
the first $n-1$ coordinates “tangential”, whereas we will refer to the
coordinate $x_{n}$ as the normal direction.

Associated to $\kappa\in[\frac{-1}{4},\infty)$, we define the quantities

$\theta_{+}=\frac{1+\sqrt{1+4\kappa}}{2},\qquad\theta_{-}=\frac{1-\sqrt{1+4\kappa}}{2},$
(1.14)

which arise in the study of the linearized operator (1.8) as discussed in the
introduction (see also Lemma 3.1).

When $\beta\notin\mathbb{N}$ we use the single index notation $C^{\beta}$ for
the Hölder space $C^{\lfloor\beta\rfloor,\beta-\lfloor\beta\rfloor}$ where
$\lfloor\cdot\rfloor$ denotes the integer part of a positive real number.
Regarding Hölder norms of domains, we will assume throughout this paper that
$B_{2}\cap\Omega$ is a graph with $C^{\beta}$ norm bounded by one. We remark
that this assumption does not incur in any loss of generality since any
solution $u$ to (1.2) can be rescaled as $u(rx)/r^{\frac{2}{2-\gamma}}$ to
obtain a new solution in a domain with smaller Hölder norm where the
graphycality assumption holds.

Finally, following a widely used notation, we will use $C$ within proofs of
propositions to indicate a unspecified constant, whose value will be allowed
to change from line to line, and whose dependences will be clearly established
in the statements of the corresponding propositions. We will make use of sub-
indices whenever we will want to underline certain hidden dependencies of the
constant.

##  2\. Regularized distance and first estimates

In our analysis we will use some classical properties of the distance function
to the boundary and some of its generalizations. It is a classical result
that, for $\beta\geq 1$, nearby $C^{\beta}$ pieces of boundary, the distance
function is $C^{\beta}$. A priori the domains we are dealing with are
$C^{1,\alpha}$, this forces us to consider regularized versions of the
distance function.

######  Lemma 2.1.

Let $\beta>1$ with $\beta\notin\mathbb{N}$, and let $\partial\Omega\cap B_{2}$
be a $C^{\beta}$ graph. Then, there exist a generalized distance function
$d\in C^{\infty}(\Omega\cap B_{1})\cap C^{\beta}(\overline{\Omega}\cap B_{1})$
satisfying

  1. (1)

$\frac{1}{C}\text{dist}(x)\leq d(x)\leq C\text{dist}(x),\qquad x\in\Omega\cap
B_{1},$ (2.1)

where $\text{dist}(x):=\text{dist}(x,\partial\Omega)$.

  2. (2)

$d(x)=x_{n}+Q(x)+g(x),\qquad x\in\Omega\cap B_{1},$ (2.2)

where $Q\in\mathbf{P}_{\lfloor\beta\rfloor}$ with its zeroth and first order
terms identically equal to zero and also satisfying
$\|Q\|_{L^{\infty}(B_{1})}\leq C$, and where $g\in C^{\infty}(\Omega\cap
B_{1})$ satisfies $|\nabla g|\leq C|x|^{\beta-1}$.

  3. (3)

For any $\lambda\in\mathbb{R}$, if $\beta\in(1,2)$

$|\Delta d^{\lambda}-\lambda(\lambda-1)d^{\lambda-2}|\leq
Cd^{\beta+\lambda-3}\,\mbox{ in $\Omega\cap B_{1}$,}$ (2.3)

and if $\beta>2$, then

$\Delta
d^{\lambda}-\lambda(\lambda-1)d^{\lambda-2}=d^{\lambda-1}g(x),\quad\mbox{in
$\Omega\cap B_{1}$,}$ (2.4)

with $g\in C^{\beta-2}(\overline{\Omega}\cap B_{1})\cap C^{\infty}(\Omega\cap
B_{1})$.

  4. (4)

In particular, we have that

$\Delta\Big{(}c_{\gamma}d^{\frac{2}{2-\gamma}}\Big{)}-\Big{(}c_{\gamma}d^{\frac{2}{2-\gamma}}\Big{)}^{\gamma-1}=h\,\mbox{
in $\Omega\cap B_{1}$,}$ (2.5)

with $|h|\leq Cd^{\beta+\frac{2}{2-\gamma}-3}$ if $\beta\in(1,2)$ and
$h=d^{\frac{\gamma}{2-\gamma}}g(x)$ with $g\in
C^{\beta-2}(\overline{\Omega}\cap B_{1})$ if $\beta>2$.

  5. (5)

If $\beta\in(1,2)$

$\displaystyle-\Delta\Big{(}d^{\frac{1}{2}}|\ln(d)|^{\frac{1}{2}}\Big{)}+\frac{1}{4}\frac{d^{\frac{2}{3}}|\ln(d)|^{\frac{1}{2}}}{d^{2}}=\frac{1}{4d^{\frac{3}{2}}|\ln(d)|^{\frac{3}{2}}}\Big{(}1+h\Big{)},$
(2.6)

in $\cap\\{d<1\\}\cap B_{1}$ with $|h|\leq C(\ln(d))^{2}d^{\beta-1}$.

###### Proof.

Let us start addressing (1) and (2). We follow the construction in [29] (see
also [18, Lemma B.0.1] considering

$G(x,\rho)=\int_{B_{1}}\text{dist}\Big{(}x-\frac{\rho}{2}z\Big{)}\phi(z)dz$
(2.7)

with $\phi\in C_{c}^{\infty}(B_{1})$ a radial smooth non-negative function
satisfying $\int_{B_{1}}\phi=1$. Then $d$ is defined implicitly by
$d(x)=G(x,d(x))$. In virtue of the implicit function theorem it follows that
$d\in C^{\beta}(\overline{\Omega}\cap B_{1})\cap C^{\infty}(\Omega\cap
B_{1})$. Additionally, (1) is a consequence of the construction as discussed
in [29]. On the other hand, by differentiating the defining expression for $d$
at a point $x\in\partial\Omega\cap B_{2}$, we get

$\nabla
d(x)=\nabla\text{dist}(x)\int_{B_{1}}\phi(z)dz-\frac{1}{2}\nabla\text{dist}(x)\cdot\int_{B_{1}}z\phi(z)dz=\nabla\text{dist}(x),$
(2.8)

thanks to the properties of $\phi$. Additionally, thanks to the uniform
estimates on $d$ and the orientation assumption on $\Omega$ (2) follows. We
also remark that this construction satisfies

$|\partial^{\mu}d|\leq C_{k}d^{\beta-|\mu|},\mbox{ for $|\mu|>\beta$}.$ (2.9)

For part (3), a direct computation yields

$\Delta d^{\lambda}=\lambda(\lambda-1)|\nabla
d|^{2}\frac{d^{\lambda}}{d^{2}}+\lambda d^{\lambda-1}\Delta d.$ (2.10)

So, if $\beta\in(1,2)$, we deduce from (2.8) that $|\nabla d(x)|=1$ on
$\partial\Omega\cap B_{1}$, which combined with the $C^{\beta}$ regularity
implies that

$\Big{|}|\nabla d|^{2}-1\big{|}\leq Cd^{\beta-1}\,\,\mbox{ in $\Omega\cap
B_{1}$}.$ (2.11)

So, by combining (2.9), (2.10), and (2.11) we deduce (2.3). When $\beta>2$, we
have that the usual distance function satisfies (2.2) (see, e.g., [23, Section
14.6]), which is inherited by $d$. Hence, these considerations altogether with
(1.5) imply (4), (4) realarly, we argue by a direct computation to show part
(5). Given $\eta>0$, from (2.3) and (2.9) we deduce

$\displaystyle\Delta d^{\lambda}|\ln(d)|^{\eta}$ $\displaystyle=$
$\displaystyle\Delta(d^{\lambda})|\ln(d)|^{\eta}+2\nabla(d^{\lambda})\cdot\nabla|\ln(d)|^{\eta}+d^{\lambda}\Delta(|\ln(d)|^{\eta})$
$\displaystyle=$
$\displaystyle\Big{(}\lambda(\lambda-1)\frac{d^{\lambda}|\ln(d)|^{\eta}}{d^{2}}+\Big{(}\frac{(\eta-1)}{|\ln(d)|}-2\lambda+1\Big{)}\frac{d^{\lambda}\eta|\ln(d)|^{\eta-1}}{d^{2}}\Big{)}|\nabla
d|^{2}$
$\displaystyle+|\ln(d)|^{\eta}O(d^{\beta+\lambda-3})+|\ln(d)|^{\eta-1}O(d^{\beta+\lambda-3})$

In particular, if $\lambda=\frac{1}{2}$ and $\eta=1/2$, the previous
expression together with (2.11) yields the expansion

$\Delta\Big{(}d^{\frac{1}{2}}|\ln(d)|^{\frac{1}{2}}\Big{)}=\frac{1}{4}\frac{d^{\frac{2}{3}}|\ln(d)|^{\frac{1}{2}}}{d^{2}}-\frac{1}{4d^{\frac{3}{2}}|\ln(d)|^{\frac{3}{2}}}\Big{(}1+O(\ln(d)^{2}d^{\beta-1})\Big{)},$

which concludes the proof. ∎

In a similar vein, we construct the following barrier function that is still a
sort of generalized distance satisfying other suitable properties.

######  Lemma 2.2.

Let $\rho\in(0,\frac{1}{2})$, $\beta\in(1,2)$ and let $\partial\Omega\cap
B_{1}$ be a $C^{\beta}$ graph. Then, there exist a constant $C>0$ only
depending on $\beta$ and $\rho$, and a function $\phi\in C^{\infty}(\Omega\cap
B_{1})\cap C^{\beta}(\overline{\Omega}\cap B_{r})$ for any $r\in(0,1)$ such
that

  1. (1)

$\frac{1}{C}\text{d}(x)\leq\phi(x)\leq C\text{d}(x),\qquad x\in\Omega\cap
B_{\rho},$ (2.12)

  2. (2)

$\phi(x)\geq\frac{1}{C}\quad\mbox{for $x\in\Omega\cap B_{2\rho}^{c}$},$ (2.13)

  3. (3)

$\phi(x)=x_{n}+g(x),\qquad x\in\Omega\cap B_{1},$ (2.14)

where $g\in C^{\infty}(\Omega\cap B_{2})$ satisfies $|\nabla g|\leq
C|x|^{\beta-1}$ in $\Omega\cap B_{1}$,

  4. (4)

and for $\lambda>0$ we have that

$|\Delta\phi^{\lambda}-\lambda(\lambda-1)\phi^{\lambda-2}|\leq
C|x|^{\beta+\lambda-3}\,\mbox{ in $\Omega\cap B_{1}$.}$ (2.15)

###### Proof.

Let $\phi_{0}$ be the unique solution of

$\begin{cases}\Delta\phi_{0}=0,&\Omega\cap B_{1},\\\
\phi_{0}=\text{dist}(\cdot,B_{\rho}\cap\partial\Omega),&\partial\big{(}\Omega\cap
B_{1}\big{)},\end{cases}$ (2.16)

where dist is the standard distance function to a set. Let us notice that
$\phi_{0}\in C^{\infty}(\Omega\cap B_{1})\cap C^{\beta}(\overline{\Omega}\cap
B_{r})$ for any $r\in(0,1)$. By the maximum principle $\phi_{0}>0$ in
$\Omega\cap B_{1}$. Thus, since $\phi_{0}$ vanishes tangentially along
$\partial\Omega$, Hopf’s lemma together with the assumption
$\nu_{\Omega}(0)=-e_{n}$ imply that $\nabla\phi(0)=C_{\Omega}e_{n}$ for some
$C_{\Omega}$ satisfying $\frac{1}{C(\beta)}\leq C_{\Omega}\leq C(\beta)$
thanks to the uniform $C^{\beta}$ bound on $\Omega\cap B_{1}$. Putting
$\phi=\frac{1}{C_{\Omega}}\phi_{0}$, we deduce (2.13) from standard regularity
for harmonic functions [1, Lemma A.2], whereas (2.13) follows directly from
the definition of $\phi_{0}$.

Part (3) follows from boundary Schauder estimates and the uniform bound on the
$C^{\beta}$ norm of the domain. Lastly, for part (4) we argue as in Lemma 2.1

$\Delta\phi^{\lambda}=\lambda(\lambda-1)|\nabla\phi|^{2}\frac{d^{\lambda}}{d^{2}},$
(2.17)

where we have used the harmonicity of $\phi$. So, by combining (2.17), with
the fact that $|\nabla\phi(0)|=1$ and the expanssion (2.14), we obtain (2.15).
∎

In the following lemma, we recall some properties of solutions to (1.2) nearby
regular points.

######  Proposition 2.3.

Let $u$ be a solution to (1.2) and let $0\in\partial\Omega$ be a regular free
boundary point. Then, there exists $r_{0}>0$ such that $\partial\Omega$ is
$C^{1,\alpha}$ for any $\alpha\in(0,1)$. Moreover, there exist $C,\rho>0$
(depending only on $\alpha,\gamma$, and $n$) such that

$\Big{|}u(x)-c_{\gamma}d^{\frac{2}{2-\gamma}}(x)\Big{|}\leq
C|x|^{{\frac{2}{2-\gamma}+\alpha}}\quad\mbox{for $x\in\Omega\cap B_{\rho}$}.$
(2.18)

Additionally,

$d(x)^{{\frac{\gamma}{2-\gamma}}}\leq Cu_{n}(x)\quad\mbox{for $x\in\Omega\cap
B_{\rho}$}.$ (2.19)

###### Proof.

Let us start recalling that nearby $0$, $\partial\Omega$ is $C^{1,\alpha_{0}}$
for some $\alpha_{0}$ in virtue of the results in [4, 15]. Our goal is to show
that $\alpha_{0}$ can be taken arbitrarily close to 1. We will show that this
can be inferred from Appendix B in [19] \- whose results, in that appendix,
even if stated for $\gamma\in(0,1)$, hold true for the full range
$\gamma\in(0,2)$ with exactly the same proofs.

Let us start deriving (2.18). Set $w:=\sqrt{\frac{\gamma}{2}}\beta
v^{\frac{2-\gamma}{2}}$, which is normalized so that $|\nabla w(0)|=1$, see
[15]. Also, let us extend $w$ as zero outside of $\Omega$. Since $0$ is a
regular point for $u$ and $\nu(0)=-e_{n}$, given any $\varepsilon>0$, we can
find $\rho_{\varepsilon}$ such that

$(x_{n}-C\rho_{\varepsilon}\varepsilon)_{+}\leq
w(x)\leq(x_{n}+C\rho_{\varepsilon}\varepsilon)_{+},\quad\mbox{ $x\in B_{r}$}$
(2.20)

where $C>0$ is a universal constant. Let $\alpha\in(0,1)$, in virtue of [19,
Proposition B.2], if we select $\varepsilon_{0}>0$ (and in consequence
$\rho_{0}$) small enough in (2.20), we have that for any $r\in(0,\rho_{0}/2]$

$|w(x)-(x\cdot e_{0})_{+}|\leq C\varepsilon_{0}r^{1+\alpha},\quad\mbox{ $x\in
B_{r}$}$

where $C=C(\alpha,\gamma)$ and for some $e_{0}\in\mathbb{S}^{n-1}$.
Equivalently, we have that

$|w_{r}(x)-(x\cdot e_{0})_{+}|\leq C\varepsilon r^{\alpha},\quad\mbox{ $x\in
B_{1}$}$

where $w_{r}(x)=\frac{w(rx)}{r}$. However, since $0$ is a regular point for
$u$ and $\nu(0)=-e_{n}$, we have that $w_{r}(x)\to(x_{n})_{+}$ as $r\to
0^{+}$, implying that $e_{0}=e_{n}$ and therefore

$|w_{r}(x)-(x_{n})_{+}|\leq C\varepsilon_{0}r^{\alpha}\quad\mbox{ $x\in
B_{1}$}$

for $r\in(0,\rho_{0}/2)$. From this last inequality we easily deduce

$|w(x)-(x_{n})_{+}|\leq C|x|^{1+\alpha}\quad\mbox{ $x\in B_{\rho_{0}/2}$.}$
(2.21)

Repeating the same argument in a neighborhood of $0$, we find $\rho_{1}>0$
such that for $z\in B_{\rho_{1}}\cap\partial\Omega$

$\big{|}w(z+x)-(x\cdot(-\nu(z)))_{+}\big{|}\leq C|x|^{1+\alpha}\quad\mbox{
$x\in B_{\rho_{1}}(z)$,}$ (2.22)

where $\nu(z)$ is the outer normal of $\partial\Omega$ at $z$. Thus, by
applying (2.21) at the value $z\in B_{\rho_{1}/2}\cap\partial\Omega$ and
(2.22) for this same $z$ at the value $x-z$ for some $x\in B_{\rho_{1}/2}$, we
deduce

$\displaystyle\big{|}(x_{n})_{+}-((x-z)\cdot(-\nu(z)))_{+}\big{|}$
$\displaystyle\leq$
$\displaystyle|(x_{n})_{+}-w(x)|+|w(x)-((x-z)\cdot(-\nu(z)))_{+}|$ (2.23)
$\displaystyle\leq$ $\displaystyle
C\Big{(}|x|^{1+\alpha}+|z-x|^{1+\alpha}\Big{)}.$

On top of this, by plugging $z$ in (2.21), we have that $|z\cdot
e_{n}|=|(z_{n})_{+}|\leq C|z|^{1+\alpha}$ which combined with (2.23) yields

$\displaystyle\big{|}(x-z)\cdot e_{n}-((x-z)\cdot(-\nu(z))_{+}\big{|}\leq
C\Big{(}|x|^{1+\alpha}+|z-x|^{1+\alpha}+|z|^{1+\alpha}\Big{)}.$ (2.24)

On the other hand, in virtue of the flatness property (2.20) we have that

$\\{x\in B_{\rho_{1}}|x_{n}\geq
C\varepsilon\rho_{1}\\}\subset\\{w>0\\}\quad\mbox{and}\quad|z_{n}|\leq
C\varepsilon|z|.$ (2.25)

Thus, upon taking $\varepsilon_{0}>0$ small enough in (2.20) (and thus
shrinking further $\rho_{1}$), given $y\in\mathbb{S}^{n-1}$ such that
$y_{n}\geq\frac{1}{4}$ we can find $\\{x\in B_{\rho_{1}}|x_{n}\geq
C\varepsilon\rho_{1}\\}$ satisfying $|x|\leq C_{1}|z|\leq C_{2}|x|$ such that
$y=\frac{x-z}{|x-z|}$. We observe now that in virtue of (2.25) and since the
norms of $|x|$ and $|z|$ are comparable, we have that $|x|\leq C_{1}|z-x|\leq
C_{2}|x|$. Additionally, thanks to the $C^{1,\alpha_{0}}$ of $\partial\Omega$,
upon further shrinking $\rho_{1}$, we that $(x-z)\cdot(-\nu(z))>0$. Thus, by
incorporating these considerations into (2.24), we deduce

$\displaystyle\left|\frac{(x-z)}{|x-z|}\cdot(e_{n}-\nu(z))\right|\leq
C|z|^{\alpha},$

which in turn implies from our previous considerations

$\displaystyle
C\big{|}(e_{n}-\nu(z))\big{|}=\sup_{y\in\mathbb{S}^{n-1},y_{n}\geq\frac{1}{4}}\big{|}y\cdot(e_{n}-\nu(z))\big{|}\leq
C|z|^{\alpha}.$ (2.26)

From here we deduce that $\partial\Omega$ is $C^{1,\alpha}$ in $B_{\rho_{1}}$.

Aiming for (2.18), we can exploit the $C^{1,\alpha}$ regularity of
$\partial\Omega\cap B_{\rho_{1}}$ to deduce from Lemma 2.1 that $d\in
C^{1,\alpha}(\Omega\cap B_{\rho_{1}})$. On top of this, since $\nabla
d(0)=e_{n}$, we have that $|d(x)-(x_{n})_{+}|\leq C|x|^{1+\alpha}$ for $x\in
B_{\rho_{1}}$. This combined with (2.21) implies

$|w(x)-d(x)|\leq C|x|^{1+\alpha}\quad\mbox{ $x\in\Omega\cap B_{\rho_{1}}$.}$

From here, we can apply the mean value theorem to the function $t\to
t^{\frac{2}{2-\gamma}}$ and recall the definition of $c_{\gamma}$ in (1.5) to
deduce for $x\in B_{\rho_{1}}$

$\displaystyle\left|u(x)-c_{\gamma}d^{\frac{2}{2-\gamma}}(x)\right|$
$\displaystyle=$ $\displaystyle
C|w(x)-d(x)|\int_{0}^{1}\big{(}tw(x)+(1-t)d(x)\big{)}^{\frac{\gamma}{2-\gamma}}\leq
C|x|^{\frac{2}{2-\gamma}+\alpha}$

where we have used that $u(x)\leq Cd(x)^{\frac{2}{2-\gamma}}$ which, in turn,
is consequence of [4, Corollary 1.11] combined with the boundary regularity of
$\partial\Omega$ nearby 0.

Finally, since $w_{n}(0)=1$, we deduce by continuity $\nabla w$ that
$w_{n}\geq\frac{1}{C}u^{\frac{1-\beta}{\beta}}\geq\frac{1}{C}d^{\frac{\gamma}{2-\gamma}}$
on $\Omega\cap B_{\rho}$ for $\rho$ small enough. This shows (2.19) and
finishes the proof. ∎

##  3\. Boundary regularity for solutions of the linearized equation
$L_{\kappa}$.

As indicated in the introduction, our analysis relies on understanding the one
dimensional solutions of (1.12). That is the purpose of the following lemma.

######  Lemma 3.1.

Let $\kappa\geq-\frac{1}{4}$ and $\theta_{\pm}$ defined as in (1.14). Given
$c_{i}\in\mathbb{R}$ and $n_{i}\in\mathbb{N}\cup\\{0\\}$ for $i=1,2$, with
$c_{2}=0$ if $\theta_{-}\leq 0$. Let $y$ be a solution of

$y^{\prime\prime}+\frac{\kappa}{x^{2}}y=c_{1}x^{\theta_{+}+n_{1}-1}+c_{2}x^{\theta_{-}+n_{2}-1}\,\mbox{in
$(0,\infty)$},$ (3.1)

satisfying

$\begin{array}[]{rl}\displaystyle\lim_{x\to
0^{+}}\frac{y}{x^{\theta_{-}}}=0&\quad\textrm{if}\quad\kappa>-{\textstyle\frac{1}{4}\vspace{1mm}}\\\
\displaystyle\lim_{x\to 0^{+}}\frac{y}{x^{1/2}|\log
x|}=0&\quad\textrm{if}\quad\kappa=-{\textstyle\frac{1}{4}}.\end{array}$ (3.2)

Then

$y(x)=\lambda_{1}x^{\theta_{+}}+Ac_{1}x^{\theta_{+}+n_{1}+1}+Bc_{2}x^{\theta_{-}+n_{2}+1}$
(3.3)

where $\lambda_{1}\in\mathbb{R}$ is any number, and $A$ and $B$ are real
valued parameters fully determined by $\kappa,n_{1}$ and $n_{2}$.

###### Proof.

The homogeneous ODE associated with (3.1) is a standard Cauchy-Euler equation
whose solutions are of the form $x^{\theta}$, with $\theta$ satisfying

$\theta(\theta-1)+\kappa=0.$

Therefore, characteristic roots correspond to

$\theta_{\pm}=\frac{1\pm\sqrt{1+4\kappa}}{2},$

if $\kappa>\frac{-1}{4}$. However, $x^{\theta_{-}}$ is discarded by the
boundary condition (3.2). On the other hand, if $\kappa=-\frac{1}{4}$, we have
multiplicity and thus we must replace $x^{\theta_{-}}$ by
$\ln(x)x^{\frac{1}{2}}$. This latter solution is also ruled out by (3.2).
Finally, the structure of the particular solutions follows from standard
computations in linear ODE theory, so we skip them for the sake of brevity. ∎

###  3.1. Boundary estimates

Our goal now is to prove Lemma 1.3 and derive uniform boundary estimates for
solutions from it. We start showing an elementary form of the maximum
principle for sub/super solutions of $L_{\kappa}$ with suitable boundary
behavior.

######  Lemma 3.2.

Let $D\subset\mathbb{R}^{n}$ be any bounded domain, and let $v,\psi\in
C^{2}(D)$ satisfying the following properties

$\displaystyle L_{\kappa}v$ $\displaystyle\leq$ $\displaystyle
0,\qquad\mbox{in $D$},$ (3.4) $\displaystyle L_{\kappa}\psi$ $\displaystyle>$
$\displaystyle 0,\qquad\mbox{in $D$},$ (3.5) $\displaystyle\psi$
$\displaystyle>$ $\displaystyle 0,\qquad\mbox{in $D$},$

and such that

$\limsup_{x\to\partial D}\frac{v(x)}{\psi(x)}\leq 0.$ (3.6)

Then, $v\leq 0$ in $D$.

###### Proof.

Let us notice that by (3.6) either $v\leq 0$ in $D$ or $\frac{v}{\psi}$
attains a positive maximum inside $D$. Arguing by contradiction, we assume
that the latter holds. This means that there is $M>0$ and $z\in D$ such that

$\displaystyle v(z)$ $\displaystyle=$ $\displaystyle M\psi(z),$ $\displaystyle
v$ $\displaystyle\leq$ $\displaystyle M\psi,\quad\mbox{in D}.$

However, this latter observation yields $-\Delta v(z)\geq-M\Delta\psi(z)$, so
we deduce that

$0\geq L_{\kappa}v(z)\geq ML_{\kappa}\psi(z)>0,$

a contradiction. ∎

As a corollary, we derive Lemma 1.3.

###### Proof Lemma 1.3.

Let $\beta=1+\alpha$ so that $\Omega\cap B_{1}$ is a $C^{\beta}$ domain with
$\beta\in(1,2)$. Let $\rho>0$ such that $d<1$ on $\Omega\cap B_{\rho}$ and
consider the function $\phi_{1}=d^{\frac{1}{2}}(-\ln(d))^{\frac{1}{2}}$,
which, thanks to (2.6), satisfies

$\displaystyle L_{\kappa}\phi_{1}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{1}{4}+\kappa\Big{)}\frac{\phi_{1}}{d^{2}}+\frac{\phi_{1}}{4d_{1}^{2}(-\ln(d_{1}))^{2}}\Big{(}1+O(\ln(d)^{2}d^{\beta-1})\Big{)}.$
(3.7)

Since $d\leq C\rho$ in $B_{\rho}\cap\Omega$, we can choose $\rho$ such that
$1+O(\ln(\rho)^{2}\rho^{\min\\{\beta-1,1\\}})\geq\frac{1}{2}$ in $\Omega\cap
B_{\rho}$, depending only on the $C^{\beta}$ norm of $\Omega$. Hence, from
(3.7) we deduce

$L_{\kappa}\phi_{1}\geq\Big{(}\frac{1}{4}+\kappa\Big{)}\frac{\phi_{1}}{d^{2}}+\frac{\phi_{1}}{d^{2}(-\ln(d_{1}))^{2}}(C-\rho^{\beta-1}(-\ln(\rho))^{2})>0\quad\mbox{
in $\Omega\cap B_{\rho}$,}$ (3.8)

by shrinking $\rho$ further (if necessary) to make the last term positive. On
the other hand, if $\kappa>-\frac{1}{4}$ (in which case we have that
$\theta_{-}<\frac{1}{2}$), we can combine (2.3) with the identity
$\theta_{-}(\theta_{-}-1)=\kappa$ to derive the bounds

$-Cd^{\theta_{-}-2}\rho^{\beta-1}\leq L_{\kappa}(d^{\theta_{-}})\leq
Cd^{\theta_{-}-2}\rho^{\beta-1}\quad\mbox{ in $\Omega\cap B_{\rho}$}.$ (3.9)

Regardless of the value of $\kappa$, we can combine (3.8) and (3.9) to deduce
that the barrier $\phi_{2}=\phi_{1}+(1/4+\kappa)d^{\theta_{-}}$ satisfies

$L_{\kappa}(\phi_{2})>\big{(}d^{1/2-\theta_{-}}(-\ln(d))^{\frac{1}{2}}+O(\rho^{\beta-1})\big{)}\frac{(1/4+\kappa)d^{\theta_{-}}}{d^{2}}>0\quad\mbox{
in $\Omega\cap B_{\rho}$.}$

Hence, if for some $r\leq\rho$, we have that $v\leq 0$ on $\Omega\cap\partial
B_{r}$, we deduce that

$\limsup_{x\to\partial(\Omega\cap B_{r})}\frac{v(x)}{\phi_{2}(x)}\leq 0,$

in virtue of the boundary condition (1.13). Finally, the result follows
directly from Lemma 3.4. ∎

We apply now the previous lemma to prove a first step towards Theorem 1.2,
which will be improved later on. In virtue of this lemma, we will derive
boundary estimates that will endow us with the required compactness to carry
out our blow-up arguments.

######  Lemma 3.3.

Let $\kappa\in[-\frac{1}{4},\infty)$, let $\beta\in(1,2)$ and let
$\Omega\subset\mathbb{R}^{n}$ be a $C^{\beta}$ domain. Let $d$ be the
regularized distance to $\partial\Omega$, given by Lemma 2.1.

Let $v\in C(\Omega\cap B_{1})$ be any solution to

$-\Delta v+\kappa\,\frac{v}{d^{2}}=f\quad\text{in}\quad\Omega\cap B_{1},$
(3.10)

satisfying

$\begin{array}[]{rl}\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{\theta_{-}}}=0&\quad\textrm{if}\quad\kappa>-{\textstyle\frac{1}{4}\vspace{1mm}}\\\
\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{1/2}|\log
d|}=0&\quad\textrm{if}\quad\kappa=-{\textstyle\frac{1}{4}},\end{array}$ (3.11)

where $f\in C(\Omega\cap B_{1})$ satisfies

$fd^{2-\theta_{+}-\varepsilon}\in L^{\infty}(\Omega\cap B_{1})$ (3.12)

for some $\varepsilon>0$.
Then, there exists $\rho\in(0,1)$ such that

$|v|\leq C\Big{(}\|v\|_{L^{\infty}(\Omega\cap
B_{1})}+\|f\phi^{2-\theta_{+}-\varepsilon}\|_{L^{\infty}(\Omega\cap
B_{1})}\Big{)}d^{\theta_{+}},\,\,\mbox{ in $\Omega\cap B_{\rho}$,}$ (3.13)

where $\phi$ is the barrier constructed in Lemma 2.2 and $\rho$ depends only
on $\beta$, $\varepsilon$ and $n$.

Furthermore, if $\kappa<0$, then $v\in C^{\theta_{+}}(\overline{\Omega}\cap
B_{1/2})$ and

$[v]_{\theta^{+},\Omega\cap B_{\rho}}\leq C\Big{(}\|v\|_{L^{\infty}(\Omega\cap
B_{1})}+\|f\phi^{2-\theta_{+}-\varepsilon}\|_{L^{\infty}(\Omega\cap
B_{1})}\Big{)}.$ (3.14)

Whilst, if $\kappa\geq 0$, we have the Lipschitz estimate

$[v]_{1,\Omega\cap B_{\rho}}\leq C\Big{(}\|v\|_{L^{\infty}(\Omega\cap
B_{1})}+\|f\phi^{2-\theta_{+}-\varepsilon}\|_{L^{\infty}(\Omega\cap
B_{1})}\Big{)}.$ (3.15)

###### Proof.

Let $\rho\in(0,1/2)$ to be determined later in the proof and let $\phi$ be the
barrier constructed in Lemma 2.2. Thanks to (2.15), we can use the identity
$(\theta_{+}-1)\theta_{+}=\kappa$ as in (3.9) to deduce

$|L_{\kappa}(\phi^{\theta_{+}})|\leq C\phi^{\theta_{+}-2}\rho^{\beta-1}$
(3.16)

in $B_{2\rho}\cap\Omega$.

Let $\varepsilon\in(0,\beta-1)$ such that (3.12) holds. Since
$(\theta_{+}+\varepsilon-1)(\theta_{+}+\varepsilon)=\kappa+\varepsilon(2\theta_{+}-1+\varepsilon)$
and $\theta_{+}\geq\frac{1}{2}$, we can proceed as in (3.16) to get

$\Big{|}L_{\kappa}(\phi^{\theta_{+}+\varepsilon})+\varepsilon(2\theta_{+}-1+\varepsilon)\phi^{\theta_{+}+\varepsilon-2}\Big{|}\leq
C\phi^{\theta_{+}+\varepsilon-2}\rho^{\beta-1},$ (3.17)

$B_{2\rho}\cap\Omega$.

So, we can fix $\rho>0$ small enough depending on $\varepsilon$ and $\beta$
such that for some $\eta>0$

$L_{k}\Big{(}\phi^{\theta_{+}+\varepsilon}-\phi^{\theta_{+}}\Big{)}\leq-\eta\phi^{\theta^{+}+\varepsilon-2}$
(3.18)

in $B_{2\rho}\cap\Omega$. Thus, after taking $\phi$ subordinated to $\rho$, we
also have from (2.13) that

$\phi^{\theta_{+}+\varepsilon}-\phi^{\theta_{+}}\geq\eta\mbox{ on
$\Omega\cap\partial B_{2\rho}$}.$ (3.19)

Let us divide both sides of (3.10) by

$A=\frac{\|v\|_{L^{\infty}(\Omega\cap
B_{1})}+\|\phi^{2-\theta_{+}-\varepsilon}f\|_{L^{\infty}(\Omega\cap
B_{1})}}{\eta},$

and let us notice that, by linearity, $v_{1}=\frac{1}{A}v$ solves (3.10) with
right hand side $f_{1}=\frac{1}{A}f$ and with the same boundary condition
(3.11). Furthermore, by construction $\|v_{1}\|_{L^{\infty}(\Omega\cap
B_{1})}\leq\eta$ and $|f_{1}|\leq\eta\phi^{\theta_{+}+\varepsilon-2}$ in
$\Omega\cap B_{1}$.

Thus, the function
$u=v_{1}-\Big{(}d^{\theta_{+}}+d^{\theta_{+}+\varepsilon}\Big{)}$ satisfies
the hypotheses of Lemma 1.3, implying that
$v_{1}\leq\phi^{\theta_{+}}-\phi^{\theta_{+}+\varepsilon}$ in $\Omega\cap
B_{\rho}$. By applying the previous argument to $-v$ and combining it with
(2.12) we derive (3.13).

We complete the proof deducing (3.14). Let us consider first the case when
$\kappa<0$. Given $x_{0}\in\Omega$ with
$\text{dist}(x_{0},\partial\Omega)=2r<\rho$, we can combine (3.13) with
interior Schauder estimates to deduce

$\displaystyle[v]_{C^{\theta_{+}}(B_{\frac{r}{2}}(x_{0}))}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\frac{1}{r^{\theta_{+}}}\|v\|_{L^{\infty}(B_{r}(x_{0}))}+r^{2-\theta_{+}}\left\|\frac{\kappa
v}{d^{2}}-f\right\|_{L^{\infty}(B_{r}(x_{0}))}\Big{)}$ $\displaystyle\leq$
$\displaystyle C\Big{(}\|v\|_{L^{\infty}(\Omega\cap
B_{1})}+\|\phi^{2-\theta_{+}-\varepsilon}f\|_{L^{\infty}(\Omega\cap
B_{1})}\Big{)}.$

The estimate (3.14) follows from the previous inequality combined with a
covering argument. See, e.g., Corollary 3.5 and Lemma B.2 in [28]. In the case
$\kappa\geq 0$, we can argue analogously to derive (3.15) see, e.g., [28,
Corollary 3.5]. ∎

######  Lemma 3.4.

Let $M\in\mathbb{N}$, $\kappa\geq-\frac{1}{4}$, $\beta\in(1,2)$, and let
$\Omega\subset\mathbb{R}^{n}$ be a $C^{\beta}$ domain. Let $d$ be the
regularized distance to $\partial\Omega$, given by Lemma 2.1. Let $v\in
C(\Omega\cap B_{1})$ be a solution of

$L_{\kappa}(v)(x)=\big{(}R_{1}(x)+g_{1}(x)\big{)}\left(\frac{d(rx)}{r}\right)^{\theta_{+}-1}+\big{(}R_{2}(x)+g_{2}(x)\big{)}\left(\frac{d(rx)}{r}\right)^{\theta_{-}-1}\quad\textrm{in}\quad{\textstyle\frac{1}{r}}\Omega\cap
B_{1},$ (3.20)

with $R_{i}\in\mathbf{P}_{M}$, and $g_{i}\in
L^{\infty}({\textstyle\frac{1}{r}}\Omega\cap B_{1})$ for $i=1,2$ and with
$R_{2}=g_{2}=0$ if $\theta_{-}\leq 0$. Then, there exist $\rho_{0}>0$ and
$C>0$ only depending on $M$ and $\beta$ such that then for any
$r\in(0,\rho_{0})$,

$\|R_{i}\|_{L^{\infty}\big{(}{\textstyle\frac{1}{r}}\Omega\cap
B_{1}\big{)}}\leq
C\big{(}\|g_{1}\|_{L^{\infty}\big{(}{\textstyle\frac{1}{r}}\Omega\cap
B_{1}\big{)}}+\|g_{2}\|_{L^{\infty}\big{(}{\textstyle\frac{1}{r}}\Omega\cap
B_{1}\big{)}}+\|v\|_{L^{\infty}\big{(}{\textstyle\frac{1}{r}}\Omega\cap
B_{1}\big{)}}\Big{)},$ (3.21)

for $i=1,2$.

###### Proof.

Arguing by contradiction, we assume sequences of domains $\Omega_{l}$ with
$\|\partial\Omega_{l}\cap B_{2}\|_{C^{\beta}}\leq 1$, a sequence of radii
$\\{r_{l}\\}_{l\in\mathbb{N}}$ with $r_{l}\to 0$, and of polynomials
$\\{R_{i}^{l}\\}_{l\in\mathbb{N}}$, functions $g_{i}^{l}\in
L^{\infty}\big{(}\frac{1}{r_{l}}\Omega\cap B_{1}\big{)}$ for $i=1,2$ and
solutions $u_{l}$ to (3.20) such that (3.21) does not hold. Thus, by dividing
both sides of (3.20) and (3.21) by
$\|R_{1}^{l}\|_{L^{\infty}\big{(}\frac{1}{r_{l}}\Omega\cap
B_{1}\big{)}}+\|R_{2}^{l}\|_{L^{\infty}\big{(}\frac{1}{r_{l}}\Omega\cap
B_{1}\big{)}}$ we obtain a sequence of functions $\\{\tilde{v}_{l}\\}$
satisfying

$L_{\kappa}(\tilde{v}_{l})(x)=\Big{(}\tilde{R}_{1}^{l}(x)+\tilde{g}_{1}^{l}(x)\Big{)}\Big{(}\frac{d_{l}(rx)}{r}\Big{)}^{\theta_{+}-1}+\Big{(}\tilde{R}_{2}^{l}(x)+\tilde{g}_{2}^{l}(x)\Big{)}\Big{(}\frac{d_{l}(rx)}{r}\Big{)}^{\theta_{-}-1},\quad\text{in}\,\,\frac{1}{r_{l}}\Omega\cap
B_{1},$

with $\|\tilde{g}_{i}^{l}\|_{L^{\infty}(\Omega\cap B_{\rho})}\to 0$ as
$l\to\infty$ for $i=1,2$,

$\limsup_{l\to\infty}\|u_{l}\|_{L^{\infty}(B_{1}\cap\frac{1}{r_{l}}\Omega)}\to
0,$ (3.22)

and

$\|\tilde{R}_{1}^{l}\|_{L^{\infty}\big{(}\frac{1}{r_{l}}\Omega\cap
B_{1}\big{)}}+\|\tilde{R}_{2}^{l}\|_{L^{\infty}\big{(}\frac{1}{r_{l}}\Omega\cap
B_{1}\big{)}}=1.$

This last bound on the polynomials implies, bearing in mind that
$\mathbf{P}_{M}$ is a finite dimensional space, that, up to substracting a
further subsequence, $\tilde{R}_{i}^{l}\to R^{*}_{i}$ for $i=1,2$ with at
least one $Q_{i}^{*}\neq 0$. On the other hand, in virtue of Lemma 2.1, we
have that, up to subsequences,

$\frac{d_{l}(r_{l}x)}{r_{l}}\to x_{n},$

locally uniformly in $\mathbb{R}^{n}$ -by extending $d$ by zero outside
$\Omega\cap B_{1}$. Therefore, by the stability of viscosity solutions under
locally uniform convergence, we deduce that for
$\lim_{l\to\infty}u_{l}=u_{*}=0$

$0=\Delta
u_{*}-\frac{\tilde{C_{\gamma}}}{x_{n}^{2}}u=Q_{1}^{*}x_{n}^{\theta_{+}-1}+Q_{2}^{*}x_{n}^{\theta_{-}-1},$

which yields a contradiction since at least one of the $Q_{i}^{*}$ is not zero
and since $\theta_{+}-\theta_{-}\notin\mathbb{N}$. ∎

We prove now a Liouville-type theorem.

######  Lemma 3.5.

Let $M\in\mathbb{N}$, $\kappa\geq-\frac{1}{4}$, $\theta_{\pm}$ as in (1.14),
and $\beta>1$ with $\beta\notin\mathbb{N}$. Let $Q_{i}\in\mathbf{P}_{M}$ for
$i=1,2$ with $Q_{2}=0$ if $\theta_{-}\leq 0$. If $u$ is such that

$\begin{cases}L_{\kappa}u=Q_{1}(x)x_{n}^{\theta_{+}-1}+Q_{2}(x)x_{n}^{\theta_{-}-1}\quad\textrm{in}\quad\\{x_{n}>0\\}\\\
\|u\|_{L^{\infty}(B_{R})}\leq CR^{{\theta_{+}}+\beta-1}.\end{cases}$ (3.23)

Let also us assume that $u$ satisfies one of the following conditions

  1. (1)

either

$\begin{array}[]{rl}\displaystyle\lim_{x_{n}\to
0^{+}}\frac{u(x^{\prime},x_{n})}{x_{n}^{\theta_{-}}}=0,&\quad\textrm{if}\quad\kappa>-{\textstyle\frac{1}{4}\vspace{1mm}}\\\
\displaystyle\lim_{x_{n}\to
0^{+}}\frac{u(x^{\prime},x_{n})}{\ln(x_{n})x_{n}^{\frac{1}{2}}}=0,&\quad\textrm{if}\quad\kappa=-{\textstyle\frac{1}{4}}.\end{array}$
(3.24)

  2. (2)

or

$\lim_{x_{n}\to 0^{+}}\frac{u(x^{\prime},x_{n})}{x_{n}^{\theta_{+}}}=0.$
(3.25)

Then, $u=P_{1}(x)x_{n}^{\theta_{+}}+P_{2}(x)x_{n}^{\theta_{-}+1}$, where
$P_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$P_{2}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta-\rfloor-2}$. Furthermore,
if (3.24) holds, then $P_{2}=0$ whenever $Q_{2}=0$. On the other hand, if
(3.25) holds, $P_{1}=0$ whenever $Q_{1}=0$.

###### Proof.

We will assume throughout the proof $\kappa<0$, since the case $\kappa\geq 0$
is completely analogous.

Given any $R>1$, let us define $u_{R}(x)=R^{-\theta_{+}-\beta+1}u(Rx)$. Thanks
to the growth bound in (3.23), we have that $\|u_{R}\|_{L^{\infty}(B_{1})}\leq
C$. Noticing that

$L_{k}u_{R}=R^{2-\beta}Q_{1}(Rx)x_{n}^{\theta_{+}-1}+R^{2-\beta-\theta_{+}+\theta_{-}}Q_{2}(Rx)x_{n}^{\theta_{-}-1},$

we can use Lemma 3.4 in the flat domain $\Omega=\mathbb{R}^{n}_{+}$, to deduce

$R^{2-\beta}\|Q_{1}(Rx)\|_{L^{\infty}(B_{1})}+R^{2-\beta-\theta_{+}+\theta_{-}}\|Q_{2}(Rx)\|_{L^{\infty}(B_{1})}\leq
C,$ (3.26)

for any $R>1$, implying that

$Q_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-2},\quad
Q_{2}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}.$ (3.27)

Now, taking $\varepsilon\in(0,1-\theta_{+}-\theta_{-})$ in (3.14), we deduce
from (3.26) and Lemma 3.3 that $[u_{R}]_{C^{\theta_{+}}(B_{\frac{1}{2}})}\leq
C(1+\|Q_{1}\|_{L^{\infty}(B_{1})}+\|Q_{2}\|_{L^{\infty}(B_{1})})$. This
implies that

$[u]_{C^{\theta_{+}}\big{(}B_{R/2}\big{)}}\leq CR^{\beta-1}.$ (3.28)

Arguing as in the proof of [1, Theorem 3.10], we consider the quotients

$u_{1}(x)=\frac{u(x+\tau h)-u(x)}{h^{\theta_{+}}}$ (3.29)

with $h\in(0,R/2)$, $x\in\mathbb{R}^{n}_{+}$, and $\tau\in\mathbb{S}^{n-1}$
with $\tau_{n}=0$. From (3.28), we have that
$\|u_{1}\|_{L^{\infty}(B_{R})}\leq CR^{\beta-1}$. On the other hand, since
$\tau$ is orthogonal to $e_{n}$, we have that $u_{1}$ satisfies the boundary
condition (3.24) together with

$\begin{cases}L_{\kappa}u_{1}=\tilde{Q}_{1}(x)x_{n}^{\theta_{+}-1}+\tilde{Q}_{2}(x)x_{n}^{\theta_{-}-1},\quad\\{x_{n}>0\\}\\\
\|u_{1}\|_{L^{\infty}(B_{R})}\leq CR^{\beta-1},\\\ \end{cases}$ (3.30)

where $\tilde{Q}_{i}=\frac{Q_{i}(x+\tau h)-Q_{i}(x)}{h^{\theta_{+}}}$. Arguing
as before, we conclude that $[u_{1}]_{C^{\theta_{+}}\big{(}B_{R/2}\big{)}}\leq
CR^{\beta-1-\theta_{+}}$. By repeating this argument inductively $k$ times,
until $k\theta_{+}>\beta-1$, we deduce that

$[u_{k}]_{C^{\theta_{+}}\big{(}B_{R/2}\big{)}}\leq CR^{\beta-1-k\theta_{+}}\to
0,$

as $R\to\infty$, which, by the boundary condition, implies that $u_{k}=0$.
Therefore, we conclude iterating backwards the difference quotients (see the
proof of [1, Theorem 3.10]) that

$u(x^{\prime},x_{n})=\sum_{|\lambda|\leq
M}C_{\lambda}(x_{n})(x^{\prime})^{\lambda},$ (3.31)

with $M=\lfloor\beta-1\rfloor$ for some functions $C_{\lambda}(x_{n})$. Our
next step consists in proving by induction that

$C_{\lambda}(x_{n})=P_{1,\lambda}(x_{n})(x_{n})_{+}^{\theta_{+}}+P_{2,\lambda}(x_{n})(x_{n})_{+}^{\theta_{-}+1},$

for polynomials $P_{1,\lambda}$ and $P_{2,\lambda}$ satisfying, furthermore,
that $P_{2,\lambda}(x_{n})=0$ if $Q_{2}=0$ whenever (3.24) holds, and
$P_{1,\lambda}(x_{n})=0$ if $Q_{1}=0$ when we have (3.25) instead.

For the base case, if we take tangential derivatives of maximal order
$\lambda_{0}$, i.e., $|\lambda_{0}|=M$, we conclude, bearing in mind (3.27),
that $C_{\lambda_{0}}(x_{n})$ satisfies

$\begin{cases}L_{\kappa,1}(C_{\lambda_{0}})=0\\\
C_{\lambda_{0}}(x_{n})=0,\end{cases}$

where $L_{\kappa,1}(y)=-y^{\prime\prime}+\frac{\kappa}{x_{n}^{2}}y$, i.e., the
$1$ dimensional version of $L_{\kappa}$.

Combining Lemma 3.1, and noticing from the decomposition (3.31) that
$C_{\lambda}$ inherits the corresponding boundary condition from $u$, we
deduce that

$C_{\lambda_{0}}(x_{n})=C_{1,\lambda_{0}}x^{\theta_{+}},$

with $C_{1,\lambda_{0}}=0$ if (3.24) holds instead. This proves the base case.

Let us assume now the result for $|\lambda|>j$ and show that it holds for
$\beta$ with $|\beta|=j$. Taking derivatives on both sides of (3.31) with
respect to the tangential variables $x^{\prime}$, and setting up the notation
$v=\partial^{\beta}u$, we deduce from the inductive hypothesis that

$u(x^{\prime},x_{n})=\beta!C_{\beta}(x_{n})+\sum_{\\{\lambda:\lambda>\beta\\}}(x^{\prime})^{\lambda-\beta}\Big{(}P_{1,\lambda}(x_{n})(x_{n})_{+}^{\theta_{+}}+P_{2,\lambda}(x_{n})_{+}^{\theta_{-}+1}\Big{)},$
(3.32)

with $P_{i,\lambda}=0$ if $Q_{i}=0$ for $i=1,2$, provided that the right
corresponding boundary behavior is enforced. Thus, differentiating (3.23) and
applying $L_{\kappa,1}$ to the representation (3.32) yields

$\displaystyle\beta!L_{k,1}(C_{\beta}(x_{n}))$ $\displaystyle=$
$\displaystyle\partial^{\beta}Q_{1}(x)x_{n}^{{\theta_{-}}-1}+\partial^{\beta}Q_{2}(x)x_{n}^{{\theta_{+}}-1}$
(3.33)
$\displaystyle-\sum_{\\{\lambda:\lambda>\beta\\}}(\Delta(x^{\prime})^{\lambda-\beta})\Big{(}P_{1,\lambda}(x_{n})x_{n}^{{\theta_{+}}}+P_{2,\lambda}(x_{n})x_{n}^{\theta_{-}+1}\Big{)}$
$\displaystyle-\sum_{\\{\lambda:\lambda>\beta\\}}((x^{\prime})^{\lambda-\beta})\Big{(}P_{1,\lambda}^{\prime\prime}(x_{n})x_{n}^{\theta_{+}}+P_{2,\lambda}^{\prime\prime}(x_{n})x_{n}^{\theta_{-}}\Big{)}.$

Therefore, since $L_{\kappa,1}(C_{\beta}(x_{n}))$ does not depend on
$x^{\prime}$, we deduce from (3.33) that

$L_{\kappa,1}(C_{\beta}(x_{n}))=\tilde{P}_{1}(x_{n})x_{n}^{{\theta_{+}}-1}+\tilde{P}_{2}(x_{n})(x_{n})x_{n}^{{\theta_{-}}-1},$
(3.34)

for some polynomials ${\tilde{P}_{1}}$ and $\tilde{P}_{2}$, with
$\tilde{P}_{i}=0$ if $Q_{i}=0$ for $i=1,2$. Then, combining again Lemma 3.1,
with either (3.24) or (3.25), the decomposition (3.31), and the linearity of
$L_{\kappa,1}$ we deduce

$C_{\beta}(x_{n})=P_{1,\beta}(x_{n})x_{n}^{\theta_{+}}+P_{2,\beta}(x_{n})x_{n}^{\theta_{-}+1},$

with $P_{i,\beta}=0$ if $Q_{i}=0$ for $i=1,2$.

This completes the inductive argument implying

$u(x)=P_{1}(x)x_{n}^{\theta_{+}}+P_{2}(x)x_{n}^{\theta_{-}+1}$ (3.35)

for some polynomials $P_{1}$ and $P_{2}$, with $P_{i}=0$ if $Q_{i}=0$.

Lastly, since $\theta_{+}-\theta_{-}\notin\mathbb{N}$, each term in (3.35)
must satisfy the same growth bound as $u$. This implies that
$P_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and that
$P_{2}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta-\rfloor-2}$, finishing the
proof. ∎

##  4\. Higher order boundary Schauder estimates for solutions to the
linearized equation

This section is devoted to the proof of a generalized version of Theorem 1.2,
which is carried out using a blow-up in the same spirit of [1]. We divide the
argument into two parts, addressing first the case where $\partial\Omega\cap
B_{1}$ is a $C^{\beta}$ surface with $\beta\in(1,2)$ and then proving the
higher order version of this result.

######  Theorem 4.1.

Let $\kappa\geq-\frac{1}{4}$, $\theta_{\pm}$ as in (1.14), and $\beta\in(1,2)$
with $\beta+\theta_{+}-\theta_{-}\notin\mathbb{N}$. Let
$\Omega\subset\mathbb{R}^{n}$ such that $\partial\Omega\cap B_{2}$ a
$C^{\beta}$ surface, let $f$ and $g$ satisfying
$\|f\|_{C^{\beta-1}(\overline{\Omega}\cap
B_{1})},\|g\|_{C^{\beta-1+\theta_{+}-\theta_{-}}(\overline{\Omega}\cap
B_{1})}\leq 1$ and $f,g|_{\partial\Omega}\equiv 0$, where $g=0$ if
$\theta_{-}\leq 0$. Let $v\in C(\Omega\cap B_{1})$ be a solution to

$L_{\kappa}(v)=fd^{\theta_{+}-2}+gd^{\theta_{-}-2},\quad\text{in}\,\,\Omega\cap
B_{1},$ (4.1)

satisfying

$\begin{array}[]{rl}\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{\theta_{-}}}=0&\quad\textrm{if}\quad\kappa>-{\textstyle\frac{1}{4}\vspace{1mm}}\\\
\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{1/2}|\log
d|}=0&\quad\textrm{if}\quad\kappa=-{\textstyle\frac{1}{4}}.\end{array}$ (4.2)

Then, there exists a constant $C>0$ and a radius $\rho>0$, such that for every
$x_{0}\in\partial\Omega\cap B_{\rho}$, there are constants
$\lambda_{1,x_{0}},\lambda_{2,x_{0}}\in\mathbb{R}$ such that

$\displaystyle|v(x)-\lambda_{1,x_{0}}d^{\theta_{+}}(x)-\lambda_{2,x_{0}}(x)d^{\theta_{-}+1}(x)|\leq
C|x-x_{0}|^{{{\theta_{+}}+\beta-1}}\quad\mbox{for $x\in
B_{\rho}(x_{0})\cap\Omega$,}$ (4.3)

with $\lambda_{2,x_{0}}=0$ if $g=0$. Here the constants $C$ and $\rho$ depend
only on $\kappa$, $\beta$, and $n$.

###### Proof.

Preparations: Without loss of generality, let us take $x_{0}=0$ and let us
extend $v$ and $d$ by zero outside $\Omega\cap B_{1}$. So, if we contradict
the inequality (4.3), we deduce the existence of a sequence of domains
$\Omega_{j}$ with $\|\Omega_{j}\|_{C^{\beta}}\leq 1$ and, in consequence, of
generalized distance functions $d_{j}$ such that for any sequence of constants
$\lambda_{i,j}\in\mathbb{R}$ (with $k_{2,j}=0$ if $\theta_{-}\leq 0$), we have
that

$\sup_{r>0}\sup_{j\in\mathbb{N}}\frac{1}{r^{{{\theta_{+}}+\beta-1}}}\|v_{j}-\lambda_{1,j}d_{j}^{\theta_{+}}-\lambda_{2,j}d_{j}^{\theta_{-}+1}\|_{L^{\infty}(B_{r})}=\infty,$
(4.4)

where $v_{j}$ satisfies (4.2) and

$L_{\kappa}(v_{j})=f_{j}d^{\theta_{+}-2}+g_{j}d^{\theta_{-}-2},\quad\text{in}\,\,\Omega\cap
B_{1},\\\ $ (4.5)

with
$\|f_{j}\|_{C^{\beta-1}(B_{1}\cap\overline{\Omega_{j}})},\|g_{j}\|_{C^{\beta-1+\theta_{+}-\theta_{-}}(B_{1}\cap\overline{\Omega_{j}})}\leq
1$ with $f_{j},g_{j}|_{\partial\Omega_{j}}\equiv 0$ for $j\in\mathbb{N}$ and
with $g_{j}=0$ if $\theta_{-}\leq 0$.

Step one: We start selecting $\lambda_{1,j,r},\lambda_{2,j,r}$ as the the
least square coefficients of the projection of $v_{j}$ onto the subspace
spanned by $\\{d_{j}^{\theta_{+}},d_{j}^{\theta_{-}+1}\\}$ in $L^{2}(B_{r})$.
In this case, they are characterized by the orthogonality conditions

$\int_{B_{r}}\Big{(}v_{j}-\lambda_{1,j,r}d_{j}^{\theta_{+}}-\lambda_{2,j,r}d_{j}^{\theta_{-}+1}\Big{)}(c_{1}d_{j}^{\theta_{+}}+c_{2}d_{j}^{\theta_{-}+1})=0,$
(4.6)

for any $(c_{1},c_{2})\in\mathbb{R}^{2}$.

Let us define

$\theta(r)=\sup_{\rho>r}\sup_{j\in\mathbb{N}}\frac{1}{r^{{{\theta_{+}}+\beta-1}}}\big{\|}v_{j}-\mu_{1,j}d_{j}^{\theta_{+}}-\mu_{2,j}d_{j}^{\theta_{-}+1}\big{\|}_{L^{\infty}(B_{r})},$

with $\mu_{1,j}=\lambda_{1,j,r}$ and $\mu_{2,j}=\lambda_{2,j,r}$.

By Lemma 6.2 we have that $\lim_{r\to 0^{+}}\theta(r)=\infty$. Thus, we can
pick a sequence $\\{r_{j}\\}_{j\in\mathbb{N}}$ with $r_{j}\to 0^{+}$ as
$j\to\infty$ such that

$w_{j}(x):=\frac{v_{j}(r_{j}x)-\mu_{1,j}d_{j}^{\theta_{+}}(r_{j}x)-\mu_{2,j}d_{j}^{\theta_{-}+1}(r_{j}x)}{r_{j}^{{{\theta_{+}}+\beta-1}}\theta(r_{j})},$
(4.7)

satisfies $\|w_{j}\|_{L^{\infty}(B_{1})}\in[\frac{1}{2},1]$ for
$l\in\mathbb{N}$.

Additionally, Lemma 6.3 provides the growth bound

$\|w_{j}\|_{L^{\infty}(B_{R})}\leq CR^{{\theta_{+}}+\beta-1}.$ (4.8)

Step two: We proceed to show that the sequence $\\{w_{j}\\}_{j\in\mathbb{N}}$
satisfies, up to a small perturbation, a PDE of the form (3.10). Applying
Lemma 2.1, we can easily compute the Laplacian of the auxiliary term
$\mu_{1,j}d_{j}^{\theta_{+}}(r_{j}x)+\mu_{2,j}d_{j}^{\theta_{-}+1}(r_{j}x)$ as
follows

$\displaystyle\frac{1}{r_{j}^{{\theta_{+}}+\beta-1}}\Delta\Big{(}\mu_{1,j}d_{j}^{\theta_{+}}(r_{j}x)+\mu_{2,j}d_{j}^{\theta_{-}+1}(r_{j}x)\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{r_{j}^{\theta_{+}+\beta-1}}\frac{r_{j}^{2}\kappa}{d_{j}^{2}(rx)}\Big{(}\mu_{1,j}d_{j}^{\theta_{+}}(r_{j}x)+\mu_{2,j}d_{j}^{\theta_{-}+1}(r_{j}x)\Big{)}$
(4.9)
$\displaystyle+l_{j}\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{-}-1}+h_{1,j}(x)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{+}+\beta-3}$
$\displaystyle+h_{2,j}(x)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{-}+\beta-2}$

where $l_{j}\in\mathbb{R}$,
$\|h_{1,j}\|_{L^{\infty}(B_{2}\cap\overline{\Omega_{j}})}\leq\frac{C}{\theta(r_{j})}$
and
$\|h_{2,j}\|_{L^{\infty}(B_{2}\cap\overline{\Omega_{j}})}\leq\frac{Cr_{j}^{\theta_{-}-\theta_{+}+1}}{\theta(r_{j})}$.
We remark that the latter quantity goes to zero as $j\to\infty$ since
$\theta_{+}-\theta_{-}\leq 1$.

By combining (4.9) with (4.5), we deduce

$\displaystyle L_{\kappa}(w_{j})$ $\displaystyle=$ $\displaystyle
l_{j}\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{-}-1}+$ (4.11)
$\displaystyle+h_{1,j}(x)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{+}+\beta-3}+h_{2,j}(x)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{-}+\beta-2}$
$\displaystyle+\tilde{f}_{j}(x)\Big{(}\frac{d_{j}(r_{j}x)}{r_{j}}\Big{)}^{\theta_{+}-2}+\tilde{g}_{j}(x)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{-}-2}$

where $\tilde{f}_{j}=\frac{f_{j}(r_{j}x)}{\theta(r_{j})r_{j}^{\beta-1}}$ and
$\tilde{g}_{j}=\frac{g_{j}(r_{j}x)}{\theta(r_{j})r_{j}^{\beta+\theta_{+}-\theta_{-}-1}}$
and with both vanishing uniformly as $j\to\infty$ thanks to the hypotheses on
$f_{j}$ and $g_{j}$. In particular, we are fundamentally exploiting that
$|f_{j}|\leq Cd_{j}^{\beta-1}$ and that $|g_{j}|\leq
Cd_{j}^{\beta-1+\theta_{+}-\theta_{-}}$.

On the other hand, let us notice that the hypotheses (4.2) force $w_{j}$ to
satisfy the boundary conditions (3.11). With the aim of applying Lemma 3.3, we
take $\varepsilon\in(0,\min\\{\beta-1,1+\theta_{-}-\theta_{+}\\})$ in (3.13)
so that the right hand side in (4.11) remains uniformly bounded after
multiplying by $d_{j}^{2-\theta_{+}-\varepsilon}$. Thus, in virtue of Lemma
3.3 and Lemma 3.4, the solutions $\\{w_{j}\\}_{j\in\mathbb{N}}$ to the
sequence of problems (4.11) are equicontinuous and, therefore, $w_{j}\to
w_{0}$ locally uniformly as $j\to\infty$ -up to subsequences. So, thanks to
Lemma 3.4, and (4.8) we deduce that

$\begin{cases}L_{\kappa}w_{0}=lx_{n}^{\theta_{-}-1},\quad\mbox{ in
$\\{x_{n}>0\\}$},\\\ \|w_{0}\|_{L^{\infty}(B_{R}\cap\mathbb{R}^{n}_{+})}\leq
CR^{{\theta_{+}}+\beta-1}.\end{cases}$ (4.12)

Additionally, thanks to the uniform convergence, the inequality (3.13) is
satisfied by each $w_{j}$ implies that

$|w_{0}|\leq Cx_{n}^{\theta_{+}}.$ (4.13)

Step 3: Since $w_{0}$ satisfies (4.12) and (4.13), Lemma 3.5 applies, implying
that

$w_{0}=\lambda_{1}(x_{n})_{+}^{\theta_{+}}+\lambda_{2}(x_{n})_{+}^{\theta_{-}+1}$

with $\lambda_{1},\lambda_{2}\in\mathbb{R}$. However, taking limits in the
orthogonality conditions (4.6) as $j\to\infty$, yields

$\int_{B_{1}}w_{0}\Big{(}c_{1}(x_{n})_{+}^{\theta_{+}}+c_{2}(x_{n})_{+}^{\theta_{-}+1}\Big{)}=0,$

for any $(c_{1},c_{2})\in\mathbb{R}^{2}$. Thus, $w_{0}=0$ contradicting that
$\|w_{0}\|_{L^{\infty}(B_{1})}\geq\frac{1}{2}$. ∎

In the context of in Theorem 1.2, if $\partial\Omega\cap B_{1}\in C^{\beta}$
with $\beta>2$, since $f|_{\partial\Omega}\equiv 0$, we have that
$fd^{\mu-2}=\tilde{f}d^{\mu-1}$ where $\tilde{f}=f/d$ is one degree less
regular than $f$ up to the boundary. Based on this observation, we prove the
following higher order version of Theorem 4.1.

######  Theorem 4.2.

Let $\kappa\geq-\frac{1}{4}$, $\theta_{\pm}$ as in (1.14), let $\beta>2$ with
$\beta\notin\mathbb{N}$. Let $\Omega\subset\mathbb{R}^{n}$ with
$\partial\Omega\cap B_{1}$ a $C^{\beta}$ surface, let $f$ and $g$ satisfying
$\|f\|_{C^{\beta-2}(\overline{\Omega}\cap
B_{1})},\|g\|_{C^{\beta-2+\theta_{+}-\theta_{-}}(\overline{\Omega}\cap
B_{1})}\leq 1$, with $g=0$ if $\theta_{-}\leq 0$. Let $v\in
C(\overline{\Omega}\cap B_{1})$ be a solution to

$L_{\kappa}(v)=fd^{\theta_{+}-1}+gd^{\theta_{-}-1},\quad\text{in}\,\,\Omega\cap
B_{1},$ (4.14)

satisfying

$\begin{array}[]{rl}\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{\theta_{-}}}=0&\quad\textrm{if}\quad\kappa>-{\textstyle\frac{1}{4}\vspace{1mm}}\\\
\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{1/2}|\log
d|}=0&\quad\textrm{if}\quad\kappa=-{\textstyle\frac{1}{4}}.\end{array}$ (4.15)

Then, there exists a constant $C>0$ and a radius $\rho>0$, such that for every
$x_{0}\in\partial\Omega\cap B_{\rho}$, there exists polynomials
$Q_{1,x_{0}}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$Q_{2,x_{0}}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}$ such
that

$\displaystyle\Big{|}v(x)-Q_{1,x_{0}}(x)d^{\theta_{+}}(x)-Q_{2,x_{0}}(x)d^{\theta_{-}+1}(x)\Big{|}\leq
C|x-x_{0}|^{{{\theta_{+}}+\beta-1}}\quad\mbox{for $x\in
B_{\rho}(x_{0})\cap\Omega$},$ (4.16)

with $Q_{2,x_{0}}=0$ if $g=0$.

###### Proof.

Preparations: Let us assume again $x_{0}=0$ and let us extend $v$ and $d$ by
zero in $\Omega^{c}\cap B_{1}$. By contradicting the inequality (4.16), we
deduce the existence of a sequence of domains $\Omega_{j}$ with
$\|\Omega_{j}\|_{C^{\beta}}\leq 1$ and, in consequence, of generalized
distance functions $d_{j}$ such that for any sequence of polynomials
$P_{1,j}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$P_{2,j}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-1}$

$\sup_{r>0}\sup_{j\in\mathbb{N}}\frac{1}{r^{{{\theta_{+}}+\beta-1}}}\Big{\|}v_{j}-P_{1,j}d_{j}^{\theta_{+}}-P_{2,j}d_{j}^{\theta_{-}+1}\Big{\|}_{L^{\infty}(B_{r})}=\infty,$
(4.17)

with $v_{j}$ satisfying (4.15) and

$L_{\kappa}(v_{j})=f_{1,j}d_{j}^{\theta_{+}-1}+f_{2,j}d_{j}^{\theta_{-}-1}\quad\text{in}\,\,\Omega\cap
B_{1}$ (4.18)

and where $\|f_{1,j}\|_{C^{\beta-2}(B_{2}\cap\overline{\Omega_{j}})}\leq 1$
and
$\|f_{2,j}\|_{C^{\beta+\theta_{+}-\theta_{-}-2}(B_{2}\cap\overline{\Omega_{j}})}\leq
1$ for $j\in\mathbb{N}$. Furthermore, we take $f_{2,j}=P_{2,j}=0$ if $g=0$.

Step one: We select the least squares polynomials
${(Q_{1,j,r},Q_{2,j,r})\in\mathbf{P}_{\lfloor\beta\rfloor-1}\times\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}}$,
which are characterized by the orthogonality conditions

$\int_{B_{r}}(v_{j}-Q_{1,j,r}d_{j}^{\theta_{+}}-Q_{2,j,r}d_{j}^{\theta_{-}+1})(P_{1}(x)d_{j}^{\theta_{+}}+P_{2}(x)d_{j}^{\theta_{-}+1})=0,$
(4.19)

for any
$(P_{1},P_{2})\in\mathbf{P}_{\lfloor\beta\rfloor-1}\times\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}$
for $r>0$ and $j\in\mathbb{N}$.

Let us define

$\theta(r)=\sup_{\rho>r}\sup_{j\in\mathbb{N}}\frac{1}{r^{{{\theta_{+}}+\beta-1}}}\Big{\|}v_{j}-Q_{1,j,r}d_{j}^{\theta_{+}}-Q_{2,j,r}d_{j}^{\theta_{-}+1}\Big{\|}_{L^{\infty}(B_{r})}.$

By Lemma 6.2 we have that $\lim_{r\to 0^{+}}\theta(r)=\infty$. Thus, we can
pick a sequence $\\{r_{j}\\}_{j\in\mathbb{N}}$ with $r_{j}\to 0^{+}$ as
$j\to\infty$ such that

$w_{j}(x):=\frac{v_{j}(r_{j}x)-Q_{1,j,r_{j}}(r_{j}x)d_{j}^{\theta_{+}}(r_{j}x)-Q_{2,j,r_{j}}(r_{j}x)d_{j}^{\theta_{-}+1}(r_{j}x)}{r_{j}^{{{\theta_{+}}+\beta}}\theta(r_{j})},$
(4.20)

satisfies $\|w_{j}\|_{L^{\infty}(B_{1})}\in[\frac{1}{2},1]$ for
$j\in\mathbb{N}$.

Additionally, Lemma 6.3 provides the growth bound

$\|w_{j}\|_{L^{\infty}(B_{R})}\leq CR^{{\theta_{+}}+\beta-1}.$ (4.21)

Step two: We proceed to show that the sequence $\\{w_{j}\\}_{j\in\mathbb{N}}$
satisfies, up to a small perturbation, a PDE of the form (3.10). Let us start
noticing that since we fall under the hypothesis of Lemma 6.3, we can estimate
the Laplacian of the auxiliary term
$Q_{1,j,r}d_{j}^{\theta_{+}}+Q_{2,j,r}d_{j}^{\theta_{-}+1}$ as follows

$\displaystyle\frac{1}{\theta(r)r^{{\theta_{+}}+\beta-1}}\Delta\Big{(}Q_{1,r}(rx)d_{j}^{\theta_{+}}(rx)+Q_{2,r}(rx)d_{j}^{\theta_{-}+1}(rx)\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\theta(r)r^{{\theta_{+}}+\beta-1}}\frac{\kappa
r^{2}}{d_{j}^{2}(rx)}Q_{1,r}(rx)d_{j}^{\theta_{+}}(rx)$ (4.22)
$\displaystyle+\frac{1}{\theta(r)r^{{\theta_{+}}+\beta-1}}\frac{\kappa
r^{2}}{d_{j}^{2}(rx)}Q_{2,r}(rx)d_{j}^{\theta_{-}+1}(rx)$
$\displaystyle+P_{1,r}(x)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{+}-1}$
$\displaystyle+P_{2,r}(rx)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{-}-1}$
$\displaystyle+m_{*}(r,x)$

where $P_{i,r}\in\mathbf{P}_{M}$ for $i=1,2$ for some $M\in\mathbb{N}$ only
depending on $\beta$ and $\kappa$ and where $|m_{*}(x,r)|\leq Cm(r)$ with $m$
depending only on $\beta$ and $\kappa$ and such that $m(r)\to 0$ as $r\to
0^{+}$.

By combining (4.18), (4.22), and exploiting the uniform regularity of the
functions $f_{i,j}$ we deduce

$\displaystyle L_{\kappa}(w_{j})$ $\displaystyle=$
$\displaystyle\Big{(}\tilde{Q}_{1,r}(x)+g_{1,j}(x)\Big{)}\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{+}-1}+\Big{(}\tilde{Q}_{2,r}(rx)+g_{2,j}(x)\Big{)}\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\theta_{-}-1}$
(4.23) $\displaystyle+g_{3,j}(x),$

where $\tilde{Q}_{i,j}\in\mathbf{P}_{M}$ for some $M\in\mathbb{N}$ and
$\|g_{i,j}\|_{L^{\infty}(B_{2}\cap\overline{\Omega_{j}})}\to 0$ as
$j\to\infty$ for $i=1,2,3$.

On the other hand, let us notice that the hypotheses (4.15) force $w_{j}$ to
satisfy the boundary conditions (3.11). With the aim of applying Lemma 3.3, we
take $\varepsilon\in(0,1+\theta_{-}-\theta_{+})$ in (3.13) so that the right
hand side in (4.23) remains uniformly bounded after multiplying by
$d_{j}^{2-\theta_{+}-\varepsilon}$. Thus, in virtue of Lemma 3.4 and Lemma
3.3, the solutions $\\{w_{j}\\}_{j\in\mathbb{N}}$ to the sequence of problems
(4.11) are equicontinuous and, therefore, satisfy that $w_{j}\to w_{0}$
locally uniformly. So, thanks to Lemma 3.4, and (4.8) we deduce that

$\begin{cases}L_{\kappa}w_{0}=Q_{1}(x)(x_{n})_{+}^{\theta_{+}-1}+Q_{2}(x)(x_{n})_{+}^{\theta_{-}-1}\quad\mbox{
in $\\{x_{n}>0\\}$ },\\\
\|w_{0}\|_{L^{\infty}(B_{R}\cap\mathbb{R}^{n}_{+})}\leq
CR^{{\theta_{+}}+\beta-1}\end{cases}$ (4.24)

with $Q_{i}\in\mathbf{P}_{M}$ for some $M\in\mathbb{N}$ and for $i=1,2$ and
with $Q_{2}=0$ if $g=0$. Additionally, thanks to the uniform convergence, the
inequality (3.13) satisfied by each $w_{j}$ implies that

$|w_{0}|\leq Cx_{n}^{\theta_{+}}.$ (4.25)

Step 3: Since $w_{0}$ satisfies (4.24) and (4.25), Lemma 3.5 applies, implying
that

$w_{0}=P_{1}(x)(x_{n})_{+}^{{\theta_{+}}}+P_{2}(x)(x_{n})_{+}^{{\theta_{-}+1}},$

with $P_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$P_{2}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}$, with
$P_{2}=0$ if $g=0$. However, taking limits in the orthogonality condition
(4.19) as $j\to\infty$, yields

$\int_{B_{1}}w_{0}\Big{(}Q_{1}(x)x_{n}^{{\theta_{+}}}+Q_{2}(x)x_{n}^{{\theta_{-}+1}}\Big{)}=0,$

for all $Q_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$Q_{2}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}$. Thus,
$w_{0}=0$ contradicting $\|w_{0}\|_{L^{\infty}(B_{1})}\geq\frac{1}{2}.$ ∎

The following technical lemma shows how to translate estimates of the form
(4.16) into regularity for the quotients $v/d^{\mu}$.

######  Lemma 4.3.

Let $\mu>0$, let $\beta>1$ with $\beta\notin\mathbb{N}$, and let
$\Omega\subset\mathbb{R}^{n}$ with $\partial\Omega\cap B_{1}$ a $C^{\beta}$
surface. Let $v\in C^{\beta}(\Omega\cap B_{1})$ and let us assume that for
every $z\in\Omega\cap B_{1}$ there exists
$Q_{z}\in\mathbf{P}_{\lfloor\beta-1\rfloor}$ such that for every $y_{0}\in
B_{\frac{1}{2}}(z)\cap\Omega$ satisfying $\text{dist}(y_{0})=|y_{0}-z|=2r$

$[v-Q_{z}d^{\mu}]_{C^{\beta-1}(\overline{B_{r}(y_{0})})}\leq Cr^{\mu},$ (4.26)

and

$\|v-Q_{z}d^{\mu}\|_{L^{\infty}({B_{r}(y_{0})})}\leq Cr^{\mu+\beta-1},$ (4.27)

Then,

$\Big{\|}\frac{v}{d^{\mu}}\Big{\|}_{C^{\beta-1}(\overline{\Omega}\cap
B_{1/2})}\leq C.$ (4.28)

More in general, if there is an integer $k>\beta-1$ such that $v\in
C^{k}(\Omega\cap B_{1})$ and such that for every $\lambda\in\mathbb{N}^{n}$
with $|\lambda|=k$,

$\|\partial^{\lambda}(v-Q_{z}d^{\mu})\|_{L^{\infty}(B_{r}(y_{0}))}\leq
C_{k}r^{\mu+\beta-1-k},$ (4.29)

then

$|\partial^{\lambda}\big{(}v/d^{\mu}\big{)}\big{|}\leq
C_{k}d^{\beta-1-k}\quad\textrm{in}\quad\Omega\cap B_{\frac{1}{2}}.$ (4.30)

###### Proof.

The proof of the first part of this lemma is essentially contained in the
proof of [1, Theorem 1.4]. Let $x_{1},x_{2}\in B_{r}(y_{0})$ and let
$\lambda\in\mathbb{N}^{n}$ be a multi-index with
$|\lambda|=\lfloor\beta\rfloor-1$. Let $Q_{z}$ as in (4.26), arguing as in the
proof of [1, Theorem 1.4], we have the identity

$\displaystyle\partial^{\lambda}(d^{-\mu}u)(x_{1})-\partial^{\lambda}(d^{-\mu}u)(x_{2})$
$\displaystyle=$
$\displaystyle\sum_{\alpha\leq\lambda}[\partial^{\alpha}(u-Q_{z}d^{\mu})(x_{1})-\partial^{\alpha}(u-Q_{z}d^{\mu})(x_{2})]\partial^{\lambda-\alpha}d^{-\mu}(x_{1})$
(4.31)
$\displaystyle+\partial^{\alpha}(u-Q_{z}d^{\mu})(x_{2})[\partial^{\lambda-\alpha}d^{-\mu}(x_{1})-\partial^{\lambda-\alpha}d^{-\mu}(x_{2})],$

where we have used that
$\partial^{\lambda}Q_{z}(x_{1})=\partial^{\lambda}Q_{z}(x_{2})$ since $Q_{z}$
has degree $\lfloor\beta-1\rfloor=\lambda$.

We start bounding the terms in the right hand side of (4.31), recalling from
Lemma 2.1 that

$|\partial^{\lambda-\alpha}d^{-\mu}(x_{1})|\leq Cr^{|\alpha|-|\lambda|-\mu},$
(4.32)

for any $\alpha\leq\lambda$. On top of this, we can combine (4.32) with the
mean value theorem to deduce

$|\partial^{\lambda-\alpha}d^{-\mu}(x_{1})-\partial^{\lambda-\alpha}d^{-\mu}(x_{2})|\leq
Cr^{|\alpha|-\mu-\beta}|x_{1}-x_{2}|^{\beta-\lfloor\beta\rfloor}.$ (4.33)

On the other hand, we can use interpolation [23, Lemma 6.32] to deduce from
(4.26) and (4.27) that

$[\partial^{\alpha}(v-Q_{z}d^{\mu})]_{C^{\beta-\lfloor\beta\rfloor}(\overline{B_{r}(y_{0})})}\leq
Cr^{\mu-|\alpha|+\lfloor\beta\rfloor-1}$ (4.34)

for any $\alpha\leq\lambda$. In the same vein, we have that

$\sum_{|\alpha|\leq\lfloor\beta\rfloor-1}r^{|\alpha|}\|\partial^{\alpha}(v-Q_{z}d^{\mu})\|_{L^{\infty}(B_{r}(y_{0}))}\leq
Cr^{\mu+\beta-1}.$ (4.35)

Hence, by combining (4.31), (4.32), (4.34), and (4.35) we deduce

$\displaystyle|\partial^{\lambda}(d^{-\mu}u)(x_{1})-\partial^{\lambda}(d^{-\mu}v)(x_{2})|\leq
C|x_{1}-x_{2}|^{\beta-\lfloor\beta\rfloor},$

from where (4.28) follows.

Let us assume now that (4.29) holds for some $k>\lfloor\beta\rfloor-1$. Let
$x\in B_{r}(y_{0})$ and $\lambda\in\mathbb{N}^{n}$ with $|\lambda|=k$. Since
$\partial^{\lambda}Q_{z}=0$, we can proceed as in (4.31) to get

$\displaystyle\partial^{\lambda}(d^{-\mu}u)(x)=\partial^{\lambda}(d^{-\mu}v-Q_{z})(x)=\sum_{\alpha\leq\lambda}\partial^{\alpha}(v-d^{\mu}Q)(x)\,\partial^{\lambda-\alpha}d^{-\mu}.$
(4.36)

Using again [23, Lemma 6.32], we can interpolate (4.27) and (4.29) to deduce

$\sum_{|\alpha|\leq
k}r^{|\alpha|}\|\partial^{\alpha}(v-Q_{z}d^{\mu})\|_{L^{\infty}({B_{r}(y_{0})})}\leq
Cr^{\mu+\beta-1-k}.$ (4.37)

Altogether (4.32), (4.36), and (4.37) implies

$|\partial^{\lambda}(d^{-\mu}v)(x)|\leq Cr^{\beta-1-k}\leq Cd^{\beta-1-k}(x),$

where the last inequality follows from the comparability of $d(y_{0})$ and
$r$. ∎

###### Proof of Theorem 1.2.

Let us start by normalizing both sides of (1.9) dividing by
$\|f\|_{C^{k,\alpha}(\overline{\Omega}\cap
B_{1})}+\|v\|_{L^{\infty}(\Omega\cap B_{1})}$. Thus, in virtue of Lemma 4.3,
the proof will follow from verifying (4.26). Let $\beta=k+1+\alpha$ and let us
invoke Theorem 4.1 if $k=1$ and Theorem 4.2 when $k\geq 2$, to guarantee the
existence of a polynomial $Q_{z}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ for
each $z\in\partial\Omega\cap B_{1}$ such that

$\displaystyle\Big{|}v(x)-Q_{z}(x)d^{\theta_{+}}(x)\Big{|}\leq
C|x-z|^{{{\theta_{+}}+\beta-1}}\quad\mbox{for $x\in\Omega\cap B_{1}(z)$}.$
(4.38)

For any $y_{0}\in\Omega\cap B_{\frac{1}{2}}(z)$ such that
$\text{dist}(y_{0})=|y_{0}-z|=2r$ we define the rescaling

$v_{r}(x)=r^{{{-\theta_{+}}-\beta+1}}v(y_{0}+rx)-r^{1-\beta}Q_{z}(y_{0}+rx)b_{r}(x)^{\theta_{+}},$

where $b_{r}(x)=\frac{d(y_{0}+rx)}{r}$. Notice that (4.38) implies the bound
$\|v_{r}\|_{L^{\infty}(B_{1})}\leq C$. On the other hand, the equation for
$v_{r}$ can be written as

$-\Delta
v_{r}+a_{r}(x)v_{r}=r^{1-\beta}\Big{(}f(y_{0}+rx)b_{r}(x)^{\theta_{+}-2}-r^{2}L_{\kappa}\Big{(}Q_{z}(y_{0}+r\cdot)b_{r}^{\theta_{+}}\Big{)}(x)\Big{)}$
(4.39)

with $a_{r}(x)=\frac{\kappa}{b_{r}(x)^{2}}$. On the other hand, Lemma 2.1
implies that $\frac{1}{C}\leq|b_{r}|\leq C$ and $|D^{\lambda}b_{r}|\leq
C_{\lambda}$ for $x\in B_{1}$ and for any multi-index $\lambda$. So, if
$\beta\in(1,2)$, classical gradient estimates yield

$\displaystyle[v_{r}]_{C^{\beta-1}(B_{1/2})}$ $\displaystyle\leq$
$\displaystyle
C\Big{(}\|v_{r}\|_{L^{\infty}(B_{1})}+\|r^{-\beta+1}f(y_{0}+r\cdot)b_{r}^{\theta_{+}-2}\|_{L^{\infty}(B_{1})}\Big{)}$
(4.40)
$\displaystyle+Cr^{{3-\beta}}\|L_{\kappa}(Q_{z}(y_{0}+r\cdot)b_{r}^{\theta_{+}})\|_{L^{\infty}(B_{1})}.$

Moreover, we have from Lemma 2.1 that

$|L_{\kappa}(Q_{z}(y_{0}+r\cdot)b_{r})|\leq Cb_{r}(x)^{\beta+\theta_{+}-3}\leq
C$ (4.41)

and, by the $C^{\beta-1}$ regularity on $f$, $|f(y_{0}+rx)|\leq Cr^{\beta-1}$.
Thus, by rescaling (4.40), we have

$\displaystyle r^{-\theta}[v]_{C^{\beta-1}(B_{r/2})}\leq C.$ (4.42)

If $\beta>2$, since $f$ vanishes identically on the boundary, we can write
$fd^{\theta_{+}-2}=f_{0}d^{\theta_{+}-1}$ with $f_{0}\in
C^{\beta-2}(\overline{\Omega}\cap B_{1})$. Additionally, in virtue of Lemma
2.1, we have that

$L_{\kappa}(Q_{z}d^{\theta_{+}})(x)=-g(x)d(x)^{\theta_{+}-1}$ (4.43)

for some $g\in C^{\beta-2}(\overline{\Omega}\cap B_{1})$. Altogether, these
observations allow us to rewrite (4.39) as follows

$-\Delta
v_{r}+a_{r}(x)v_{r}=r^{2-\beta}\Big{(}f_{0}(y_{0}+rx)+g(y_{0}+rx)\Big{)}b_{r}(x)^{\theta_{+}-1},$

which can be rewritten in the form

$\frac{-1}{b_{r}(x)^{\theta_{+}-1}}\Delta
v_{r}+\frac{a_{r}(x)}{b_{r}(x)^{\theta_{+}-1}}v_{r}=r^{2-\beta}\Big{(}f_{0}(y_{0}+rx)+g(y_{0}+rx)\Big{)}.$
(4.44)

Thus, since $b_{r}\in C^{\beta}(B_{1})$ and $\frac{1}{C}\leq|b_{r}|\leq C$, we
can apply standard interior Schauder estimates to (4.44) (see, e.g., [31]) to
deduce

$\displaystyle[v_{r}]_{C^{\beta-1}(B_{1/2})}\leq
C\Big{(}\|v_{r}\|_{L^{\infty}(B_{1})}+r^{2-\beta}[f(y_{0}+r\cdot)+g(y_{0}+r\cdot)]_{C^{\beta-2}(B_{1})}\Big{)}$

After rescaling, we deduce

$\displaystyle r^{-\theta}[v]_{C^{\beta-1}(B_{r/2})}\leq
C\Big{(}1+[f+g]_{C^{\beta-2}(B_{r}(y_{0}))}\Big{)}\leq C,$ (4.45)

which concludes the proof. ∎

We finish this section deriving a modified version of Theorem 1.2 that will be
applied to $u-c_{\gamma}d^{\frac{2}{2-\gamma}}$ in the next section.

######  Corollary 4.4.

Let $\kappa\geq-\frac{1}{4}$, let $\mu=\theta_{+}$ if $\kappa\geq 0$ and
$\mu\in\\{\theta_{+},\theta_{-}\\}$ when $\kappa<0$. Let
$\Omega\subset\mathbb{R}^{n}$ be a $C^{\beta}$ domain with $\beta>2$, $f\in
C^{\beta-2}(\overline{\Omega}\cap B_{1})$, and let $d$ be the regularized
distance to $\partial\Omega$ given by Lemma 2.1.

Let $v\in C(\Omega\cap B_{1})$ be any solution to

$-\Delta v+\kappa\,\frac{v}{d^{2}}=fd^{\mu-1}\quad\text{in}\quad\Omega\cap
B_{1}.$ (4.46)

If it satisfies

$\displaystyle\lim_{x\to\partial\Omega}\frac{v}{d^{\theta_{+}}}=0,$ (4.47)

then, $v\in C^{\beta-2}(\overline{\Omega}\cap B_{1/2})$ and

$\left\|v/{d^{\mu+1}}\right\|_{C^{\beta-2}(\overline{\Omega}\cap B_{1/2})}\leq
C\big{(}\|f\|_{C^{\beta-2}(\overline{\Omega}\cap
B_{1})}+\|v\|_{L^{\infty}(\Omega\cap B_{1})}\big{)},$

where $C>0$ depends only on $\Omega$, $k$, $\alpha$, $n$, and $\kappa$.

Furthermore, if there is $k>\beta-2$ such that $v\in C^{k}(\Omega\cap B_{1})$,
$f\in C^{k}(\Omega\cap B_{1})$ and

$|\partial^{\alpha}f|\leq C_{\alpha}d^{\beta-2-|\alpha|},\quad\mbox{ in
$\Omega\cap B_{1}$},$ (4.48)

for every $|\alpha|\in\\{\lfloor\beta\rfloor-1,\cdots,k-2\\}$, then

$|\partial^{\lambda}\big{(}v/d^{\mu+1}\big{)}\big{|}\leq
C_{k}d^{\beta-2-|\lambda|}\quad\textrm{in}\quad\Omega\cap B_{\frac{1}{2}}.$
(4.49)

for every $|\lambda|\in\\{\lfloor\beta\rfloor-1,\cdots,k\\}$.

###### Proof.

If $\mu=\theta_{+}$, we can apply directly Theorem 1.2 to conclude that
$v=hd^{\theta_{+}}$ for some $h\in C^{\beta-1}$. On top of this, thanks to the
boundary condition (4.47), we have that $h|_{\partial\Omega}\equiv 0$ implying
that $h/d\in C^{\beta-2}$, proving the result in this case.

If $\mu=\theta_{-}$, we exploit the fact that $\theta_{+}>\theta_{-}$ to apply
Theorem 4.2 which combined with the boundary condition (4.47) implies that
(4.16) holds with $Q_{1,x_{0}}=0$ for any $x_{0}$. The rest of the argument
thus follow proceeding as in the proof of Theorem 1.2.

Assuming now $v\in C^{k}(\Omega\cap B_{1})$ and $f\in C^{k}(\Omega\cap B_{1})$
for some $k>\beta-2$ together with the condition (4.48), we notice that as in
the proof of Theorem 1.2, we have that for every $z\in\partial\Omega\cap
B_{1}$, there exists a polynomial $Q_{z}\in\mathbf{P}_{\lfloor\beta\rfloor-2}$
such that

$\displaystyle\Big{|}v(x)-Q_{z}(x)d^{\mu+1}(x)\Big{|}\leq
C|x-z|^{{{\mu+1}+\beta-2}}\quad\mbox{for $x\in B_{1}(z)\cap\Omega$}.$ (4.50)

So, as in the proof of Theorem 1.2, for any $y_{0}\in\Omega\cap
B_{\frac{1}{2}}(z)$ such that $\text{dist}(y_{0})=|y_{0}-z|=2r$ we define the
rescaling

$v_{r}(x)=r^{{{-(\mu+1)}-\beta+2}}v(y_{0}+rx)-r^{2-\beta}Q_{z}(y_{0}+rx)b_{r}(x)^{\mu+1},$

with $b_{r}(x)=\frac{d(y_{0}+rx)}{r}$. Since we are assuming $\beta>2$, we can
argue as in the proof of Theorem 1.2 to show that

$\frac{-1}{b_{r}(x)^{\mu-1}}\Delta
v_{r}+\frac{a_{r}(x)}{b_{r}(x)^{\mu-1}}v_{r}=r^{2-\beta}\Big{(}f_{0}(y_{0}+rx)+g(y_{0}+rx)\Big{)},$
(4.51)

with $a_{r}(x)=\frac{\kappa}{b_{r}(x)^{2}}$ and where $g\in
C^{\beta-2}(\overline{\Omega}\cap B_{1})$ is given by the identity
$L_{\kappa}(Q_{z}d^{\mu})=-g(z)d^{\mu-1}$ and thanks to (2.9), satisfies

$|\partial^{\alpha}g|\leq C_{\alpha}d^{\beta-2-|\alpha|},\quad\mbox{ in
$\Omega\cap B_{1}$},$ (4.52)

for $|\alpha|>\beta-2$. Thus, by applying interior Schauder estimates to
(4.51), we have that for any multi-index $\lambda$ with
$|\lambda|\in\\{\lfloor\beta\rfloor-1,\cdots,k\\}$

$\displaystyle\|\partial^{\lambda}v_{r}\|_{L^{\infty}(B_{\frac{1}{2}})}\leq
C\Big{(}\|v_{r}\|_{L^{\infty}(B_{1})}+r^{2-\beta}\|D^{k-2}(f(y_{0}+r\cdot)+g(y_{0}+r\cdot))\|_{L^{\infty}(B_{1})}\Big{)}$

which by (4.48), after rescaling and combining (4.48) with (4.52) , implies

$\displaystyle
r^{{{-(\mu+1)}-\beta+2+k}}\|\partial^{\lambda}v\|_{L^{\infty}(B_{\frac{r}{2}})}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}1+r^{k-\beta}\|D^{k-2}(f(y_{0}+r\cdot)+g(y_{0}+r\cdot))\|_{L^{\infty}(B_{1})}\Big{)}\leq
C.$

This shows that the hypotheses of Lemma 4.3 are fulfilled from where (4.49)
follows. ∎

##  5\. Higher order free boundary regularity

This section is devoted to the proof of Theorem 1.1. For this, we first show
step (a) in (1.11).

######  Theorem 5.1.

Let $\gamma\in(0,2)$ and let $\Omega\subset\mathbb{R}^{n}$ be a $C^{\beta}$
domain, with $\beta>1$ and $\beta\notin\mathbb{N}$. Let $0\leq u\in
C^{\frac{2}{2-\gamma}}$ be a nontrivial solution of

$\begin{split}\Delta u&=u^{\gamma-1}\quad\textrm{in}\quad\Omega\cap B_{1}\\\
u&=0\qquad\ \textrm{on}\quad\partial\Omega\cap B_{1}.\end{split}$ (5.1)

Then, we have

$u/d^{\frac{2}{2-\gamma}}\in C^{\beta-1}(\overline{\Omega}\cap
B_{r})\qquad\textrm{and}\qquad u_{i}/d^{\frac{\gamma}{2-\gamma}}\in
C^{\beta-1}(\overline{\Omega}\cap B_{r})$

for any $r<1$.

###### Proof.

The proof is by induction on $\lfloor\beta\rfloor$ and it is done through
steps.

Step 1: We find that $v:=u-c_{\gamma}d^{\frac{2}{2-\gamma}}$ satisfies

$L_{\kappa}v=hd^{\frac{2}{2-\gamma}-2},$ (5.2)

for some function $h$ whose regularity will be improved along the proof.

Computing directly, we have that

$\Delta
v=u^{\gamma-1}-\big{(}c_{\gamma}d^{\frac{2}{2-\gamma}}\big{)}^{\gamma-1}+gd^{\frac{2}{2-\gamma}-2},$
(5.3)

where $c_{\gamma}$ is given by (1.5), and $g$ is given by the identity

$gd^{\frac{2}{2-\gamma}-2}=\Delta\big{(}c_{\gamma}d^{\frac{2}{2-\gamma}}\big{)}-\big{(}c_{\gamma}d^{\frac{2}{2-\gamma}}\big{)}^{\gamma-1},$
(5.4)

Now, it follows from Taylor’s Theorem (applied to the function
$t^{\gamma-1}-1$ at $t=1$) that

$p^{\gamma-1}-q^{\gamma-1}=(\gamma-1)(p-q)q^{\gamma-2}+(p/q-1)^{2}q^{\gamma-1}\int_{0}^{1}(\gamma-1)(\gamma-2)\big{(}1+tp/q\big{)}^{\gamma-3}(1-t)dt$

for all positive numbers $p,q>0$. Hence, we deduce that

$u^{\gamma-1}-\big{(}c_{\gamma}d^{\frac{2}{2-\gamma}}\big{)}^{\gamma-1}=\kappa\frac{v}{d^{2}}+fd^{\frac{2}{2-\gamma}-2}$
(5.5)

where $\kappa$ is given by (1.7) and

$f:=\big{(}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{)}^{2}\int_{0}^{1}(\gamma-1)(\gamma-2)\big{(}c_{\gamma}+tu/d^{\frac{2}{2-\gamma}}\big{)}^{\gamma-3}(1-t)dt.$
(5.6)

Combining (5.3) and (5.5) we deduce

$L_{\kappa}v=fd^{\frac{2}{2-\gamma}-2}+gd^{\frac{2}{2-\gamma}-2}.$ (5.7)

Notice that $f$ has at least the same regularity up to the boundary as
$u/d^{\frac{2}{2-\gamma}}$.

Step 2: We prove the base case. Moreover, we show that for any $\beta\in(1,2)$
we have that

$u/d^{\frac{2}{2-\gamma}}\in C^{\beta-1}\qquad\textrm{and}\qquad
u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{\beta-1}\qquad\textrm{for all}$ (5.8)

for $i=1,\cdots n.$

In light of Proposition 2.3, it suffices to show
$u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{\beta-1}$. On the other hand, we
claim that

$|D^{1}\big{(}u/d^{\frac{2}{2-\gamma}}\big{)}\big{|}\leq
Cd^{\beta-2}\quad\textrm{in}\quad\Omega\cap B_{r}.$ (5.9)

for $r\in(0,1)$.

Indeed, arguing as in Corollary 4.4, given any $z\in\partial\Omega\cap B_{1}$,
for any $y_{0}\in B_{\frac{1}{2}}\cap\Omega$ such that
$\text{dist}(y_{0})=|x_{0}-z|=2r$ we can consider the rescaling

$v_{r}(x)=r^{-\beta+1-\frac{2}{2-\gamma}}v(y_{0}+rx)$

which is bounded in $B_{1}$ in virtue of the regularity of
$u/d^{\frac{2}{2-\gamma}}$. Hence, we can rewrite (5.7) as

$\frac{-1}{b_{r}(x)^{\frac{2}{2-\gamma}-2}}\Delta
v_{r}+\frac{a_{r}(x)}{b_{r}(x)^{\frac{2}{2-\gamma}-2}}v_{r}=r^{2-\beta}\Big{(}f(y_{0}+rx)+g(y_{0}+rx)\Big{)},$
(5.10)

with $b_{r}(x)=\frac{d(y_{0}+rx)}{r}$ and
$a_{r}(x)=\frac{\kappa}{b_{r}(x)^{2}}$. Let notice now that since $|g|\leq\in
Cd^{\beta-1}$ thanks to Lemma 2.1 and $|f|\leq Cd^{2(\beta-1)}$ we have that
both functions are bounded. Thus, we can use standard gradient estimates to
deduce

$\displaystyle\|\nabla v_{r}\|_{L^{\infty}(B_{\frac{1}{2}})}\leq
C\Big{(}\|v_{r}\|_{L^{\infty}(B_{1})}+r^{2-\beta}\|f(y_{0}+r\cdot)+g(y_{0}+r\cdot))\|_{L^{\infty}(B_{1})}\Big{)}\leq
C$

implying

$\displaystyle\|\nabla v\|_{L^{\infty}(B_{\frac{r}{2}})}\leq
r^{\frac{2}{2-\gamma}+\beta-2}.$

From here, we can deduce (5.9) from Lemma 4.3.

With (5.9) at hand, we notice that the elementary identity

$u_{i}/d^{\frac{\gamma}{2-\gamma}}=d\partial_{i}\left(u/{d^{\frac{2}{2-\gamma}}}\right)+cu/{d^{\frac{2}{2-\gamma}}},$

yields

$u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{\beta-1}(\overline{\Omega}\cap
B_{r}),$

completing the base case.

We shall assume from now on $\beta>2$ and, by inductive hypothesis

$u/d^{\frac{2}{2-\gamma}}\in C^{\beta-2}(\overline{\Omega}\cap B_{1}).$ (5.11)

In particular, from (5.6) we also have that $f\in C^{\beta-2}$; whereas

$g\in C^{\beta-1}(\overline{\Omega}\cap B_{1}),$ (5.12)

thanks to Lemma 2.1.

Step 3: We claim that

$|D^{k}\big{(}u/d^{\frac{2}{2-\gamma}}\big{)}\big{|}\leq
C_{k}d^{\beta-2-k}\quad\textrm{in}\quad\Omega\cap B_{r},$ (5.13)

for all $k>\beta-2$.

The proof of this claim will follow from an inductive application of Corollary
4.4. Let us start notice that the base case when $k=\lfloor\beta\rfloor-1$
follows immediately from the boundedness of $f$ and $g$ arguing as in the
previous step, i.e., we have that

$|D^{1}\big{(}u/d^{\frac{2}{2-\gamma}}\big{)}\big{|}\leq
Cd^{\beta-3}\quad\textrm{in}\quad\Omega\cap B_{r}.$

for $r\in(0,1)$.

Let us assume now that (5.13) holds for $k\leq l$ and let is show its validity
for $l+1$. Since $g$ is defined by the identity (5.4), Lemma 2.1 implies that

$|\partial^{\alpha}g|\leq C_{\alpha}d^{\beta-1-|\alpha|},\quad\mbox{ in
$\Omega\cap B_{1}$},$ (5.14)

for $|\alpha|>\beta-1$. On the other hand, from (5.6) we have that
$f=H\Big{(}u/d^{\frac{2}{2-\gamma}}\Big{)}$ with $H\in C^{\infty}$ -bearing in
mind also that $u/d^{\frac{2}{2-\gamma}}$ is uniformly bounded from below. So,
the inductive hypotheses implies that for
$|\alpha|\in\\{\lfloor\beta\rfloor-1,\cdots,l\\}$

$|\partial^{\alpha}f|\leq C_{\alpha}d^{\beta-2-|\alpha|},\quad\mbox{ in
$\Omega\cap B_{1}$}.$ (5.15)

Hence, (5.14) and (5.15) together with (5.11) allows us to invoke Corollary
4.4 and show that (5.13) holds with $k=l+1$, completing the inductive step.

Step 4: Our next goal is to show that $f\in C^{\beta-1}(\overline{\Omega}\cap
B_{r})$ for all $r<1$. We first prove it at all boundary points on
$\partial\Omega$. Namely, for any $z\in\partial\Omega\cap B_{r}$. Let us start
observing that by inductive hypothesis we have

$u/d^{\frac{2}{2-\gamma}}-c_{\gamma}=P_{z}(x)+O(|x-z|^{\beta-2}),$

for all $x\in\Omega\cap B_{1}$, for some polynomial $P_{z}$ satisfying
$P_{z}(z)=0$, and where the (implicit) constants are independent of
$z\in\partial\Omega\cap B_{r}$. Now, since $P_{z}(x)=O(|x-z|)$ we obtain

$\left(P_{z}+O(|x-z|^{\beta-2})\right)^{2}=P_{z}^{2}+O(|x-z|^{\beta-1}),$

provided that $\beta>3$, and therefore

$\big{(}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{)}^{2}=P_{z}^{2}+O(|x-z|^{\beta-1}).$

When $\beta<3$ we use Proposition 2.3 instead, to get

$u/d^{\frac{2}{2-\gamma}}-c_{\gamma}=O(|x-z|^{\alpha})$

and choosing $2\alpha=\beta-1$ we deduce

$\big{(}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{)}^{2}=O(|x-z|^{\beta-1}).$

In both cases, we also have that

$\int_{0}^{1}(\gamma-1)(\gamma-2)\big{(}c_{\gamma}+tu/d^{\frac{2}{2-\gamma}}\big{)}^{\gamma-3}(1-t)dt\in
C^{\beta-2},$

and therefore

$f(x)=Q_{z}(x)+O(|x-z|^{\beta-1})$

for all $x\in\Omega\cap B_{1}$, for some polynomial $Q_{z}$ satisfying
$Q_{z}(z)=0$, and where the (implicit) constants are independent of
$z\in\partial\Omega\cap B_{r}$.

This means that $f$ is pointwise $C^{\beta-1}$ on $\partial\Omega\cap B_{r}$.
We now want to combine this information with interior regularity estimates, to
deduce that $f\in C^{\beta-1}(\overline{\Omega}\cap B_{r})$.

Indeed, thanks to step 3 and the Leibniz Rule, choosing
$k=\lfloor\beta\rfloor$, and using that
$\big{|}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{|}\leq Cd$, we have

$\begin{split}\big{|}D^{k}\big{(}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{)}^{2}\big{|}&\leq
C\sum_{i=0}^{k/2}\big{|}D^{i}\big{(}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{)}\big{|}\cdot\big{|}D^{k-i}\big{(}u/d^{\frac{2}{2-\gamma}}-c_{\gamma}\big{)}\big{|}\\\
&\leq
Cd^{\beta-2-k}d+C\sum_{i=1}^{k/2}\big{|}D^{i}\big{(}u/d^{\frac{2}{2-\gamma}}\big{)}\big{|}\cdot\big{|}D^{k-i}\big{(}u/d^{\frac{2}{2-\gamma}}\big{)}\big{|}\\\
&\leq Cd^{\beta-1-k}+C\sum_{i=1}^{k/2}(1+d^{\beta-2-k+i})\\\ &\leq
Cd^{\beta-1-k}+Cd^{\beta-1-k}.\end{split}$

Applying again the Leibniz Rule, we find

$\big{|}D^{k}f\big{|}\leq Cd^{\beta-1-k}\quad\textrm{in}\quad\Omega\cap
B_{r},$

and in particular

$f\in C^{\beta-1}(\overline{\Omega}\cap B_{r}),$

as wanted.

Step 5: Our next step consists in running a blow-up argument as in the proof
of Theorem 4.2 and show the following:

Claim: There exists a constant $C>0$ and a radius $\rho>0$, such that for
every $x_{0}\in\partial\Omega\cap B_{\rho}$, there exists a polynomial
$Q_{x_{0}}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ such that

$\displaystyle\Big{|}v(x)-Q_{1,x_{0}}(x)d^{\frac{2}{2-\gamma}}(x)\Big{|}\leq
C|x-x_{0}|^{\frac{2}{2-\gamma}+\beta-1}\quad\mbox{for $x\in
B_{\rho}(x_{0})\cap\Omega$}.$ (5.16)

Proof of the claim: This proof follows the same lines of the proof of Theorem
4.2, for this reason we just sketch the common features of the argument,
stressing only on the (slight) variations of our reasoning.

Let us assume $x_{0}=0$ and argue by contradiction assuming a sequence of
domains $\Omega_{j}$ with $\|\Omega_{j}\|_{C^{\beta}}\leq 1$ and, in
consequence, of distance functions $d_{j}$ such that for any sequence of
polynomials $P_{j}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$

$\sup_{r>0}\sup_{j\in\mathbb{N}}\frac{1}{r^{\frac{2}{2-\gamma}+\beta-1}}\Big{\|}v_{j}-P_{j}d_{j}^{\frac{2}{2-\gamma}}\Big{\|}_{L^{\infty}(B_{r})}=\infty,$
(5.17)

with

$L_{\kappa}(v_{j})=h_{j}d_{j}^{\frac{2}{2-\gamma}-2},\quad\text{in}\,\,B_{2}\cap\Omega$
(5.18)

with $\|h_{j}\|_{C^{\beta-1}(\overline{\Omega_{j}}\cap B_{1})}\leq 1$ for
$j\in\mathbb{N}$.

We select the least squares polynomial
$Q_{j,r}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ characterized by the
orthogonality condition

$\int_{B_{r}}\Big{(}v_{j}-Q_{1,j,r}d_{j}^{\frac{2}{2-\gamma}}\Big{)}P_{1}(x)d_{j}^{\frac{2}{2-\gamma}}=0,$
(5.19)

for any $P_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ for $r>0$ and
$j\in\mathbb{N}$.

Let us define

$\theta(r)=\sup_{\rho>r}\sup_{j\in\mathbb{N}}\frac{1}{r^{{\frac{2}{2-\gamma}+\beta-1}}}\Big{\|}v_{j}-Q_{j,r}d_{j}^{\frac{2}{2-\gamma}}\Big{\|}_{L^{\infty}(B_{r})}.$

Arguing as in Lemma 6.2, we have that $\lim_{r\to 0^{+}}\theta(r)=\infty$.
Thus, we can pick a sequence $\\{r_{j}\\}_{j\in\mathbb{N}}$ with $r_{j}\to
0^{+}$ as $j\to\infty$ such that

$w_{j}(x):=\frac{v_{j}(r_{j}x)-Q_{j,r_{j}}(r_{j}x)d_{j}^{\frac{2}{2-\gamma}}(r_{j}x)}{r_{j}^{{\frac{2}{2-\gamma}+\beta-1}}\theta(r_{j})},$
(5.20)

satisfies $\|w_{j}\|_{L^{\infty}(B_{1})}\in[\frac{1}{2},1]$ for
$j\in\mathbb{N}$.

Additionally, the same reasoning of Lemma 6.3 provides the growth bound

$\|w_{j}\|_{L^{\infty}(B_{R})}\leq CR^{\frac{2}{2-\gamma}+\beta-1}.$ (5.21)

One of the main differences with Theorem 4.2 appears when we compute the
Laplacian of the auxiliary term $Q_{1,j,r}d_{j}^{\frac{2}{2-\gamma}}$. More
precisely, using (2.5) to deduce that

$\Delta
d_{j}^{\frac{2}{2-\gamma}}(x)=d_{j}^{\frac{2}{2-\gamma}}(x)(R_{j}(x)d_{j}(x)+c_{\gamma}^{\gamma-2}+d_{j}(x)E(x)),$

with $|E(x)|\leq C|x|^{\beta-2}$ and
$R_{j}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ with
$\|R_{j}\|_{L^{\infty}(B_{1})}\leq C_{\beta}$, we can argue as in Lemma 6.3 to
obtain the expansion

$\displaystyle\frac{1}{\theta(r)r^{\frac{2}{2-\gamma}+\beta-1}}\Delta\Big{(}Q_{r}(rx)d_{j}^{\frac{2}{2-\gamma}}(rx)\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\theta(r)r^{\frac{2}{2-\gamma}+\beta-1}}\frac{\kappa
r^{2}}{d_{j}^{2}(rx)}\Big{(}Q_{1,r}(rx)d_{j}^{\frac{2}{2-\gamma}}(rx)\Big{)}$
(5.23)
$\displaystyle+P_{r}(x)\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\frac{2}{2-\gamma}-2}+m_{*}(r,x)$

where $P_{r}\in\mathbf{P}_{M}$ for some $M\in\mathbb{N}$ depending only on
$\beta$ and $\kappa$ and where $|m_{*}(x,r)|\leq Cm(r)$ with $m$ only
depending on $\beta$ and $\gamma$ and such that $m(r)\to 0$ as $r\to 0^{+}$.

By combining (4.18), (4.22), and exploiting the uniform regularity of the
functions $h_{j}$ we deduce

$\displaystyle L_{\kappa}(w_{j})$ $\displaystyle=$
$\displaystyle\Big{(}\tilde{Q}_{j}(x)+g_{j}(x)\Big{)}\Big{(}\frac{d_{j}(rx)}{r}\Big{)}^{\frac{2}{2-\gamma}-2}+h_{j}(x),$
(5.24)

where $\tilde{Q}_{j}\in\mathbf{P}_{M}$ for some $M\in\mathbb{N}$ and
$\|g_{j}\|_{L^{\infty}(B_{2}\cap\overline{\Omega_{j}})}+\|h_{j}\|_{L^{\infty}(B_{2}\cap\overline{\Omega_{j}})}\to
0$ as $j\to\infty$. Additionally, in virtue of Lemma 3.4, the sequence of
polynomials $\tilde{Q}_{j}$ are uniformly bounded in $j$.

We consider two cases: if $\gamma\geq\frac{2}{3}$, then
$\frac{2}{2-\gamma}=\theta_{+}+1$; whereas, if $\gamma<\frac{2}{3}$, we have
that $\frac{2}{2-\gamma}=\theta_{-}+1$. In both cases, we invoke Corollary 4.4
to obtain uniform boundary estimates that guarantee equicontinuity for
$w_{j}/d_{j}^{\frac{2}{2-\gamma}}$. All in all, the solutions
$\\{w_{j}\\}_{j\in\mathbb{N}}$ to the sequence of problems (4.11) are
equicontinuous and, therefore, satisfy that $w_{j}\to w_{0}$ locally
uniformly. So, thanks to Lemma 3.4, and (4.8) we deduce that

$\begin{cases}L_{\kappa}w_{0}=Q(x)(x_{n})_{+}^{\frac{2}{2-\gamma}-2},\quad\\{x_{n}>0\\}\\\
\|w_{0}\|_{L^{\infty}(B_{R}\cap\mathbb{R}^{n}_{+})}\leq
CR^{{\frac{2}{2-\gamma}}+\beta-1}\end{cases}$ (5.25)

with $Q\in\mathbf{P}_{M}$ for some $M\in\mathbb{N}$. Additionally, thanks to
Corollary 4.4 we have that

$|w_{0}|\leq Cx_{n}^{\frac{2}{2-\gamma}}.$ (5.26)

So, since $w_{0}$ satisfies (5.25) and (5.26), we can apply Lemma 3.5 to
conclude that

$w_{0}=P_{1}(x)(x_{n})_{+}^{\frac{2}{2-\gamma}}$

with $P_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$. However, taking limits in
the orthogonality condition (5.19) as $j\to\infty$, yields

$\int_{B_{1}}w_{0}Q_{1}(x)x_{n}^{\frac{2}{2-\gamma}}=0,$

for all $Q_{1}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$. Thus, $w_{0}=0$
contradicting $\|w_{0}\|_{L^{\infty}(B_{1})}\geq\frac{1}{2}.$

Step 6: We conclude the induction argument.

Once we know that (5.16) holds, it follows arguing as in the proof of Theorem
1.2 that

$u/d^{\frac{2}{2-\gamma}}\in C^{\beta-1}(\overline{\Omega}\cap B_{r}).$

Moreover, exactly as in step 3, we now have

$\big{|}D^{k}\big{(}u/d^{\frac{2}{2-\gamma}}\big{)}\big{|}\leq
Cd^{\beta-1-k}\quad\textrm{in}\quad\Omega\cap B_{r},$

for all $k>\beta-1$, which combined with

$u_{i}/d^{\frac{\gamma}{2-\gamma}}=d\partial_{i}\left(u/{d^{\frac{2}{2-\gamma}}}\right)+cu/{d^{\frac{2}{2-\gamma}}},$

yields

$\big{|}D^{k}\big{(}u_{i}/d^{\frac{\gamma}{2-\gamma}}\big{)}\big{|}\leq
Cd^{\beta-1-k}\quad\textrm{in}\quad\Omega\cap B_{r}.$

This means that

$u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{\beta-1}(\overline{\Omega}\cap
B_{r}),$

and we are done.

∎

We can finally give the:

###### Proof of Theorem 1.1.

Assume that $x_{0}=0$ and that the inner unit normal vector at $0$ is
$\nu=e_{n}$, and let $\Omega=\\{u>0\\}$. Since $\Delta
u_{i}=(\gamma-1)u^{\gamma-2}u_{i}$ in $\Omega$ for all $i=1,...,n$, then it
follows that

$w:=\frac{u_{i}}{u_{n}}$

solves

${\rm div}(u_{n}^{2}\nabla w)=0\quad\textrm{in}\quad\Omega\cap B_{1}.$

Thanks to Proposition, we have that

$u_{n}^{2}\asymp d^{s}\qquad\textrm{and}\qquad u_{n}^{2}/d^{s}\in
C^{\alpha}(B_{r_{\circ}}\cap\overline{\Omega})\qquad\textrm{for
any}\quad\alpha\in(0,1),$

where $r_{\circ}>0$ and

$s:={\frac{2\gamma}{2-\gamma}}>0.$

Hence, we can write the PDE for $w$ as

${\rm div}\Big{(}d^{s}a(x)\nabla w\Big{)}=0\quad\textrm{in}\quad\Omega\cap
B_{r_{\circ}},$ (5.27)

with $a(x)\in C^{\alpha}(B_{1/2}\cap\overline{\Omega})$ and $\lambda\leq
a(x)\leq\Lambda$ for some constants $\lambda,\Lambda>0$.

Let us see now what the boundary condition for $w$ on $\partial\Omega$ is.
Again by Propositions 2.3, we have that

$w=\frac{u_{i}}{u_{n}}\in
C^{\alpha}(B_{r_{\circ}}\cap\overline{\Omega})\quad\textrm{for
any}\quad\alpha\in(0,1),$

and combining this with interior estimates, a standard argument yields
$|\nabla w|\leq Cd^{\alpha-1}$ in $\Omega\cap B_{r_{\circ}}$. In particular,
taking $\alpha>1-s$, we deduce that

$\lim_{t\downarrow 0}\frac{w(z+t\nu)-w(z)}{t^{1-s}}=0\qquad\textrm{for
all}\quad z\in\partial\Omega\cap B_{r_{\circ}}$

or, equivalently,

$\lim_{t\downarrow 0}t^{s}\,\nabla w(z+t\nu)\cdot\nu=0\qquad\textrm{for
all}\quad z\in\partial\Omega\cap B_{r_{\circ}}.$ (5.28)

This means that $w$ is a weak solution of (5.27) with Neumann-type boundary
conditions (5.28), and therefore it follows from [32, Theorem 1.1] that

$w\in C^{1+\alpha}(B_{r}\cap\overline{\Omega})\quad\textrm{for any}\quad
r\in(0,r_{\circ}).$

But then, exactly as in (1.3), we deduce that $\nu\in C^{1,\alpha}$ and
$\partial\Omega\in C^{2,\alpha}$.

Now, since $\Omega$ is $C^{2,\alpha}$, we can apply Theorem 5.1 to find that

$u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{1,\alpha}(\overline{\Omega}\cap
B_{r}),$

and in particular

$a(x)=u_{n}^{2}/d^{s}\in C^{1,\alpha}(\overline{\Omega}\cap B_{r}).$

Hence, we can apply again [32, Theorem 1.1] to deduce that

$w\in C^{2+\alpha}(\overline{\Omega}\cap B_{r})\quad\textrm{for any}\quad
r\in(0,r_{\circ}).$

Iterating this procedure, we find that

$\partial\Omega\in C^{\infty}\qquad\textrm{and}\qquad
u_{i}/d^{\frac{\gamma}{2-\gamma}}\in C^{\infty}.$

This yields $u/d^{\frac{2}{2-\gamma}}\in C^{\infty}$ and thus
$u^{\frac{2-\gamma}{2}}=\Big{(}u/d^{\frac{2}{2-\gamma}}\Big{)}^{\frac{2-\gamma}{2}}d\in
C^{\infty}$ , and the theorem is proved. ∎

##  6\. Polynomial lemmas

In this section we gather technical lemmas that provide control over the
polynomials used in the least squares approximation in our various blow-up
arguments. All of these results can be seen as a generalization to those in
[1] regarding the control of their least square polynomials, in the sense that
we deal with more than one element in the kernel of the limiting linearized
operator.

######  Lemma 6.1.

Let $\kappa\geq-\frac{1}{4}$ and $\theta_{\pm}$ as in (1.14). Let
$k_{1},k_{2}\in\mathbb{N}$, $\beta>1$, and let $\Omega$ be such that
$\|\partial\Omega\cap B_{2}\|_{C^{\beta}}\leq 1$. There exists $\rho>0$ such
that for any $r\in(0,\rho)$ and for any $Q_{i}\in\mathbf{P}_{k_{i}}$ for
$i=1,2$, if

$\|Q_{1}d^{\theta_{+}}+Q_{2}d^{\theta_{-}+1}\|_{L^{\infty}(\partial\Omega\cap
B_{r})}\leq\theta,$ (6.1)

where $\theta>0$ and with $Q_{2}=0$ if either $\theta_{-}\leq 0$ or if
$\kappa=-\frac{1}{4}$, then

$\|Q_{1}\|_{L^{\infty}(\partial\Omega\cap
B_{r})}r^{\theta_{+}}+\|Q_{2}\|_{L^{\infty}(\partial\Omega\cap
B_{r})}r^{\theta_{-}+1}\leq C\theta,$ (6.2)

and

$\sum_{|\mu|\leq
k_{1}}r^{|\mu|+\theta_{+}}\big{|}q^{(\mu)}_{1}\big{|}+\sum_{|\mu|\leq
k_{2}}r^{|\mu|+\theta_{-}+1}\big{|}q^{(\mu)}_{2}\big{|}\leq C\theta,$ (6.3)

with $C=C(\rho,n,k_{1},k_{2},\kappa)$.

###### Proof.

Let us notice first that by the equivalence of norms in finite dimensional
spaces, there exists a constant $C=C(k_{1},k_{2},\kappa)$ such that

$\displaystyle\sum_{|\mu|\leq
k_{1}}\big{|}p^{(\mu)}_{1}\big{|}+\sum_{|\mu|\leq
k_{2}}\big{|}p^{(\mu)}_{2}\big{|}$ $\displaystyle\leq$ $\displaystyle
C(\|P_{1}(x)\|_{L^{\infty}(\mathbb{R}^{n}_{+}\cap
B_{1})}+\|P_{2}(x)\|_{L^{\infty}(\mathbb{R}^{n}_{+}\cap B_{1})})$ (6.4)
$\displaystyle\leq$ $\displaystyle
C\Big{(}\|P_{1}(x)x_{n}^{\theta_{+}}\|_{L^{\infty}(\mathbb{R}^{n}_{+}\cap
B_{1})}+\|P_{2}(x)x_{n}^{\theta_{-}+1}\|_{L^{\infty}(\mathbb{R}^{n}_{+}\cap
B_{1})}\Big{)}$ $\displaystyle\leq$ $\displaystyle
C\|P_{1}(x)x_{n}^{\theta_{+}}+P_{2}(x)x_{n}^{\theta_{-}1+1}\|_{L^{\infty}(\mathbb{R}^{n}_{+}\cap
B_{1})},$

where the last inequality uses fundamentally the linear independence of
$x_{n}^{\theta_{+}}$ and $x_{n}^{\theta_{-}+1}$, i.e.,
$\theta_{+}-\theta_{-}\notin\mathbb{N}$.

Let $F_{r}:=\frac{1}{r}\Omega$. By compactness, we can deduce from (6.4) and
from the $C^{\beta}$ bound in $\partial\Omega\cap B_{2}$, we can find $\rho>0$
such that for $r\in(0,\rho)$ and for any $P_{i}\in\mathbf{P}_{k_{i}}$,

$\displaystyle\sum_{|\mu|\leq
k_{1}}\big{|}p^{(\mu)}_{1}\big{|}+\sum_{|\mu|\leq
k_{2}}\big{|}p^{(\mu)}_{2}\big{|}$ $\displaystyle\leq$
$\displaystyle\Big{\|}P_{1}(x)\Big{\|}_{L^{\infty}(F_{r}\cap
B_{1})}+\|P_{2}(x)\|_{L^{\infty}(F_{r}\cap B_{1})}$ (6.5) $\displaystyle\leq$
$\displaystyle
C\Big{\|}P_{1}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{+}}+P_{2}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{-}+1}\Big{\|}_{L^{\infty}(F_{r}\cap
B_{1})}.$

Let $Q_{i}(x)\in\mathbf{P}_{k_{i}}$ for $i=1,2$ and let $r\in(0,r_{0})$. Let
us consider the polynomials

$P_{1}(x)=\sum_{|\mu|\leq
k_{1}}r^{|\mu|+\theta_{+}}q_{1}^{(\mu)}x^{\alpha},\qquad
P_{2}(x)=\sum_{|\mu|\leq k_{2}}r^{|\mu|+\theta_{-}+1}q_{2}^{(\mu)}x^{\alpha}.$

With this choosing of $P_{1}$ and $P_{2}$ in (6.5), we deduce, after a change
of variables, that

$\displaystyle\sum_{|\mu|\leq
k_{1}}r^{|\mu|+\theta_{+}}\big{|}q^{(\mu)}_{1}\big{|}+\sum_{|\mu|\leq
k_{2}}r^{|\mu|+\theta_{-}+1}\big{|}q^{(\mu)}_{2}\big{|}$ $\displaystyle\leq$
$\displaystyle r^{\theta_{+}}\Big{\|}Q_{1}\Big{\|}_{L^{\infty}(\Omega\cap
B_{r})}+r^{\theta_{-}+1}\|Q_{2}(x)\|_{L^{\infty}(\Omega\cap B_{r})}$
$\displaystyle\leq$ $\displaystyle
C\Big{\|}Q_{1}d^{\theta_{+}}+Q_{2}d^{{\theta_{-}+1}}\Big{\|}_{L^{\infty}(\Omega\cap
B_{r})},$

which yields (6.2) and (6.3). ∎

######  Lemma 6.2.

Let $\kappa\geq-\frac{1}{4}$ and $\theta_{\pm}$ as in (1.14). Let $\beta>1$
with $\beta\notin\mathbb{N}$, and let $\Omega$ be such that
$\|\partial\Omega\cap B_{2}\|_{C^{\beta}}\leq 1$. Let $v\in L^{\infty}(B_{2})$
and let us assume that for some $\rho>0$ and each $r\in(0,\rho)$ there exists
$Q_{1,r}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$Q_{2,r}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor}-2$ such that

$\|v-Q_{1,r}d^{\theta_{+}}-Q_{2,r}d^{\theta_{-}}\|_{L^{\infty}(B_{r})}\leq
c_{0}r^{\theta_{+}+\beta-1},$ (6.6)

where $d$ is extended by $0$ outside of $\Omega\cap B_{2}$, and with
$Q_{2,r}=0$ if $\theta_{-}\leq 0$ or if $\kappa=-\frac{1}{4}$.

Then there exists $\tilde{Q_{1}}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$\tilde{Q_{2}}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor}-2$
such that

$\|v-\tilde{Q_{1}}d^{\theta_{+}}-\tilde{Q_{2}}d^{{\theta_{-}+1}}\|_{L^{\infty}(B_{r})}\leq
Cc_{0}r^{\beta+\theta_{+}-1},$ (6.7)

for $r\in(0,\rho)$.

###### Proof.

Let us consider (6.7) for the values $r$ and $2r$. Subtracting them and then
applying the triangular inequality yields

$\Big{\|}(Q_{1,2r}-Q_{1,r})d^{\theta_{+}}+(Q_{2,2r}-Q_{2,r})d^{\theta_{-}+1}\Big{\|}_{{L^{\infty}(B_{r})}}\leq
Cc_{0}r^{\theta_{+}+\beta-1}.$

Therefore, Lemma 6.1 implies

$|q^{(\mu_{1})}_{i,r}-q^{(\mu_{1})}_{i,2r}|\leq Cc_{0}r^{\beta-1-|\mu_{1}|},$
(6.8)

with $|\mu_{1}|\leq\lfloor\beta\rfloor-1$; whereas

$|q^{(\mu_{2})}_{2,r}-q^{(\mu_{2})}_{i,2r}|\leq
Cc_{0}r^{\beta-2+\theta_{+}-\theta_{-}-|\mu_{2}|},$ (6.9)

for $|\mu_{2}|\leq\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2$. Let us remark
that the latter inequality implies the uniform boundedness in $r$ for the
differences $|q^{(\mu_{i})}_{i,r}-q^{(\mu_{i})}_{i,2r}|$, for $i=1,2$.

On the other hand, applying the triangular inequality in (6.6) for $r=1$ and
using again Lemma 6.1, we obtain the bound

$\|Q_{1,1}\|_{L^{\infty}(B_{1})}+\|Q_{2,1}\|_{L^{\infty}(B_{1})}\leq
Cc_{0}+\|v\|_{L^{\infty}(B_{\rho})}.$ (6.10)

The latter bound combined with (6.8) and (6.9) implies the convergence of the
sequences $\\{q^{(\mu)}_{i,2^{-j}}\\}_{j\in\mathbb{N}}$ for $i=1,2$ and for
each $\mu$. Denoting the corresponding limits by $q^{(\mu)}_{i}$, we deduce,
namely for $i=2$, adding up dyadically and applying the bound that for any
$r\in(0,\rho)$

$\displaystyle|q^{(\mu)}_{2}-q^{(\mu)}_{2,r}|$ $\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\infty}|q^{(\mu)}_{2,2^{-j}r}-q^{(\mu)}_{2,2^{-j-1}r}|\leq
Cc_{0}\sum_{j=0}^{\infty}(r2^{-j})^{\beta-2+\theta_{+}-\theta_{-}-|\mu|}$
$\displaystyle\leq$ $\displaystyle
Cc_{0}r^{\beta-2+\theta_{+}-\theta_{-}-|\mu|}\sum_{j=0}^{\infty}(2^{-j})^{\beta+\theta_{+}-\theta_{-}-2-|\mu|}\leq
Cc_{0}r^{\beta+\theta_{+}-\theta_{-}-2-|\mu|}.$

The same estimate holds, mutatis mutandis, for $i=1$.

Let us define
$\tilde{Q_{1}}=\sum_{|\mu|\leq\lfloor\beta\rfloor-1}q_{1}^{\mu}x^{\mu}$ and
$\tilde{Q_{2}}=\sum_{|\mu|\leq\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}q_{2}^{\mu}x^{\mu}$.
Applying the triangular inequality together with the previous bound (and its
analogue for $\tilde{Q_{1}}$) yields

$\displaystyle\|v-\tilde{Q_{1}}d^{\theta_{+}}-\tilde{Q_{2}}d^{{\theta_{-}+1}}\|_{L^{\infty}(B_{r})}$
$\displaystyle\leq$
$\displaystyle\|v-\tilde{Q_{1,r}}d^{\gamma_{1}}-\tilde{Q_{2,r}}d^{{\gamma_{2}}}\|_{L^{\infty}(B_{r})}$
$\displaystyle+\|(Q_{1,r}-\tilde{Q_{1}})d^{\theta_{+}}\|_{L^{\infty}(B_{r})}+\|(Q_{2,r}-\tilde{Q_{2}})d^{\theta_{-}+1}\|_{L^{\infty}(B_{r})}$
$\displaystyle\leq$ $\displaystyle Cc_{0}r^{\beta+\gamma_{1}},$

which proves the result. ∎

######  Lemma 6.3.

Let $\kappa\geq-\frac{1}{4}$ and $\theta_{\pm}$ as in (1.14). Let $\beta>1$
with $\beta\notin\mathbb{N}$, and let $\Omega$ be such that
$\|\partial\Omega\cap B_{2}\|_{C^{\beta}}\leq 1$. Let $v\in L^{\infty}(B_{2})$
and let us assume that for some $\rho>0$ and each $r\in(0,\rho)$ there exists
$Q_{1,r}\in\mathbf{P}_{\lfloor\beta\rfloor-1}$ and
$Q_{2,r}\in\mathbf{P}_{\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2}$ such that

$\|v-Q_{1,r}d^{\theta_{+}}-Q_{2,r}d^{\theta_{-}+1}\|_{L^{\infty}(B_{s})}\leq\theta(s)r^{\beta-1+\theta_{+}},$
(6.11)

where $\theta:(0,\rho)\to(0,\infty)$ is a non-increasing function, with

$\lim_{r\to 0^{+}}\theta(r)=+\infty,$ (6.12)

and with $Q_{2,r}=0$ if $\theta_{-}\leq 0$ or if $\kappa=-\frac{1}{4}$.

Then

  1. (1)

$\lim_{r\to 0^{+}}\frac{q_{i,r}^{(\mu)}}{\theta(r)}=0,$ (6.13)

for $i=1,2$.

  2. (2)

For $r\in(0,\rho)$ the functions

$w_{r}(x):=\frac{v(rx)-Q_{1,r}(rx)d^{\theta_{+}}(rx)-Q_{2,r}(rx)d^{\theta_{-}+1}(rx)}{r^{{\theta_{+}+\beta-1}}\theta(r)},$
(6.14)

satisfy the uniform bound

$\|w_{r}\|_{L^{\infty}(B_{R})}\leq CR^{\beta-1+\theta_{+}},$ (6.15)

for $R>1$.

  3. (3)

Lastly, if $\beta>2$,

$\displaystyle\frac{1}{\theta(r)r^{{\theta_{+}}+\beta-1}}\Delta\Big{(}Q_{1,r}(rx)d^{\theta_{+}}(rx)+Q_{2,r}(rx)d^{\theta_{-}+1}(rx)\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\theta(r)r^{{\theta_{+}}+\beta-1}}\frac{\kappa
r^{2}}{d^{2}(rx)}Q_{1,r}(rx)d^{\theta_{+}}(rx)$
$\displaystyle+\frac{1}{\theta(r)r^{{\theta_{+}}+\beta-1}}\frac{\kappa
r^{2}}{d^{2}(rx)}Q_{2,r}(rx)d^{\theta_{-}+1}(rx)$
$\displaystyle+P_{1,r}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{+}-1}$
$\displaystyle+P_{2,r}(rx)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{-}-1}$
$\displaystyle+m_{*}(r,x)$ (6.16)

where $P_{i,r}\in\mathbf{P}_{M}$ for $i=1,2$ and $|m_{*}(x,r)|\leq Cm(r)$ such
that $m(r)\to 0$ as $r\to 0^{+}$. Here $m$ and $M$ only depends on $\beta$,
$\kappa$, and $n$.

###### Proof.

By subtracting (6.11) at the values $r$ and $2r$ and then applying the
triangular inequality, we obtain

$\Big{\|}(Q_{1,2r}-Q_{1,r})d^{\theta_{+}}+(Q_{2,2r}-Q_{2,r})d^{\theta_{-}+1}\Big{\|}_{{L^{\infty}(B_{r})}}\leq
C\theta(r)r^{\beta+{\gamma_{1}}}$

Therefore, Lemma 6.1 implies

$\|Q_{1,2r}-Q_{1,r}\|_{L^{\infty}(B_{2r})}r^{\theta_{+}}+\|Q_{2,2r}-Q_{2,r}\|_{L^{\infty}(B_{2r})}r^{\theta_{-}+1}\leq
C\theta(r)r^{\beta-1+\theta_{+}}$ (6.17)

Let $i\in\\{1,2\\}$. In virtue of (6.17), we deduce

$|q^{(\mu)}_{1,r}-q^{(\mu)}_{1,2r}|\leq C\theta(r)r^{\beta-|\mu|-1},$

and

$|q^{(\mu)}_{2,r}-q^{(\mu)}_{2,2r}|\leq
C\theta(r)r^{\beta-2+\theta_{+}-\theta_{-}-|\mu|}.$

By using the triangular inequality, it follows that for $i=1$,

$|q^{(\mu)}_{1,r}-q^{(\mu)}_{1,2^{J}r}|\leq\sum_{l=0}^{J-1}|q^{(\mu)}_{1,2^{l}r}-q^{(\mu)}_{1,2^{l+1}r}|\leq
C\sum_{l=0}^{J-1}\theta(r2^{l})(r2^{l})^{\beta-|\mu|-1},$ (6.18)

where $|\mu|\leq\lfloor\beta\rfloor-1$. Reasoning similarly, we obtain

$|q^{(\mu)}_{2,r}-q^{(\mu)}_{2,2^{J}r}|\leq
C\sum_{l=0}^{J-1}\theta(r2^{l})(r2^{l})^{\beta-2+\theta_{+}-\theta_{-}-|\mu|},$
(6.19)

where $|\mu|\leq\lfloor\beta+\theta_{+}-\theta_{-}\rfloor-2$.

Let us notice also that $q^{(\mu)}_{i,s}$ is bounded for $s\in[\rho/4,\rho/2]$
and for $i=1,2$. Indeed, since $v\in L^{\infty}(B_{\rho})$ and $\theta$ is
bounded on $[s/4,s/2]$, we deduce from (6.11) and Lemma 6.1 that for
$s\in[\rho/4,\rho/2]$

$\|Q_{1,s}\|_{L^{\infty}(B_{s})}s^{\theta_{+}}+\|Q_{2,s}\|_{L^{\infty}(B_{s})}s^{\theta_{-}+1}\leq
C_{\rho}s^{\beta-1+\theta_{+}}+\|v\|_{L^{\infty}(B_{\rho})}.$

From here, the boundedness claim readily follows.

On the other hand, if we take $J$ such that
$2^{J}r\in[\frac{\rho}{4},\rho/2)$, we can use the montonicity of $\theta$
combined with (6.18) to obtain the estimate

$\displaystyle\frac{|q^{(\mu)}_{1,r}-q^{(\mu)}_{1,2^{J}r}|}{\theta(r)}\leq
C\sum_{l=0}^{J-1}\frac{\theta(\rho
2^{l-J-1})}{\theta(r)}(2^{l-J})^{\beta-|\mu|-1}\leq
C\sum_{l=1}^{J}\frac{\theta(\rho
2^{-i-1})}{\theta(r)}(2^{-i})^{\beta-|\mu|-1}.$

So, since, $|\mu|<{\beta-1}$ and (6.13) holds, we deduce from the dominated
convergence theorem that the right hand side of the latter inequality goes to
zero as $r\to 0^{+}$. By applying an analogous reasoning to (6.19), we obtain
(6.13).

Focusing on (6.15), let us notice that the monotonicity of $\theta$ combined
with (6.18) and (6.19) yields

$|q^{(\mu)}_{1,r}-q^{(\mu)}_{1,2^{J}r}|\leq
C\theta(r)r^{\beta-|\mu|-1}\sum_{l=0}^{J-1}(2^{l})^{\beta-|\mu|-1}\leq
C\theta(r)(2^{J}r)^{\beta-|\mu|-1},$ (6.20)

and

$|q^{(\mu)}_{2,r}-q^{(\mu)}_{2,2^{J}r}|\leq
C\theta(r)(2^{J}r)^{\beta-2+\theta_{+}-\theta_{-}-|\mu|},$ (6.21)

respectively.

It follows that for any $R>1$,

$\|(Q_{1,r}-Q_{1,Rr})d^{\theta_{+}}\|_{L^{\infty}(B_{Rr})}+\|(Q_{2,r}-Q_{2,Rr})d^{\theta_{-}+1}\|_{L^{\infty}(B_{Rr})}\\\
\leq C\theta(r)(Rr)^{\beta-1+{\theta_{+}}}.$ (6.22)

Hence, using the triangular inequality and combining it with (6.11) and (6.22)

$\displaystyle r^{{\theta_{+}}+\beta-1}\theta(r)\|v\|_{L^{\infty}(B_{rR})}$
$\displaystyle=$
$\displaystyle\|v-Q_{1,r}d^{\theta_{+}}-Q_{1,2,r}d^{\theta_{-}+1}\|_{L^{\infty}(B_{Rr})}$
$\displaystyle\leq$
$\displaystyle\|v-Q_{1,rR}d^{\theta_{+}}-Q_{2,rR}d^{\theta_{-}+1}\|_{L^{\infty}(B_{Rr})}$
$\displaystyle+\|(Q_{1,r}-Q_{1,Rr})d^{\theta_{+}}\|_{L^{\infty}(B_{Rr})}$
$\displaystyle+\|(Q_{2,r}-Q_{2,Rr})d^{\theta_{-}+1}\|_{L^{\infty}(B_{Rr})}$
$\displaystyle\leq$ $\displaystyle
C\theta(Rr)(Rr)^{\beta-1+{\theta_{+}}}+C\theta(r)(Rr)^{\beta-1+{\theta_{+}}}.$

Therefore, by the monotonicity of $\theta$, (6.15) follows.

We conclude the proof of the Lemma showing (6.16). Let
$f(x)=Q_{1,r}(x)d^{\theta_{+}}(x)+Q_{2,r}(x)d^{\theta_{-}+1}(x)$, computing
directly we have that

$\displaystyle\Delta f(x)$ $\displaystyle=$ $\displaystyle
Q_{1,r}(x)\Delta(d^{\theta_{+}}(x))+Q_{2,r}(x)\Delta(d^{\theta_{-}+1}(x))$
(6.23) $\displaystyle+2\theta_{+}d^{\theta_{+}-1}\nabla Q_{1,r}(x)\cdot\nabla
d(x)+2(\theta_{-}+1)d^{\theta_{-}}\nabla Q_{2,r}(x)\cdot\nabla d(x)$
$\displaystyle+d^{\theta_{+}}(x)\Delta Q_{1,r}(x)+d^{\theta_{-}+1}(x)\Delta
Q_{2,r}(x)$

Since $\|\Omega\cap B_{2}\|_{C^{\beta}}\leq 1$, Lemma 2.1 provides the
expansion

$d(x)=P(x)+g(x)=\sum_{|\lambda|\leq\lfloor\beta\rfloor}p^{(\lambda)}x^{\lambda}+g(x),$
(6.24)

where $P\in\mathbf{P}_{\lfloor\beta\rfloor}$ with
$\|P\|_{L^{\infty}(\Omega\cap B_{1})}\leq C_{\beta}$, and where $g\in
C^{\infty}(\Omega\cap B_{2})$ satisfies $|\nabla g|\leq
C_{\beta}|x|^{\beta-1}$. Therefore,

$\displaystyle\frac{r^{2}d^{\theta_{+}-1}}{\theta(r)r^{\theta_{+}+\beta-1}}\nabla
Q_{1,r}(rx)\cdot\nabla d(rx)$ $\displaystyle=$
$\displaystyle\frac{r^{2}d^{\theta_{+}-1}}{\theta(r)r^{\theta_{+}+\beta-1}}\left(\nabla
Q_{1,r}(rx)\cdot\nabla P(rx)+\nabla Q_{1,r}(x)\cdot\nabla g(rx)\right)$ (6.25)
$\displaystyle+R_{1,r}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{+}-1}+m_{1}(x,r)$

where $R_{1,r}\in\mathbf{P}_{2\lfloor\beta\rfloor-3}$ and $|m_{1}(x,r)|\leq
Cm(r)$ with $m$ only depending on $\beta$ and $\kappa$ and such that $m(r)\to
0$ as $r\to 0^{+}$. Analogously,

$\displaystyle\frac{r^{2}d^{\theta_{-}}}{\theta(r)r^{\theta_{+}+\beta-1}}\nabla
Q_{2,r}(rx)\cdot\nabla d(rx)$ $\displaystyle=$
$\displaystyle\frac{r^{2}d^{\theta_{-}}}{\theta(r)r^{\theta_{+}+\beta-1}}\left(\nabla
Q_{2,r}(rx)\cdot\nabla P(rx)+\nabla Q_{2,r}(x)\cdot\nabla g(rx)\right)$ (6.26)
$\displaystyle+R_{1,r}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{-}}+m_{2}(x,r)$

where $R_{2,r}\in\mathbf{P}_{\lfloor 2\beta+\theta_{+}-\theta_{-}\rfloor-4}$
and $|m_{2}(x,r)|\leq Cm(r)$.

In a similar vein, using (6.24), we deduce that

$\displaystyle\frac{r^{2}}{\theta(r)r^{\theta_{+}+\beta-1}}\Big{(}d^{\theta_{+}}(rx)\Delta
Q_{1,r}(rx)+d^{\theta_{-}+1}(rx)\Delta Q_{2,r}(rx)\Big{)}=$ $\displaystyle
m_{3}(x,r)+R_{3,r}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{+}-1}+R_{4,r}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{-}},$
(6.27)

with $R_{i,r}\in\mathbf{P}_{M}$ for $i=3,4$ for some $M\in\mathbb{N}$ only
depending on $\beta$ and $\kappa$ and where $|m_{3}(x,r)|\leq Cm(r)$.

On the other hand, thanks to Lemma 2.1, we also have that

$\Delta
d^{\theta_{+}}(x)=\frac{\kappa}{d^{2}(x)}d^{\theta_{+}}+d^{\theta_{+}-1}\Big{(}P_{0}(x)+g_{2}(x)\Big{)},$
(6.28)

with $P_{0}\in\mathbf{P}_{M}$ for some $M$ depending on $\beta$ and
$|g_{2}(x)|\leq C|x|^{\beta-1}$. Combining (6.23), (6.25), (6.26), (6.27),
(6.28), and (2.4) applied to $d^{\theta_{-}+1}$, we deduce

$\displaystyle\frac{r^{2}}{\theta(r)r^{\theta_{+}+\beta-1}}\Big{(}\Delta
f(rx)-\frac{\kappa}{d(rx)^{2}}f(rx)\Big{)}=P_{1,r}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{+}-1}+P_{2,r}(x)\Big{(}\frac{d(rx)}{r}\Big{)}^{\theta_{-}}+m_{*}(r,x)$

with $P_{i,r}\in\mathbf{P}_{M}$ for $i=1,2$ for some $M\in\mathbb{N}$ only
depending on $\beta$ and $\kappa$ and where $|m_{*}(x,r)|\leq Cm(r)$. This
proves (6.16). ∎

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