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\def\rightheadline{EJDE--1999/23\hfil Dirichlet problem
\hfil\folio}
\def\leftheadline{\folio\hfil Michail Borsuk \& Dmitriy Portnyagin
 \hfil EJDE--1999/23}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999), No.~23, pp.~1--25.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp ejde.math.swt.edu (login: ftp)\bigskip} }

\topmatter
\title On the Dirichlet problem for quasilinear elliptic second order
equations with triple degeneracy and singularity in a domain with 
a boundary  conical point
\endtitle

\thanks 
{\it 1991 Mathematics Subject Classifications:} 35B45, 35B65,
35D10, 35J25, 35J60, 35J65, 35J70.\hfil\break\indent
{\it Key words and phrases:} quasilinear elliptic degenerate equations, 
      barrier functions, \hfil\break\indent conical points.
\hfil\break\indent
\copyright 1999 Southwest Texas State University  and
University of North Texas.\hfil\break\indent
Submitted April 23, 1999. Published June 24, 1999.
\endthanks

\author Michail Borsuk \& Dmitriy Portnyagin \endauthor

\address Michail Borsuk \hfil\break
Department of Applied Mathematics \hfil\break
Olsztyn University of Agriculture and  Technology \hfil\break 
10-957 Olsztyn-Kortowo, Poland 
\endaddress
\email borsuk\@art.olsztyn.pl
\endemail

\address Dmitriy Portnyagin \hfil\break
Department of Physics \hfil\break
Lvov State University \hfil\break
290602 Lvov, Ukraine 
\endaddress
\email mitport\@hotmail.com
\endemail

\abstract
   In this article we prove boundedness and H\"older
continuity of weak solutions to the Dirichlet problem for a second
order quasilinear elliptic equation with triple degeneracy and
singularity. In particular, we study equations of the form
 $$\displaylines{
 -\frac{d}{dx_i} (|x|^\tau |u|^q |\nabla u|^{m-2} u_{x_i})+
\frac{a_0|x|^\tau }{(x_{n-1}^2+x_n^2)^{m/2}} u|u|^{q+m-2}
   -\mu |x|^\tau u |u| ^{q-2} |\nabla u|^m =\cr
  \hfill =f_0(x)-\frac{\partial f_i}{\partial x_i}, } 
$$
with $a_0\ge0$,  $q\ge0$, $0\le \mu <1$, $1<m\le n$,  and $\tau >m-n$
in a domain with a boundary conical point.  We obtain the exact
H\"older exponent of the solution near the conical point.
\endabstract
\endtopmatter

\document

\def\meas{\operatorname{meas}}


\head 0. Introduction \endhead

Lately many mathematicians have considered nonlinear problems with 
elliptic degenerate equations (see e.g. [6] and its extensive bibliography). 
In the present paper, we continue the investigation on the behaviour of 
solutions of the first boundary value problem for a quasilinear elliptic
second order equation with triple degeneracy (see [18]). Namely, 
we derive an exact H\"older exponent for weak solutions,
 in a neighbourhood of a conical boundary point, for the Dirichlet problem
$$\gather
    -\frac{d}{dx_i}(|x|^\tau {|u|}^q{|\nabla u|}^{m-2}u_{x_i})+
\frac{a_0|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}u|u|^{q+m-2}
   -\mu |x|^\tau u{|u|}^{q-2}{|\nabla u|}^m =\\
 \hskip 7cm =f_0(x)-\frac{\partial f_i}{\partial x_i}, \quad  x\in G,  \tag 0.1 \cr
 u(x)=0,\quad x\in\partial G; \tag 0.2\\
a_0\ge0,\quad q\ge0,\quad  \mu \ge0,\quad 1<m\le n,\quad \tau >m-n;   
   \tag 0.3  \endgather 
$$
(summation over repeated indices from $1$ to $n$ is understood), 
where $G$ is a $n$-dimensional {\it bounded convex} circular cone with the 
vertex at the origin of coordinates $O$. We shall construct functions playing 
a fundamental role in the study of the behaviour of solutions to elliptic 
boundary-value problems in the neighbourhood of irregular boundary points
(see [5, 7, 13-15]). 
The special structure of the solution near a conical point is of particular 
interest for physical applications ([2, 8, 11]). It can be used also to improve
numerical algorithms ([1, 4, 12]). However, the behaviour of solutions near 
conical  boundary points is treated only in special cases (see [5, 13-17]).

Let $\Omega =G\cap S^{n-1}$ be a domain
on the unit sphere with smooth boundary $\partial \Omega$. For $x\in {\Bbb R}^n $ we denote 
the spherical coordinates by  
$(r,\omega)=(r, \omega_1, \dots, \omega_{n-1})$ 
with $r=|x|$, $\omega \in \Omega$. We also set $G_0^d=G\cap \{|x|<d\}$ and 
$\Omega_d=G\cap \{|x|=d\}$ for all $d>0$.

Let $L_p(G)$ and $W^{k,p}(G), p>1$ be the usual Lebesgue's and
Sobolev's spaces. $W^{k,p}_0(G)$ denotes the space of functions
in $W^{k,p}(G)$ that vanish on $\partial G$ in the sense of traces.
For $k$ a non-negative integer and $\tau$ a real number, 
 we define the space $V^k_{p,\tau}(G)$ 
as the closure of $C_0^\infty (\overline {G}\setminus \{O\})$
with respect to the norm
$$
\| u \| _{V^k_{p,\tau}(G)}=\big(\int_{G}\sum_{|\beta|=0}^{k}
r^{p(|\beta|-k)+\tau}|D^\beta u|^p dx\big)^{1/p}.$$
We will denote by ${\goth N}^1_{m,\tau,q}(G)$ the set of functions
$u(x)\in V^1_{m,\tau}(G)\cap L_{\infty}(G)$ such that
$$
\int_{G} \frac{|x|^\tau |u|^{q+m}} {(x_{n-1}^2+x_n^2)^{m/2}}dx<\infty,
\quad q\ge 0,\; \tau >m-n, \; 1<m\le n
$$
(i.e. the integral is finite).

Through this paper we assume that $f_0(x), f_1(x), \dots ,f_n(x)$ are measurable 
functions such that
$$f_0(x)\in L_p(G),\quad |x|^{-\tau/m}f_i(x)\in L_{\frac{mp}{m-1}}(G),
\quad (i=1,\dots ,n),\tag 0.4$$
where 
$$\frac{1}{p}<\frac{m}{n}-\frac{1}{t},\quad \frac{1}{t}<\frac{m}{n}<1+\frac{1}{t}<m,\quad 
m-n<\tau <\min (m-1;\frac{n}{t}). \tag 0.5 $$
We also set $|f|=(\sum_{i=1}^{n}f^2_i)^{1/2}$ and $(|u|-k)_+=\max {(|u|-k;0)}$.     

\head 1. Boundedness of weak solutions \endhead

 \definition{Definition}
A function $u(x)$ is called a  weak solution of (0.1)-(0.2), if 
$u(x)$ belongs to ${\goth N}^1_{m,\tau,q}(G)$ and 
for all $\phi(x)\in {\goth N}^1_{m,\tau,q}(G)$, it satisfies
 $$\multline \int_{G}\{|x|^\tau |u|^q|\nabla u|^{m-2}u_{x_i}\phi_{x_i}+
\frac{a_0|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}u|u|^{q+m-2}\phi-
\mu |x|^\tau u|u|^{q-2}|\nabla u|^m\phi\}dx \\
                   =  \int_{G}\{f_0(x)\phi+f_i\phi_{x_i}\}dx \,.\endmultline
$$
 \enddefinition
Our goal in this section is to obtain an $L_{\infty}(G)$-a priori estimate 
of weak solutions of (0.1)-(0.2). For this end, we need the following statements
([3, Lemma 2.1]).

\proclaim{Lemma 1.1} Let $\varkappa>0$, and 
$$ \eta(x)= \cases
          e^{\varkappa x}-1 &x\ge 0 \\
         -e^{-\varkappa x}+1&x<0\,.
            \endcases $$
Let $a, b$ be positive constants, and $m>1$. If $\varkappa>(2b/a)
 +m$, then 
  $$\gather
            a\eta'(x)-b|\eta(x)|\ge\frac{a}{2}\text{e }^{\varkappa x}
                                            \quad\forall x\ge0 \,,\tag1.1\\
            \eta(x)\ge[\eta(\frac{x}{m})]^m \quad\forall x\ge0\,.\tag1.2
     \endgather$$
Moreover, there exist $ d\ge 0$ and $M>0$ such that
  $$\gather
    \eta(x)\le M[\eta(\frac{x}{m})]^m, \quad \eta'(x)\le
 M[\eta(\frac{x}{m})]^m \quad\forall x\ge d\,,\tag1.3\\
      |\eta(x)|\ge x,\quad\forall x\in{\Bbb R}.\tag1.4
     \endgather
$$
\endproclaim
The following statement is due to Stampacchia ([9, Lemma B.1]).

\proclaim{Lemma 1.2}  Let $\alpha$, $\beta$, $\gamma$, $k_0$ be real positive numbers,
$\gamma>1$,  and $\psi:{\Bbb R}_+\rightarrow{\Bbb R}_+$ be a decreasing function such that
   $$\psi(l)\le\frac{\alpha}{(l-k)^{\beta}}[\psi(k)]^{\gamma},\quad
\forall l>k\ge k_0.$$
Then $\psi(k_0+\delta)=0$, where
 $\delta^{\beta}=\alpha\psi(k_0)^{\gamma-1}2^{\beta(\frac{\gamma}{\gamma-1})}$.
\endproclaim

We will also use the following properties.
\proclaim{Lemma 1.3} ([6, Examples 1.5, 1.6, p. 29)
Let $m^{\#}$ denote the number associated to $m$ by
     $$\frac{1}{m^\#}=\frac{1}{m}(1+\frac{1}{t})-\frac{1}{n}\tag 1.5$$
and assume that (0.5) hold. Then there exist constants 
$c_1>0$, $c_2>0$ (depending only on $\meas G, n, m, t, \tau $) such that
$$\gather
\int_{G}|x|^{\tau -m}|u|^mdx\leq c_1 \int_{G}|x|^\tau |\nabla u|^mdx, \tag1.6 \\
(\int_{G}|u|^{m^{\#}}dx)^{m/m^\#} \leq c_2 \int_{G}(|x|^{\tau -m}|u|^m+
|x|^\tau |\nabla u|^m)dx \tag1.7
\endgather$$
for any $u(x)\in V^1_{m,\tau}(G)$.
\endproclaim

Our main statement in this section is as follows.

\proclaim{ Theorem 1.4} Let $u(x)$ be a weak solution of (0.1)-(0.5).
 Then there exists a constant $M_0>0$ 
depending only on  $\meas G$, $n$, $m$, $\tau$, $\mu$, $q$,
$a_0$, $\|f_0(x)\|_{L_{p}(G)}$, and  
$\||x|^{-\tau/m}|f(x)|\|_{L_{\frac{mp}{m-1}}(G)}$ such that 
$||u||_{L^{\infty}(G)}\le M_0$.
\endproclaim

\demo{Proof}
We shall follow the proof of in [3, Theorem 3.1]. Let 
$A(k)=\{ x\in G: |u(x)|>k\}$ and  $\chi_{A(k)}$ be the characteristic function
for the set $A(k)$. We remark that $A(k+d)\subseteq A(k)$ for all $d>0$.
By setting $\phi (x)=\eta((|u|-k)_+){\chi_{A(k)}}
\operatorname{sgn}u$ in the definition of weak solution, with $\eta$ defined 
by Lemma 1 and $k\ge k_0$
(without less of generality we can assume $k_0>1$), we get the inequality
   $$\multline
     \int_{A(k)} |x|^\tau |u|^q|\nabla u|^m \eta'((|u|-k)_+)+
     a_0\int_{A(k)} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}|u|^{q+m-1}\eta((|u|-k)_+)\le\\
 \le\int_{A(k)} |f_0||\eta((|u|-k)_+)|+\mu \int_{A(k)} |x|^\tau |\nabla u|^m|u|^{q-1}|\eta((|u|-k)_+)|+\\
     +\int_{A(k)} |f||\nabla u||\eta'((|u|-k)_+)|.
           \endmultline\tag1.8$$
By Young's inequality
   $$\multline |f||\nabla u|
                =(|x|^{\tau /m}|u|^{q/m}|\nabla u|)
               (|x|^{-\tau /m}|u|^{-q/m}
            |f|)\le\\
       \le \frac{1}{m}|x|^\tau |u|^{q}|\nabla u|^m+
        \frac{m-1}{m}
|x|^{\frac{-\tau }{m-1}} |u|^{\frac{-q}{m-1}}|f|^{\frac{m}{m-1}}.
                 \endmultline$$
Substituting this expression in  $(1.8)$, we have
   $$\multline
     \int_{A(k)} |x|^\tau |\nabla u|^m[\frac{m-1}{m}|u|^{q}\eta'-\mu
     |u|^{q-1}|\eta|]+
   a_0\int_{A(k)} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}|u|^{q+m-1}|\eta|\le\\
\le\int_{A(k)} |f_0||\eta|+\frac{m-1}{m} \int_{A(k)}
|x|^{\frac{-\tau }{m-1}} |u|^{\frac{-q}{m-1}}|f|^{\frac{m}{m-1}}{|\eta'|}.   
       \endmultline$$
Because  $|u|>k$ on $A(k)$,  we have
   $$\multline
     \int_{A(k)} |x|^\tau |u|^q |\nabla u|^m[\frac{m-1}{m}\eta'-\mu
     |\eta|]+
   a_0\int_{A(k)} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}|u|^{q+m-1} |\eta|\le\\
\le\int_{A(k)} |f_0||\eta|+\frac{m-1}{m}\int_{A(k)} |x|^{\frac{-\tau }{m-1}}|u|^{\frac{-q}{m-1}}|f|^{\frac{m}
{m-1}}{|\eta'|}.   
          \endmultline$$
Transforming the first integral on the left-hand side with the respect to 
(1.1), choosing $\varkappa$ appropriately by setting in Lemma~1.1   
$a=\frac{m-1}{m}$, $b=\mu $, i.e. $\varkappa>\frac{2m\mu}{m-1}+m$, we obtain
   $$\multline
\frac{m-1}{2m}k_0^q\int_{A(k)} |x|^\tau |\nabla u|^m \text{e }^{\varkappa (|u|-k)_+}+
a_0 \int_{A(k)} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}|u|^{q+m-1}|\eta|\le\\
     \le\int_{A(k)} |f_0||\eta|+\frac{m-1}{m}k_0^{\frac{-q}{m-1}}\int_{A(k)} 
|x|^{\frac{-\tau }{m-1}} |f|^{\frac{m}{m-1}}{|\eta'|}. 
          \endmultline \tag1.9$$
Setting $w_k(x)=\eta(\frac{(|u|-k)_+}{m})$, 
 $$\text{e }^{\varkappa (|u|-k)_+}|\nabla u|^m=
        (\text{e }^{\frac{\varkappa (|u|-k)_+}{m}}|\nabla u|)^m=
       (\frac{m}{\varkappa})^m|\nabla w_k|^m.$$
Now, we can rewrite (1.9) with (1.2) and (1.3)  as  
$$\multline
\frac{m-1}{2m}\bigl({\frac{m}{\varkappa}}\bigr)^m k_0^q{{\int_{A(k)}}} 
|x|^\tau |\nabla w_k|^m+
a_0{{\int_{A(k)}}} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}
|u|^{q+m-1}|w_k|^m\le\\
\le M {\int_{A(k+d)}} h(x)|w_k|^m+c_3\text{e }^{\varkappa d}
{\int_{A(k)\setminus A(k+d)}} h(x), \endmultline \tag 1.10
$$
where
$$ h(x)=|f_0(x)|+|x|^{\frac{-\tau }{m-1}} |f|^{\frac{m}{m-1}}\,. \tag 1.11
$$
By (0.4)-(0.5) we have that $h(x)\in L_p(G)$, where $p$ is such 
that  $\frac{1}{p}<\frac{m}{n}-\frac{1}{t}$.
Using H\"older's inequality with exponents $p$ and $p'$, we obtain
$$\int_{A(k+d)} h|w_k|^m\le\|h(x)\|_{L_p(G)}(\int_{A(k+d)}|w_k|^{mp'})
^{1/p'}.\tag 1.12
$$
From $\frac{1}{p}<\frac{m}{n}-\frac{1}{t}$ it follows that $mp'<m^\#$, where
$m^\#$ is defined by  $(1.5)$. Let $s$ be a real number such that $mp'<s<m^\#$.
Using the interpolation inequality 
$$\big(\int_{A(k+d)} |w_k|^{mp'}\big)^{1/p'}
\le\big(\int_{A(k)} |w_k|^m \big)^{\theta} 
\big( \int_{A(k)} |w_k|^s\big)^{(1-\theta)m/s}$$
with $\theta\in(0,1)$ which is defined by
    $\frac{1}{mp'}=\frac{\theta}{m}+\frac{1-\theta}{s}$, by  
H\"older's inequality with exponents $\frac{m^\#}{s}$ and $\frac{m^\#}{m^\#-s}$
from (1.12) we get
 $$
 \int_{A(k+d)} h|w_k|^m\le c_4(\int_{A(k)} |w_k|^m)^{\theta}
  (\int_{A(k)} |w_k|^{m^\#})^{(1-\theta)m/m^\#}, \tag 1.13
 $$
where, $c_4=\|h(x)\|_{L_p(G)}(\meas  G)^{m(m^\#-s)(1-\theta)/sm^\#}$.
  
Now, using Young's inequality with exponents $1/\theta$ and 
$1/(1-\theta)$, from (1.13) we obtain 
$$
  \int_{A(k+d)} h|w_k|^m\le\frac{c_5}{{\varepsilon}^{1/\theta}}\int_{A(k)} |w_k|^m
   +\varepsilon^{\frac{1}{(1-\theta)}}(1-\theta)
    (\int_{A(k)} |w_k|^{m^\#})^{\frac{m}{m^\#}},\quad\forall \varepsilon>0\,,
\tag 1.14 $$

 where $c_5=\theta \|h(x)\|_{L_p(G)}^\frac{1}{\theta}(\meas  G)^
{m(m^\#-s)(1-\theta)/\theta sm^\#}$. 
Returning to (1.10), by (1.7) from Lemma 1.3 and estimate (1.14)
we obtain
   $$\multline
\frac{m-1}{2mc_2}\Bigl({\frac{m}{\varkappa}}\Bigr)^mk_0^q(\int_{A(k)}
|w_k|^{m^\#})^{\frac{m}
{m^\#}}+a_0\int_{A(k)} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}|u|^{q+m-1}
|w_k|^m\le\\
     \le\frac{c_5M}{{\varepsilon}^{1/\theta}}\int_{A(k)} |w_k|^m+
      M\varepsilon^{\frac{1}{(1-\theta)}}(1-\theta)(\int_{A(k)}
      |w_k|^{m^\#})^{\frac{m}{m^\#}}+
      c_3\text{e }^{\varkappa d}\int_{A(k)} h,\quad \forall \varepsilon>0.
          \endmultline\tag1.15$$
Now we consider two cases:  $a_0>0$ and  $a_0=0$.

\noindent{\bf Case 1): $a_0>0$. }
For this case by (0.5) and $m>\tau$,  we have
$$\multline
\int_{A(k)} |w_k|^m=\int_{A(k)} [\frac{(x_{n-1}^2+x_n^2)^\frac{m}{2}}{|x|^\tau }|u|^
{-(q+m-1)}]
\frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}|u|^{q+m-1}|w_k|^m\le \\
\le k_0^{-(q+m-1)}\int_{A(k)} (|x|^{m-\tau})\frac{|x|^\tau
}{(x_{n-1}^2+x_n^2)
^\frac{m}{2}}|u|^{q+m-1}|w_k|^m\le \\
\le \frac{(\operatorname{diam} G)^{m-\tau}}{k_0^{q+m-1}}
\int_{A(k)}\frac{|x|^\tau }{(x_{n-1}^2+x_n^2)^\frac{m}{2}}|u|^{q+m-1}|w_k|^m, \quad 
\forall k\ge k_0. \endmultline
$$
Therefore, from (1.15) we obtain
   $$\multline
c_6k_0^q(\int_{A(k)} |w_k|^{m^\#})^{\frac{m}
{m^\#}}+a_0\int_{A(k)} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)
^\frac{m}{2}}|u|^{q+m-1}|w_k|^m\le \\
   \le  \frac{c_7M}{\varepsilon^{1/\theta}k_0^{q+m-1}}
     \int_{A(k)} \frac{|x|^\tau }{(x_{n-1}^2+x_n^2)
^\frac{m}{2}}|u|^{q+m-1}|w_k|^m+ c_3\text{e }^{\varkappa d}\int_{A(k)} h+ \\
 + M\varepsilon^{\frac{1}{(1-\theta)}}(1-\theta)(\int_{A(k)}
 |w_k|^{m^\#})^{m/m^\#}\,. \endmultline \tag {$1.15_1$}    
$$
Now, we can choose  $\varepsilon>0$ and $k_0>0$ such that 
    $$
  k_0^{q+m-1}=\frac{c_7M}{a_0{\varepsilon}^{1/\theta}}, \quad
        M\varepsilon^{\frac{1}{(1-\theta)}}(1-\theta)\le\frac{c_6k_0^q}{2}\,;$$
i.e., 
$$
 k_0^{\frac{q+(m-1)\theta}{(1-\theta)}}\ge\frac{2(M)^{\frac{1}{1-\theta}}(1-\theta)}
 {c_6}(\frac{c_7}{a_0})^{\frac{\theta}{1-\theta}}; \quad
{\varepsilon}=(\frac{Mc_7}{a_0k_0^{q+m-1}})^\theta.
$$
Thus from $(1.15_1)$  for every $k>k_0$  it results
   $$(\int_{A(k)} |w_k|^{m^\#})^{\frac{m}{m^\#}}\le c_8\int_{A(k)} h
     \le c_8\|h(x)\|_{L_p(G)}\meas ^{1-\frac{1}{p}} A(k).\tag1.16$$

\noindent{\bf Case 2): $a_0=0$. }
In this case from $(1.10)$ by $(1.14)$ we have
$$\multline
 k_0^q\int_{A(k)} |x|^\tau |\nabla w_k|^m\le c_9\int_{A(k+d)} h(x)|w_k|^m+c_{10}\int_{A(k)} h(x)\le\\
 \le\frac{c_5c_9}{{\varepsilon}^{1/\theta}}\int_{A(k)} |w_k|^m+
      c_9\varepsilon^{\frac{1}{(1-\theta)}}(1-\theta)(\int_{A(k)}
      |w_k|^{m^\#})^{\frac{m}{m^\#}}+
      c_{10}\int_{A(k)} h,\quad \forall \varepsilon>0. 
\endmultline$$
Further, by (1.6) and the fact that $m>\tau$, by (0.5) we obtain
$$\multline
\int_{A(k)} |w_k|^m=\int_{A(k)} (|x|^{\tau -m}|w_k|^m)|x|^{m-\tau}\le 
(\operatorname{diam} G)^{m-\tau}
\int_{A(k)} |x|^{\tau -m}|w_k|^m\le \\
\le c_1(\operatorname{diam} G)^{m-\tau}\int_{A(k)} |x|^\tau |\nabla w_k|^m.
\endmultline$$
From the last two inequalities, for all $\varepsilon>0$ we have
$$\multline
k_0^q\int_{A(k)} |x|^\tau |\nabla w_k|^m
\le c_{11}{\varepsilon}^{-\frac{1}{\theta}}\int_{A(k)} |x|^\tau |\nabla w_k|^m
+\\
+c_9\varepsilon^{\frac{1}{(1-\theta)}}(1-\theta)(\int_{A(k)}
|w_k|^{m^\#})^{\frac{m}{m^\#}}
+ c_{10}\int_{A(k)} h,
\endmultline\tag {$1.15_2$}
$$
Now we set $c_{11}{\varepsilon}^{-\frac{1}{\theta}}=\frac{1}{2}k_0^q$. Then by
virtue of $(1.7)$ for $\forall k>k_0$ we have
$$\frac{1}{2c_2}k_0^q(\int_{A(k)} |w_k|^{m^\#})^{\frac{m}{m^\#}}\le
 c_9\varepsilon^{\frac{1}{(1-\theta)}}(1-\theta)(\int_{A(k)}
 |w_k|^{m^\#})^{\frac{m}{m^\#}}
+ c_{10}\int_{A(k)} h.
$$
If we choose $c_9\varepsilon^{1/(1-\theta)}(1-\theta)\le
\frac{1}{4c_2}k_0^q$,
then we get again $(1.16)$. In this case we choose
$$k_0\ge
(4c_2c_9(1-\theta))^{\frac{1-\theta}{q}}(2c_{11})^{\frac{\theta}{q}};
\quad {\varepsilon}=(\frac{2c_{11}}{k_0^q})^\theta.$$

Now we return to $(1.16)$. Let us now take  $l>k>k_0$. Due to $(1.4)$ and 
  $|w_k|>\frac{1}{m}(|u|-k)_+$,  we have
   $$\int_{A(l)} |w_k|^{m^\#}\ge(\frac{l-k}{m})^{m^\#}\meas A(l)\,.
                                             \tag1.17$$
Combining  (1.16) and (1.17), we obtain
    $$\meas A(l)\le(\frac{m}{l-k})^{m^\#}(c_8
    \|h(x)\|_{L_p(G)})
^{\frac{m^\#}{m}}\text{meas }^{\frac{m^\#}{m}(1-\frac{1}{p})}A(k),\quad
                      \forall l>k\ge k_0. \tag 1.18$$
Notice that from (0.5) and (1.5) it follows that  
                   $\frac{m^\#}{m}(1-\frac{1}{p})>1$.

Lemma 1.2 implies that $\meas A(k_0+\delta)=0$,
where $\delta$ depends only on the constants in $(1.18)$.
This means that  $|u(x)|<k_0+\delta$ for a.e. $x\in G$.
Theorem 1.4 is proved.
\enddemo

\head 2. H\"older continuity of the weak solution. \endhead 

In this section we prove, that the weak solution of  (0.1)-(0.5) 
is H\"older continuous in a neighbourhood of a conical point.

\proclaim{Theorem 2.1}
 Let $u(x)$ be a weak solution of (0.1)-(0.5) with $\tau =0$, and
let $d$ be a positive number.
 Then in the domain $G_0^d$, $u(x)$ is H\"older 
 continuous with exponent $\alpha$ depending only on the data
 assumptions and the domain  $G$.
\endproclaim

\demo{Proof}
Let $M_0$ be the number from Theorem 1.4. Let us introduce the cut-off
function
$\quad  \zeta(x)\in C^{\infty}_0(G_0^\rho), \quad \rho\in (0,d):$
   $$\zeta(x)=\cases 1,\quad x\in G_0^{\rho-\sigma\rho},\quad
\sigma\in(0,1) \\
                  0,\quad x\notin G_0^{\rho};
     \endcases$$
$$0<\zeta(x)<1, \quad x\in G_0^{\rho}; \quad |\nabla \zeta|\le 1/(\sigma\rho).$$
\qquad Let us define the set $A_{k,\rho}:=\{x\in G_0^{\rho}\mid u(x)>k\}$, 
for all $k\in {\Bbb R}$. Substituting \break 
$\phi(x)=\zeta^m(x)\text{max }\{u(x)-k,0\}$ in the definition
of weak solution, we get 
$$\multline \int_{A_{k,\rho}} |u|^q\zeta^m|\nabla u|^m+m
             \int_{A_{k,\rho}} \zeta^{m-1}(x)\zeta_{x_i}u_{x_i}|\nabla u|^{m-2}|u|^q(u-k)+\\
            +a_0\int_{A_{k,\rho}} (x_{n-1}^2+x_n^2)^{-\frac{m}{2}} u|u|^{q+m-2}\zeta^m(u-k)
    =\mu\int_{A_{k,\rho}} |u|^{q-2}u|\nabla u|^m\zeta^m(u-k)+\\
       +\int_{A_{k,\rho}} f_0\zeta^m(u-k)+m\int_{A_{k,\rho}} \zeta^{m-1}(u-k)f_i\zeta_{x_i}+\int_{A_{k,\rho}} f_i u_{x_i}\zeta^m.
            \endmultline\tag2.1
$$
Using Young's inequality with exponents $m/(m-1)$ and
$m$ in the second integral of the right-hand side, 
 $$\multline
    |m\zeta^{m-1}\zeta_{x_i}u_{x_i}(u-k)|\nabla u|^{m-2}|u|^q|\le
    m(|\nabla u|\zeta)^{m-1}((u-k)|\nabla \zeta|)|u|^q\le\\
    \le(m-1)\varepsilon(|\nabla u|\zeta)^m|u|^q+\frac{1}{m}
    \varepsilon^{1-m}(u-k)^m|\nabla \zeta|^m|u|^q,
    \quad\forall\varepsilon>0.
            \endmultline$$
Choosing $\varepsilon=\frac{1}{2(m-1)}$, we obtain
 $$\multline
   \frac{1}{2}\int_{A_{k,\rho}} |u|^q\zeta^m|\nabla u|^m+a_0\int_{A_{k,\rho}} 
   (x_{n-1}^2+x_n^2)^{-\frac{m}{2}}u|u|^{q+m-2}\zeta^m(u-k)\le\\
      \le C(m)\int_{A_{k,\rho}} (u-k)^m|\nabla \zeta|^m|u|^q+\mu\int_{A_{k,\rho}} |u|^{q-1}|\nabla u|^m\zeta^m(u-k)+\\
      +\int_{A_{k,\rho}} f_0\zeta^m(u-k)+m\int_{A_{k,\rho}} \zeta^{m-1}(u-k)f_i\zeta_{x_i}+\int_{A_{k,\rho}} f_i u_{x_i}\zeta^m.
            \endmultline\tag2.2$$
Now, we choose $k_0\ge \frac{1}{2}M_0$, if $\mu=0$,  
$\max_{G_0^{\rho}} (u(x)-k_0) \le \frac{1}{4\mu}$, if $\mu >0$, and 
without lost of generality  $k_0>1$.
Then for every $k\ge k_0$,
 $$
       \mu|u|^{q-1}|\nabla u|^m\zeta^m(u-k)\biggm|_{A_{k,\rho}} \le
        \mu |u|^{q-1}|\nabla u|^m\zeta^m
          {\underset{G_0^{\rho}}\to{\text{max }}}(u(x)-k)
       \le\frac{1}{4}|u|^q\zeta^m|\nabla u|^m.
 \tag2.3$$
Substituting (2.3) in (2.2), we have
 $$\multline
       \frac{1}{4}\int_{A_{k,\rho}} |u|^q\zeta^m|\nabla u|^m+a_0\int_{A_{k,\rho}} 
       (x_{n-1}^2+x_n^2)^{-\frac{m}{2}}u|u|^{q+m-2}\zeta^m(u-k)\le\\
      \le C(m)\int_{A_{k,\rho}} (u-k)^m|\nabla \zeta|^m|u|^q+\int_{A_{k,\rho}} f_0\zeta^m(u-k)+\\
      +m\int_{A_{k,\rho}} \zeta^{m-1}(u-k)f_i\zeta_{x_i}+\int_{A_{k,\rho}} f_i u_{x_i}\zeta^m.
            \endmultline\tag2.4$$
Let us now estimate $f_i u_{x_i}$ on $A_{k,\rho}:$
 $$\multline
      f_i u_{x_i}\le|\nabla u||f|\le\varepsilon|\nabla u|^m+c(\varepsilon)|f|^{\frac{m}{m-1}}
      \le\varepsilon|u|^q|\nabla u|^m+c(\varepsilon)|f|^{\frac{m}{m-1}},\quad\forall\varepsilon>0.
            \endmultline\tag2.5$$
 From (2.4) and (2.5) with $\varepsilon=1/8$ and taking into
account that $(u-k)\le\frac{1}{4\mu}$, if $\mu >0$, and $(u-k)\le
\frac{1}{2}M_0$, if $\mu =0$, we get:
 $$\multline
       \frac{1}{8}\int_{A_{k,\rho}} |u|^q\zeta^m|\nabla u|^m+a_0\int_{A_{k,\rho}} 
       (x_{n-1}^2+x_n^2)^{-\frac{m}{2}}u|u|^{q+m-2}\zeta^m(u-k)\le\\
      \le C(m)\int_{A_{k,\rho}} (u-k)^m|\nabla \zeta|^m|u|^q+C(m,M_0,\mu)\int_{A_{k,\rho}} 
      \{|f_0|+|f|^{\frac{m}{m-1}}\}\zeta^m+ \\
      + m\int_{A_{k,\rho}} \zeta^{m-1}(u-k)f_i\zeta_{x_i}.
            \endmultline$$
Now using Young's inequality on the last integral,
       $$m((u-k)\zeta_{x_i})(f_i\zeta^{m-1})\le(u-k)^m|\nabla \zeta|^m+
                 (m-1)|f|^{\frac{m}{m-1}}\zeta^m.$$
It results that
 $$\multline
       \frac{1}{8}\int_{A_{k,\rho}} |u|^q\zeta^m|\nabla u|^m+a_0\int_{A_{k,\rho}} 
       (x_{n-1}^2+x_n^2)^{-\frac{m}{2}}u|u|^{q+m-2}\zeta^m(u-k)\le\\
      \le C(m)\int_{A_{k,\rho}}(u-k)^m|\nabla \zeta|^m|u|^q+C(m,M_0,\mu)\int_{A_{k,\rho}} h\zeta^m,
            \endmultline$$
where we have set $h(x)=|f_0|+|f|^{\frac{m}{m-1}}$ (see $(1.11)$ with $\tau=0$).
We shall strengthen the inequality, if we use the estimate
$$\multline 
  \int_{A_{k,\rho-\sigma\rho}} |u|^q|\nabla u|^m\le \\
\le C(m)\max_{A_{k,\rho}}|u|^q
     \int_{A_{k,\rho}}(u-k)^m|\nabla \zeta|^m+
    C(m,M_0,\mu)\max_{A_{k,\rho}}|u|^q\int_{A_{k,\rho}} 
    h \endmultline \tag2.6
$$
From the definition of $\zeta(x)$ it follows that
   $$\multline
      \int_{A_{k,\rho}}(u-k)^m|\nabla \zeta|^m\le (\sigma\rho)^{-m}
      \max_{A_{k,\rho}}(u(x)-k_0)^m
      {\meas A_{k,\rho}}=\\
 =(\sigma\rho)^{-m}\max_{A_{k,\rho}}(u(x)-k_0)^m
       \meas ^{\frac{m}{s}}A_{k,\rho}
       \meas ^{1-\frac{m}{s}}A_{k,\rho}, \quad s>n.
          \endmultline\tag2.7$$
Since ${\meas A_{k,\rho}}\le{\meas G_0^{\rho}}=
\int_{G_0^{\rho}} dx=\meas \Omega\int_{0}^{\rho}r^{n-1} dr
=\frac{\rho^n}{n}\meas \Omega$, we
can rewrite (2.7) as 
$$\int_{A_{k,\rho}}(u-k)^m|\nabla \zeta|^m\le (\frac{1}{n}
\meas  \Omega)^{\frac{m}{s}}\sigma^{-m}
   {\rho}^{-m(1-\frac{n}{s})} \max_{A_{k,\rho}} (u(x)-k_0)^m
       \meas ^{1-\frac{m}{s}}A_{k,\rho},\tag2.8$$
for $s>n$. Combining  (2.6) and (2.8),
 $$\multline
  \int_{A_{k,\rho-\sigma\rho}} |u|^q|\nabla u|^m\le  \\
 \max_{A_{k,\rho}}|u|^q[
      \gamma\sigma^{-m}{\rho}^{-m(1-\frac{n}{s})}
        \max_{A_{k,\rho}}(u(x)-k)^m
         \meas ^{1-\frac{m}{s}}A_{k,\rho}+ 
     C(m,M_0,\mu)\int_{A_{k,\rho}} h ],\endmultline \tag 2.9
$$
where $\gamma=C(m)(\frac{1}{n}\meas \Omega)^{\frac{m}{s}}$, 
$s>n> m$. 
By H\"older's inequality, 
  $$\multline \int_{A_{k,\rho}} h\le(\int_{A_{k,\rho}}
  h^{\frac{s}{m}})^{\frac{m}{s}}
    \meas ^{1-\frac{m}{s}}A_{k,\rho}
     \le||h||_{L_{\frac{s}{m}}(G)}\meas ^{1-\frac{m}{s}}A_{k,\rho}\le\\
\le \meas ^{\frac{m}{s}-\frac{1}{p}}G||h||_{L_p(G)}\meas ^{1-\frac{m}{s}}
A_{k,\rho},\quad s=\frac{mnt}{mt-1}, \endmultline\tag2.10
$$
where $p$ and $t$ is defined by (0.5).
From (2.10) and (2.9) it follows that
$$
\int_{A_{k,\rho-\sigma\rho}} |u|^q|\nabla u|^m\le \max_{A_{k,\rho}} |u|^q
[\gamma\sigma^{-m}{\rho}^{-m(1-\frac{n}{s})}
\max_{A_{k,\rho}}(u(x)-k)^m
 +\gamma_1] \meas ^{1-\frac{m}{s}}A_{k,\rho}, \tag 2.11
$$
where $\gamma_1=C(m,M_0,\mu)\meas ^{\frac{m}{s}-\frac{1}{p}}G
         (\|f_0\|_{L_p(G)}+\|f\|_{L_{\frac{mp}{m-1}}(G)})$.
This proves that $u(x)$ belongs to the class 
${\Bbb{\goth {\tilde B}}}^1_{m}(\overline {G_0^d},|u|^q, M_0, \gamma,
\gamma_1, \frac{1}{s})$
(see [10, \S9 chapt. II]), where $\gamma, \gamma_1$ are defined
by (2.9), (2.11),
and also by [10, Theorem 9.1 chapt. II] 
$u(x)$ is H\"older continuous with exponent $\alpha>0$, depending only 
on  $\mu$, $q$, $m$, $n$ and domain $G$. This finishes the proof of Theorem 2.1.         
\enddemo

\head Construction of the barrier function \endhead

In this section we  construct the barrier function for the homogeneous 
Dirichlet problem (0.1)-(0.2) with $\tau =0$ in $n$-dimensional 
{\it infinite convex} circular cone $G_0$ with the vertex at the origin of 
coordinates $O$ and its lateral area $\Gamma_0$.
$$
\gather {\goth L}u\equiv\frac{d}{dx_i}({|u|}^q{|\nabla u|}
^{m-2}u_{x_i})=\frac{a_0}{(x_{n-1}^2+x_n^2)^\frac{m}{2}}u|u|^{q+m-2}-
\mu u{|u|}^{q-2}{|\nabla u|}^m, \;  x\in G_0, \tag 3.1\\ 
 u(x)=0,\quad x\in\Gamma_0 \tag 3.2 \\
a_0\ge0,\quad 0\le \mu <1,\quad q\ge0,\quad 2\le m\leq n. \tag 3.3
   \endgather
$$ 
Let us transfer to the spherical coordinates with the pole at the
point $O$.
 $$\gathered x_1=r\cos\omega_1,\quad 
  x_2=r\cos\omega_2 \sin\omega_1, \\
               \quad \ddots       \\
           x_{n-1}=r\cos\omega_{n-1} \sin\omega_{n-2}\dots\sin\omega_1, \\
           x_{n}=r\sin\omega_{n-1}\dots \sin \omega_1\,,\endgathered \tag3.4$$
where $0\le r=|x|<\infty$, $0\le\omega_k\le\pi$, $k\le n-2$, 
 $-\frac{\omega_0}{2}\le\omega_{n-1}\le\frac{\omega_0}{2}$, 
 $\omega_0\in(0,\pi)$. 
          
The lateral area $\Gamma_0$ may be obtained as the surface of
revolution of the angle
$x_{n-1}=x_n\cot \frac{\omega_0}{2}$, $0<\omega_0<\pi$,
around the axis $Ox_{n-1}$.
$$
\Gamma_0=\bigl\{x\in {\Bbb R}^n \bigl | \; x_{n-1}^2=\bigl (x_n^2+\sum_{k=1}
^{n-2}x_k^2\bigr )\cot ^2\frac{\omega_0}{2}\bigr \}. \tag3.5$$
Thus we get         
$$
G_0=\bigl\{x\in {\Bbb R}^n \bigl | \; x_{n-1}^2>\bigl (x_n^2+\sum_{k=1}
^{n-2}x_k^2\bigr )\cot ^2\frac{\omega_0}{2}\bigr \}. \tag3.6$$

\proclaim{Lemma 3.1} Let $q_{n-1}=(\sin\omega_1\dots\sin\omega_{n-2})^2$.
Then
$$q_{n-1}\bigl |_{ \Gamma_0}=1, \qquad 
q_{n-1}\bigl |_{G_0}>\cos ^2\frac{\omega_0}{2}. $$
\endproclaim

\demo{Proof}  From (3.4) we have
$$\gathered
x_{n-1}^2=r^2q_{n-1}\cos ^2\omega, \quad x_{n}^2=r^2q_{n-1}\sin ^2\omega \quad
\forall \omega\in [-\frac{\omega_0}{2},\frac{\omega_0}{2}]; \\
x_{n-1}^2+x_n^2=r^2q_{n-1}, \\
\sum_{k=1}^{n-2}x_k^2=|x|^2-(x_{n-1}^2+x_n^2)=r^2(1-q_{n-1}).
\endgathered
$$
Now by (3.5) on $ \Gamma_0$ we have
$$r^2q_{n-1}\cos ^2\omega=r^2(1-q_{n-1}+q_{n-1}\sin ^2\omega)\cot ^2\frac{\omega_0}{2}
=r^2(1-q_{n-1}\cos ^2\omega)\cot ^2\frac{\omega_0}{2},$$
whence
$$
q_{n-1}\frac{\cos ^2\omega}{\sin ^2\frac{\omega_0}{2}}=\frac{\cos ^2\frac{\omega_0}{2}}
{\sin ^2\frac{\omega_0}{2}}
$$
and since $\omega \bigl |_{ \Gamma_0}=\pm \frac{\omega_0}{2}$,  we get
first assertion of lemma. Similarly by  (3.6) in $G_0$,
$$r^2q_{n-1}\cos ^2\omega>r^2(1-q_{n-1}\cos ^2\omega)\cot ^2\frac{\omega_0}{2},$$
whence
$q_{n-1}\cos ^2\omega>\cos ^2\frac{\omega_0}{2}$ for all $\omega\in 
(-\frac{\omega_0}{2},\frac{\omega_0}{2})$.
This means that second assertion of lemma holds.
\enddemo

 We shall seek the solution to Problem (3.1)-(3.2) of the form  
   $$u=r^{\lambda}(\sin \omega_1\dots\sin \omega_{n-2})^{\lambda}\Phi(\omega_{n-1}),
           \quad\lambda>0\tag3.7
           $$
with $\Phi(\omega) \ge0$ and $\lambda$, satisfying the following condition
$$\lambda^m(q+m-1+\mu)+\lambda^{m-1}(2-m)>a_0. \tag {\bf *}$$
The differential operator ${\goth L}$ becomes            
$${\goth L}u=\frac{1}{J}\sum_{i=1}^{n}\frac{d}{d \xi_i}
({|u|}^q{|\nabla u|}^{m-2}\frac{J}{H_i^2}\frac{\partial u}{\partial
\xi_i}),$$
where 
  $$\gathered
  J=r^{n-1}\sin^{n-2}\omega_1\dots\sin\omega_{n-2};\\
H_1=1, \quad\xi_1=r,\quad\xi_{i+1}=\omega_{i};
          \quad H_{i+1}=r\sqrt{q_i}, \quad i=\{\overline{1,n-1}\};\\
          q_1=1,\quad q_i=(\sin\omega_1\dots\sin\omega_{i-1})^2,
          \quad i=\{\overline{2,n-1}\}.
          \endgathered\tag3.8$$ 
Then $\Phi(\omega)$ satisfies the equation
  $$\multline \frac{d}{d \omega_{n-1}}[(\lambda^2
               \Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}|\Phi|^q{\Phi}']+\\
   +\lambda q_{n-1}{\left\{ \sum_{k=1}^{n-2}\frac{1}{q_k}
  \langle [\lambda(q+m-1)+n-m-k+1]\text{ctg }^2{\omega_k}
  -\frac{1}{\sin^2 \omega_k}\rangle+
  \right.}\\
  {\left. \vphantom{\sum_{k=1}^{n-2}\frac{1}{q_k}}+[\lambda(q+m-1)+n-m]
                     \right\}} \Phi|\Phi|^q
         (\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}=\\
=a_0\Phi|\Phi|^{q+m-2}-\mu\Phi|\Phi|^{q-2}(\lambda^2\Phi^2+{\Phi'}^2)
^{\frac{m}{2}},
                \endmultline$$
where $\Phi=\Phi(\omega_{n-1})$ and $\omega_{n-1}\in(-\omega_0/2,\omega_0/2)$.         

Let us show that
  $$\multline    \sum_{k=1}^{n-2}\frac{1}{q_k}
   \langle [\lambda(q+m-1)+n-m-k+1]\text{ctg }^2{\omega_k}
   -\frac{1}{\sin^2 \omega_k}\rangle =\\
    =\dfrac{[\lambda(q+m-1)-m+2]}{(\sin \omega_1\dots\sin \omega_{n-2})^2}
    -[\lambda(q+m-1)-m+n]                      \endmultline\tag3.9$$
For this we use induction on $n$. In the case $n=3$, (3.9)
is obvious. Now we suppose that (3.9) is true for $n-1$, that is 
  $$\multline     \sum_{k=1}^{n-3}\frac{1}{q_k}
   \langle [\lambda(q+m-1)+n-m-k]\text{ctg }^2{\omega_k}-\frac{1}{\sin^2
   \omega_k}\rangle =\\
    =\dfrac{[\lambda(q+m-1)-m+2]}{(\sin \omega_1\dots\sin \omega_{n-3})^2}
    -[\lambda(q+m-1)-m+n-1]  \endmultline$$
It is easy to calculate that
$$\sum_{k=1}^{n-2}\frac{\text{ctg }^2{\omega_k}}{q_k}=\frac{1}{q_{n-1}}-1.\tag3.10$$
Then we have
$$\multline
      \sum_{k=1}^{n-2}\frac{1}{q_k}
   \langle [\lambda(q+m-1)+n-m-k+1]\text{ctg }^2{\omega_k}
   -\frac{1}{\sin^2 \omega_k}\rangle=\\
=\sum_{k=1}^{n-3}\frac{1}{q_k}
   \langle [\lambda(q+m-1)+n-m-k+1]\text{ctg }^2{\omega_k}
   -\frac{1}{\sin^2 \omega_k}\rangle+\\
+\frac{1}{q_{n-2}}\langle [\lambda(q+m-1)+3-m]\text{ctg }^2{\omega_{n-2}}-
\frac{1}{\sin^2 \omega_{n-2}}\rangle=\\
=\sum_{k=1}^{n-2}\frac{\text{ctg }^2{\omega_k}}{q_k}
+\frac{1}{q_{n-2}}\langle [\lambda(q+m-1)+2-m]\text{ctg }^2{\omega_{n-2}}-
\frac{1}{\sin^2 \omega_{n-2}}\rangle +\\
+\frac{\lambda(q+m-1)-m+2}{(\sin \omega_1\dots\sin \omega_{n-3})^2}-
     [\lambda(q+m-1)-m+n-1]=\\
=\langle \sum_{k=1}^{n-2}\frac{\text{ctg
}^2{\omega_k}}{q_k}-\frac{1}{q_{n-1}}\rangle      
- [\lambda(q+m-1)-m+n-1]+\\     
+\frac{1}{q_{n-2}}[\lambda(q+m-1)-m+2](1+\text{ctg
}^2{\omega_{n-2}})\quad{=\atop\text{by (3.10)}} \\     
=\frac{1}{q_{n-1}}[\lambda(q+m-1)-m+2]-[\lambda(q+m-1)-m+n],
 \endmultline$$
\quad \hfill Q.E.D.
 

Now by (3.9), Problem (3.1)-(3.2) for $\Phi(\omega),
\omega=\omega_{n-1}$ becomes 
$$\gather
   \frac{d}{d \omega}[(\lambda^2
              \Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}|\Phi|^q{\Phi}']+
     \lambda[\lambda(q+m-1)-m+2]\Phi|\Phi|^q(\lambda^2
               \Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}=\\
=a_0\Phi|\Phi|^{q+m-2}- \mu\Phi|\Phi|^{q-2}(\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m}{2}},
\quad \omega\in (-\omega_0/2,\omega_0/2),  \tag 3.11\\
\Phi(-\omega_0/2)=\Phi(\omega_0/2)=0. \tag 3.12  \endgather
$$
By setting  $\Phi'/\Phi=y$ we arrive at
$$\gather
[(m-1)y^2+\lambda^2](y^2+\lambda^2)^{\frac{m-4}{2}}y'+
    (m-1+q+\mu)(y^2+\lambda^2)^{\frac{m}{2}}+\\
 +\lambda(2-m)(y^2+\lambda^2)^{\frac{m-2}{2}}=a_0,\quad \omega\in 
         (-\omega_0/2,\omega_0/2), \tag 3.13\\
y(0)=0,\quad \lim_{\omega\to \frac{\omega_0}{2}-0}y(\omega)=-\infty. \tag 3.14
\endgather 
$$         
We explain (3.14). In fact, from (3.11)-(3.12) it follows easily
that $\Phi(-\omega)=\Phi(\omega)$,  $\omega\in [-\omega_0/2,\omega_0/2]$, 
and from this  $y(-\omega)=-y(\omega)$, $\omega\in [-\omega_0/2,\omega_0/2]$. 
Consequently, we have $y(0)=0$. 
Further, from (3.13)  we obtain
$$\gathered 
-[(m-1)y^2+\lambda^2](y^2+\lambda^2)^{\frac{m-4}{2}}y'=\\
=  (m-1+q+\mu)(y^2+\lambda^2)^{\frac{m}{2}}+
 \lambda(2-m)(y^2+\lambda^2)^{\frac{m-2}{2}}-a_0=\\
 =(y^2+\lambda^2)^{\frac{m-2}{2}}
 [(m-1+q+\mu)(y^2+\lambda^2)+\lambda(2-m)]-a_0\ge\\
 \ge (y^2+\lambda^2)^{\frac{m-2}{2}}[\lambda^2(m-1+q+\mu)
+\lambda(2-m)]-a_0\ge \\
 \ge \lambda^m(m-1+q+\mu) +\lambda^{m-1}(2-m)-a_0>0 
 \endgathered \tag3.15$$
by (*) and (3.3). Thus it is proved that $y'(\omega)<0, \omega\in [-\omega_0/2,\omega_0/2]$. 
Therefore 
$y(\omega)$ is decreasing function on  $[-\omega_0/2,\omega_0/2]$. From this
we conclude the last condition of (3.14).

{\bf Properties of the function $\Phi(\omega)$.}
We turn our attention to the properties of the function
$\Phi(\omega)$.  First of all, notice that 
the solutions to (3.11)-(3.12) are determined uniquely up to
a scalar multiple provided that $\lambda$ satisfies (*). We will consider
the solution normed by the condition
$$\Phi(0)=1.\tag3.16$$
We rewrite the (3.11) in the following form 
$$\multline
-\Phi[(m-1){\Phi'}^2+\lambda^2\Phi^2](\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-4}{2}}
{\Phi}''=-a_0\Phi^m + (q+\mu )(\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m}{2}}+\\
+\Phi^2(\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-4}{2}}\big\{\lambda[\lambda(m-1)-m+2]
(\lambda^2 \Phi^2+{\Phi'}^2)+(m-2)\lambda^2{\Phi'}^2 \big\}
\endmultline \tag3.17$$
Now, since $m\ge 2$, by virtue of (*) from the (3.17), it follows that
$$\multline
-\Phi[(m-1){\Phi'}^2+\lambda^2\Phi^2](\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-4}{2}}
{\Phi}''\ge -a_0\Phi^m +\\
+ (\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}
\big\{(q+\mu )(\lambda^2 \Phi^2+{\Phi'}^2)+\lambda[\lambda(m-1)-m+2]\Phi^2
 \big\} \ge \\
 \ge \Phi^m \big\{(q+\mu +m-1)\lambda ^m + (2-m)\lambda ^{m-1}-a_0
\big\}>0
\endmultline $$
 (here we take into account that by (*) 
$(q+\mu +m-1)\lambda ^2 + (2-m)\lambda >0)$.

Summarizing the above we obtain the following: 
$$\gathered
\Phi(\omega)\ge 0 \quad \forall \omega\in [-\omega_0/2,\omega_0/2]; \qquad \Phi(-\omega_0/2)=
\Phi(\omega_0/2)=0;\\
\Phi(-\omega)=\Phi(\omega) \quad \forall \omega\in [-\omega_0/2,\omega_0/2];\\
\Phi'(0)=0;\\
\Phi''(\omega)<0 \quad \forall \omega\in [-\omega_0/2,\omega_0/2].
\endgathered \tag3.18$$


\proclaim{Corollary} 
$$\max_{[-\omega_0/2,\omega_0/2]}\Phi(\omega)=\Phi(0)=1\; \Rightarrow \;
0\le \Phi(\omega)\le 1\quad \forall \omega\in [-\omega_0/2,\omega_0/2].
\tag3.19$$
\endproclaim

Now we solve (3.11)-(3.14). Rewriting the (3.13)
in the form  $y'=g(y,)$ we observe that by (3.15), 
$g(y)\not=0$ for all $y\in {\Bbb R}$. Moreover, 
being rational functions with nonzero denominators $g(y)$ and
$g'(y)$ are continuous functions. By the theory of ordinary
differential equations the Cauchy's problem (3.13), (3.14) is
uniquely solvable in the strip 
$$\bigl \{(\omega, y)\bigr \}\subset \bigl [-\frac{\omega_0}{2},
\frac{\omega_0}{2}\bigr ]\times (-\infty ,+\infty ).$$
Integrating (3.11)-(3.14), we obtain
$$\gathered
\Phi(\omega)=\exp \int_{0}^{\omega}y(\xi)d\xi,\\
\int_{0}^{-y}\frac{[(m-1)z^2+\lambda^2](z^2+\lambda^2)^{\frac{m-4}{2}}}
{(m-1+q+\mu)(z^2+\lambda^2)^\frac{m}{2}+
\lambda(2-m)(z^2+\lambda^2)^{\frac{m-2}{2}}-a_0}\, dz=\omega . 
\endgathered\tag3.20
$$
From this we get in particular that
  $$\int_{0}^{+\infty}\frac{[(m-1)y^2+\lambda^2](y^2+\lambda^2)^{\frac{m-4}{2}} }
        {(m-1+q+\mu)(y^2+\lambda^2)^{\frac{m}{2}}+
         \lambda(2-m)(y^2+\lambda^2)^{\frac{m-2}{2}}-a_0}\,dy
         =\frac{\omega_0}{2}.\tag3.21$$
This expression gives the equation for finding a sharp estimate for the
exponent $\lambda$ in (3.7). For the case $a_0=0$ this exponent
is calculated explicitly in [18]; we denote this value by $\lambda_0$.

{\bf Solutions of (3.21).} We set
$$ \Lambda (\lambda , a_0, y) \equiv\frac{[(m-1)y^2+\lambda ^2](y^2+\lambda ^2)^
{\frac{m-4}{2}}}{(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}+\lambda (2-m)(y^2+\lambda ^2)^
{\frac{m-2}{2}}-a_0}, \tag3.22$$
$${\Bbb{\goth F}}(\lambda , a_0, \omega_0)=-\frac{\omega_0}{2}+\int_{0}^{+\infty} \Lambda 
(\lambda , a_0, y)dy. \tag3.23$$
Then (3.21) takes the form 
$${\Bbb{\goth F}}(\lambda , a_0, \omega_0)=0.\tag3.24$$
According to what has been said above
$${\Bbb{\goth F}}(\lambda _0, 0, \omega_0)=0.\tag3.25$$
Direct calculations give
$$\multline
\frac{\partial \Lambda}{\partial\lambda}=-\lambda \frac{(y^2+\lambda ^2)^
{\frac{m-6}{2}}}{\bigl[(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}+\lambda (2-m)(y^2+\lambda ^2)^
{\frac{m-2}{2}}-a_0\bigr]^2}\times \\
\times \biggl\{(y^2+\lambda ^2)\bigl[2(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}+(m-2)a_0\bigr]+\\
+(m-2)y^2\bigl[4(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}+2\lambda (2-m)(y^2+\lambda ^2)^
{\frac{m-2}{2}}+(m-4)a_0\bigr]\biggr\} \endmultline
$$
By (*)  with $m\ge 2$, the above expression is less than or equal to 
$$ \multline
 -2\lambda (m-2)\frac{y^2(y^2+\lambda ^2)^
{\frac{m-6}{2}}}{\bigl[(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}+\lambda (2-m)(y^2+\lambda ^2)^
{\frac{m-2}{2}}-a_0\bigr]^2}\times \\
\times \biggl \{(y^2+\lambda ^2)^{\frac{m-2}{2}}[2(m-1+q+\mu)\lambda ^2+(2-m)\lambda
]-a_0+\frac{m-2}{2}a_0\biggr \}<\\
<-2\lambda (m-2)\frac{y^2(y^2+\lambda ^2)^
{\frac{m-6}{2}}}{\bigl[(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}+\lambda (2-m)(y^2+\lambda ^2)^
{\frac{m-2}{2}}-a_0\bigr]^2}\times \\
\times \biggl \{(y^2+\lambda ^2)^{\frac{m-2}{2}}[(m-1+q+\mu)\lambda ^2+(2-m)\lambda
]-a_0\biggr \}<\\
<-2\lambda (m-2)\frac{y^2(y^2+\lambda ^2)^
{\frac{m-6}{2}}}{\bigl[(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}+\lambda (2-m)(y^2+\lambda ^2)^
{\frac{m-2}{2}}-a_0\bigr]^2}\times \\
\times \biggl \{(y^2+\lambda ^2)^{\frac{m-2}{2}}\lambda ^{2-m}a_0-a_0\biggr
\}\le 0, \endmultline \tag3.26$$
for all $y$ and $\lambda$, $a_0$ satisfying (*).  
Similarly for all $y$, $\lambda$, $a_0$, we have
$$\frac{\partial \Lambda}{\partial a_0}=\frac{[(m-1)y^2+\lambda ^2](y^2
+\lambda ^2)^
{\frac{m-4}{2}}}{\bigl [(m-1+q+\mu)(y^2+\lambda ^2)^{\frac{m}{2}}
+\lambda (2-m)(y^2+\lambda ^2)^
{\frac{m-2}{2}}-a_0\bigr]^2}>0. \tag3.27$$

Hence, we can apply the Implicit Function Theorem in a neighborhood of the
point $(\lambda _0, 0)$. Then  (3.24) (and
therefore and the (3.21)) determines $\lambda =\lambda (a_0,\omega_0)$ as 
single-valued {\bf continuous} function of $a_0$,  depending continuously
on the parameter $\omega_0$ and having continuous partial derivatives
$\frac{\partial\lambda }{\partial a_0},\frac{\partial\lambda }
{\partial \omega_0}$. Applying the analytic
continuation method, we obtain the solvability of the equation
(3.21) for $\forall a_0$, satisfying (*).

Now, we analyze the properties of $\lambda$ as the function 
$\lambda (a_0, \omega_0)$. First, from (3.24) we get
$$\frac{\partial {\Bbb{\goth F}}}{\partial \lambda}\frac{\partial \lambda}
{\partial a_0}+\frac{\partial {\Bbb{\goth F}}}
{\partial a_0}=0, \quad \frac{\partial {\Bbb{\goth F}}}{\partial \lambda}
\frac{\partial \lambda}{\partial \omega_0}+
\frac{\partial {\Bbb{\goth F}}}{\partial \omega_0}=0;$$
from this it follows that
$$\frac{\partial \lambda}{\partial a_0}=-\frac{\bigl(\frac{\partial {\Bbb{\goth F}}}
{\partial a_0}\bigr)}{\bigl(\frac{\partial {\Bbb{\goth F}}}{\partial \lambda}\bigr)};
 \quad \frac{\partial \lambda}{\partial \omega_0}=-\frac{\bigl(
\frac{\partial {\Bbb{\goth F}}}
{\partial \omega_0}\bigr)}{\bigl(\frac{\partial {\Bbb{\goth F}}}{\partial 
\lambda}\bigr)}.\tag3.28$$
But by virtue of (3.26), (3.27) we have
$$
\frac{\partial {\Bbb{\goth F}}}{\partial a_0}=\int_{0}^{+\infty}
\frac{\partial \Lambda}{\partial a_0}dy>0,
\quad \frac{\partial {\Bbb{\goth F}}}{\partial \lambda}=\int_{0}^{+\infty}
\frac{\partial \Lambda}{\partial \lambda}dy<0,
\quad \frac{\partial {\Bbb{\goth F}}}{\partial a_0}=-\frac{1}{2} \qquad 
\forall (\lambda, a_0). \tag3.29
$$
From (3.28) - (3.29) we obtain
$$\frac{\partial \lambda}{\partial a_0}>0; \qquad \frac{\partial \lambda}
{\partial \omega _0}<0 \qquad 
\forall  a_0,\; \text {satisfying (*)}.\tag3.30$$

Thus we derive: {\it the function $\lambda (a_0, \omega_0)$ {\bf increases}
with respect to $a_0$ and {\bf decreases} with respect to $\omega_0$.}

Multiplying (3.11) by $\Phi(\omega)$ and integrating over 
$(-\frac{\omega_0}{2}, \frac{\omega_0}{2})$, we have
$$\multline
(1-\mu)\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^q (\lambda^2\Phi^2
+{\Phi'}^2)^{\frac{m-2}{2}}{\Phi'}^2d\omega
=-a_0\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{q+m}d\omega+\\
+[\lambda^2(m-1+q+\mu)+\lambda (2-m)]
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{q+2} 
(\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}d\omega \ge \\
\ge 
\langle \lambda^m(m-1+q+\mu) +\lambda^{m-1}(2-m) -a_0\rangle
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}
{2}}|\Phi|^{q+m}d\omega >0 \endmultline $$
by virtue of (*). This justifies the condition $0\le \mu< 1$ in (3.3).

\proclaim{Lemma 3.2} Let assumptions (3.3), (*) hold  and in addition
$$ q+\mu <1, \qquad if\quad a_0=0\tag3.31$$
$$(1-q-\mu)(\lambda m+2-m)>0,\qquad if\quad a_0>0. \tag3.32$$
Then we have
$$\int_{-\frac{\omega _0}{2}}^{\frac{\omega _0}{2}}|\Phi'|^{m}d\omega \le 
c(a_0,q,\mu,m,\lambda , \omega _0).\tag3.33$$
\endproclaim
\demo{Proof} Dividing (3.11) by $\Phi|\Phi|^{q-2},$ we have
$$\multline
\Phi\frac{d}{d \omega}[(\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}{\Phi}']+
\lambda[\lambda(q+m-1)-m+2]\Phi^2(\lambda^2 \Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}+\\
+q{\Phi'}^2(\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m-2}{2}}=a_0|\Phi|^{m}- 
\mu (\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m}{2}}.
\endmultline$$
We integrate to obtain
$$\multline
(q-1+\mu)\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}(\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m}{2}}d\omega+\lambda(\lambda m+2-m)
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\Phi^{2} (\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m-2}{2}}d\omega=\\
=a_0\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{m}d\omega.
\endmultline \tag3.34$$
At first let  $a_0=0.$ From assumptions and (*) it follows that
$$\lambda m+2-m>\lambda (1-q-\mu)>0,$$
and we have
$$(1-q-\mu)\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}(\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m}{2}}d\omega=\lambda(\lambda m+2-m)
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\Phi^{2} (\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m-2}{2}}d\omega.$$
Then, applying Young's inequality with $p=\frac{m}{m-2},\;
p'=\frac{m}{2}$, we get :
$$\multline
(1-q-\mu)\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}(\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m}{2}}d\omega \le \\
\le \eta\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}(\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m}{2}}d\omega+c_{\eta}\lambda^{\frac{m}{2}}(\lambda m+2-m)^{\frac{m}{2}}
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{m}d\omega \qquad \forall \eta>0.
\endmultline $$
From this, choosing $\eta=\frac{1}{2}(1-q-\mu)$ and taking into account
(3.16), (3.19), we obtain (3.33).

Now let $a_0>0.$ If $q+\mu<1,$ then (*) implies that $\lambda (\lambda m+2-m)>0$
and we can rewrite (3.34) in the following way
$$\multline
(1-q-\mu)\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}(\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m}{2}}d\omega+a_0\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{m}d\omega=\\
=\lambda(\lambda m+2-m)
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\Phi^{2} (\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m-2}{2}}d\omega.
\endmultline $$
If $q+\mu>1$ and $\lambda m+2-m<0,$ then we rewrite (3.34) as
$$\multline
(q-1+\mu)\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}(\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m}{2}}d\omega=\lambda(m-\lambda m-2)
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\Phi^{2} (\lambda^2\Phi^2+{\Phi'}^2)^
{\frac{m-2}{2}}d\omega+\\
+a_0\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{m}d\omega.
\endmultline $$
In  both cases we obtain (3.33) by applying Young's inequality
as above.

Finally, if $q+\mu\ge 1$ and $\lambda m+2-m\ge 0,$ then from (3.34) we have
$$(q-1+\mu)\lambda ^m\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^md\omega+\lambda ^{m-1}
(\lambda m+2-m)
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{m}d\omega \le 
 a_0\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi|^{m}d\omega,$$
what contradicts (*) by $\Phi\ge 0.$
 The lemma is proved.
\enddemo
From $(3.21)$ and $(3.7)$ we get the function 
$$w=(r\sqrt{q_{n-1}})^{\lambda}\Phi(\omega)$$
that will be {\it a barrier} of our boundary value problem  $(0.1)-(0.3).$
\proclaim{Lemma 3.3} Let assumptions of lemma 3.2 hold.
Then the function $\zeta(|x|)w(x)$ belongs to ${\Bbb{\goth N}}^{1}_{m,0,q}
(G^d_0),$ where $\zeta(r)\in C_0^{\infty}[0,d].$
\endproclaim
\demo{Proof} We must show that
$$I[w]\equiv \int_{G_0^d} (|x|^{-m}|w|^m+|\nabla w|^m+
\frac{|w|^{q+m}}{(x_{n-1}^2+x_n^2)^\frac{m}{2}})dx<\infty.\tag3.35 $$
Elementary calculations from (3.10) will give
$$
|\nabla w|^m=(r\sqrt{q_{n-1}})^{m(\lambda -1)}(\lambda^2\Phi^2
+{\Phi'}^2)^{m/2}.\tag3.36$$
By Lemma 3.1, we have
$$\multline
I[w]=\int_{G_0^d}\Bigl\{(r\sqrt{q_{n-1}})^{m(\lambda -1)}(\lambda^2\Phi^2+
{\Phi'}^2)^{\frac{m}{2}}+r^{m(\lambda -1)}q_{n-1}^{m\lambda /2}\Phi^m(\omega)+\\
+r^{m(\lambda -1)+q\lambda }q_{n-1}^{\frac{(m+q)\lambda -m}{2}}
\Phi^{m+q}(\omega)\Bigr\}dx\le \\
\le c(\lambda , m, \omega_0)\biggl \{\int_{0}^{d}r^{m\lambda 
+n-1-m}dr\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}
{2}}(\lambda^2\Phi^2+{\Phi'}^2)^{\frac{m}{2}}d\omega +\\
+\int_{0}^{d}r^{m\lambda +n-1-m}dr
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\Phi^md\omega
+\int_{0}^{d}r^{(m+q)\lambda
+n-1-m}dr\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\Phi^{q+m}d\omega \biggr \}.
\endmultline$$
By virtue of $m\le n$,  (3.19), (3.33) it is clear that
$I[w]$ is finite. 
Thus
$$I[w]\le c(m, \lambda ,q, n, \mu, a_0, \omega_0, d).$$ Thus,
Lemma 3.3 is proved. \hfill Q.E.D.
\enddemo

\noindent{\bf Example.} Let $m=2$ and consider the Dirichlet problem
  $$ \gather
- \frac{d}{dx_i}({|u|}^qu_{x_i})+
 \frac{a_0}{x_{n-1}^2+x_n^2}u|u|^q-
\mu u{|u|}^{q-2}{|\nabla u|}^{2}=0,
   \quad  x\in G_0, \\ 
  u(x)=0,\quad x\in\Gamma_0,\endgather 
$$
where $a_0\ge 0$, $0\le \mu <1$, $q\ge 0$. 
   
From (3.20), (3.21) we obtain
$$\lambda ={\sqrt {\frac{a_0}{1+q+\mu}+(\frac{\pi}{(1+q+\mu)\omega_0})^2}}\tag3.37$$
and
$$\Phi(\omega)=\biggl(\cos\frac{\pi\omega}{\omega_0}\biggr)^{\frac{1}{1+q+\mu}},
\quad \omega\in [-\omega_0/2,\omega_0/2] .\tag3.38$$
Thus the solution to the problem above is the function
   $$\gathered
   u(r,\omega)=r^{\lambda}(\sin \omega_1\dots\sin \omega_{n-2})^{\lambda}
   \biggl(\cos\frac{\pi\omega_{n-1}}{\omega_0}\biggr)^{\frac{1}{1+q+\mu}},  \\
(r, \omega)\in G_0, \quad \omega_{n-1}\in
[-\omega_0/2,\omega_0/2], \quad 0<\omega _0<\pi ,
  \endgathered  \tag3.39$$
where $\lambda $ is defined by (3.37).

Now we calculate
$$\Phi'(\omega)=-\frac{\pi}{(1+q+\mu)\omega_0}\biggl(\cos \frac{\pi\omega}{\omega_0}\biggr)^
{-\frac{q+\mu}{1+q+\mu}}\sin \frac{\pi \omega}{\omega_0}$$
and it is easy to observe that all properties  of
$\Phi(\omega)$ are fulfilled. Moreover, 
$$\multline
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}{\Phi'}(\omega)^2d\omega=(\frac{\pi}{(1+q+\mu)\omega_0})^2
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\biggl(\cos \frac{\pi\omega}{\omega_0}\biggr)^
{-\frac{2(q+\mu)}{1+q+\mu}}\sin ^2\frac{\pi \omega}{\omega_0}d\omega=\\
=\frac{2\pi}{(1+q+\mu)^2\omega_0}\int_{0}^{\frac{\pi}{2}}(\cos t)^
{-\frac{2(q+\mu)}{1+q+\mu}}\sin ^2tdt=\\
=\frac{2\pi}{(1+q+\mu)^2\omega_0}\int_{0}^{\frac{\pi}{2}}(\sin t)^
{-\frac{2(q+\mu)}{1+q+\mu}}\cos ^2tdt=\\
=\frac{\pi}{(1+q+\mu)^2\omega_0}\frac{ \Gamma (\frac{3}{2}) \Gamma \bigl
(\frac{1-q-\mu}{2(1+q+\mu)}\bigr )}{ \Gamma \bigl
(\frac{2+q+\mu}{1+q+\mu}\bigr )}
\endmultline\tag3.40$$
provided that $q+\mu <1$. This integral is {\bf non-convergent},
if $q+\mu \ge 1$. At the same time for $\forall q>0$ we have
$$\multline
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}|\Phi(\omega)|^q{{\Phi'}^2}(\omega)d\omega=(\frac{\pi}{(1+q+\mu)
\omega_0})^2
\int_{-\frac{\omega_0}{2}}^{\frac{\omega_0}{2}}\biggl(\cos\frac{\pi\omega}{\omega_0}\biggr)^
{-\frac{q+2\mu}{1+q+\mu}}\sin ^2\frac{\pi \omega}{\omega_0}d\omega=\\
=\frac{2\pi}{(1+q+\mu)^2\omega_0}\int_{0}^{\frac{\pi}{2}}(\sin t)^
{-\frac{q+2\mu}{1+q+\mu}}\cos ^2tdt=\\
=\frac{\pi}{(1+q+\mu)^2\omega_0}\frac{ \Gamma (\frac{3}{2}) \Gamma \bigl(\frac{1-\mu}{2(1+q+\mu)}\bigr )}{ \Gamma \bigl
(\frac{2+\frac{3}{2}q+\mu}{1+q+\mu}\bigr )},
\endmultline \tag3.41$$
since $\mu <1$.

\head 4. More precise definition of the H\"older exponent. \endhead

Now we return to the (0.1)-(0.5) with $\tau =0$. For its weak
solutions we make more precise the value of $\alpha$-H\"older
exponent established in the Theorem 2.1. To this end we use the
weak comparison principle ([7, \S 10.4], [13,\S 3.1]) and the
barrier function constructed in \S 3.

\proclaim{ Theorem 4.1} Let $u(x)$ be a weak solution of (0.1)-(0.5),
where
$$\tau=0,\quad 0\le \mu<1, \quad 2\le m\le n; \quad
\text{and}\; q+\mu <1,\;\text{if}\; a_0=0.  \tag4.1$$
Suppose that there exists a nonnegative constant $k_1$ such
that 
$$\biggl |f_0(x)-\sum_{i=1}^n \frac{\partial f_i(x)}{\partial x_i}\biggr 
|\le k_1|x|^{\beta},\tag4.2 $$
where
$$\beta >\lambda (q+m-1)-m,\tag4.3$$
$\lambda$ is the positive solution of (3.21) with $\omega_0 \in (0,\pi)$ such 
that (*) is fulfilled and 
$$(1-q-\mu)(m\lambda +2-m)>0,\quad 
\text{if}\; a_0>0. \tag4.4$$
Then  $\forall \varepsilon >0$ there exists a constant $c_{\varepsilon}
>0$, 
depending only on $\varepsilon $, $n$, $m$,  $\mu$, $q,$
$a_0$, $\omega_0$ such that 
$$|u(x)|\le c_{\varepsilon}|x|^{\lambda -\varepsilon}.\tag4.5$$
\endproclaim


Before proving the theorem, we make some transformations and additional 
investigations. We make the change of variables
$$u=v|v|^{t-1},\quad t=\frac{m-1}{q+m-1}. \tag4.6$$
As result,  (0.1)-(0.2) takes the form
$$\gathered
{\goth M}_0v(x)=F(x), \quad x\in G\,,\\
 v(x)=0, \quad x\in \partial G,\endgathered \tag4.7
$$
where
$${\goth M}_0v(x)\equiv -\frac{d}{dx_i}({|\nabla v|}^{m-2}v_{x_i})+
 \frac{\overline a_0}{(x_{n-1}^2+x_n^2)^\frac{m}{2}}v|v|^{m-2}-
\overline {\mu} v^{-1}{|\nabla v|}^m,\tag4.8$$
$$\overline a_0=t^{1-m}a_0, \quad \overline {\mu}=t\mu ,\quad
F(x)=t^{1-m}\biggl (f_0(x)-\sum_{i=1}^n 
\frac{\partial f_i(x)}{\partial x_i}\biggr).\tag4.9$$
Now 
$$ \gathered
\overline w=(r\sqrt {q_{n-1}})^{\overline {\lambda}}\overline { \Phi}(\omega),\\
\overline {\lambda} =\frac{1}{t}\lambda , \quad \overline
{ \Phi}(\omega)= \Phi^{1/t}(\omega)
\endgathered \tag4.10$$
plays the role of barrier function.
Because of (3.11)-(3.14) and (3.21), it is easy to verify that 
$(\overline {\lambda}, \overline { \Phi}(\omega))$ is the solution to
   $$\gather
   \frac{d}{d \omega}[(\overline {\lambda}^2
          \overline {\Phi}^2+{\overline
{\Phi}'}^2)^{\frac{m-2}{2}}{\overline {\Phi}}']+
    \overline {\lambda}[\overline {\lambda}(m-1)-m+2]\overline {\Phi}
    (\overline {\lambda}^2
  \overline {\Phi}^2+{\overline {\Phi}'}^2)^{\frac{m-2}{2}}= \\
 =\overline {a}_0\overline {\Phi}|\overline {\Phi}|^{m-2}- \overline {\mu}
\frac{1}{\overline {\Phi}}(\overline {\lambda}^2\overline
{\Phi}^2+{\overline {\Phi}'}^2)^{\frac{m}{2}},
\quad \omega\in (-\omega_0/2,\omega_0/2),    \tag 4.11\\
\overline {\Phi}(-\omega_0/2)=\overline {\Phi}(\omega_0/2)=0, \tag 4.12\\
{\int_{0}^{+\infty}}\frac{[(m-1)y^2+\overline
{\lambda}^2](y^2+\overline {\lambda}^2)^{\frac{m-4}{2}} }
        {(m-1+\overline {\mu})(y^2+\overline {\lambda}^2)^{\frac{m}{2}}+
 \overline {\lambda}(2-m)(y^2+\overline {\lambda}^2)^{\frac{m-2}{2}}-
 \overline {a}_0} \,dy =\frac{\omega_0}{2}. \tag 4.13 \endgather
$$
It is obvious that the properties of $(\lambda ,\Phi)$, established in \S~3 
remain valid for $(\overline {\lambda}, \overline { \Phi}(\omega))$.
In particular the (*) becomes 
$$
 P_m(\overline {\lambda})\equiv (m-1+\overline {\mu})\overline {\lambda}^m 
 + (2-m)\overline {\lambda}^{m-1}-\overline {a}_0>0. \tag{$\overline *$}
$$
Now we consider a perturbation of the (4.11)-(4.13). Namely, 
$\forall \varepsilon \in (0,\pi -\omega_0)$ we consider on the
segment $[-\frac{\omega_0+\varepsilon}{2},\frac{\omega_0+\varepsilon}{2}]$ the
problem for $(\lambda _{\varepsilon},\Phi_{\varepsilon}):$
   $$\gather 
   \frac{d}{d \omega}[(\lambda _{\varepsilon}^2
          \Phi_{\varepsilon}^2+\Phi_{\varepsilon}^{'2})^{\frac{m-2}{2}}
          {\Phi}'_{\varepsilon}]+
    \lambda_{\varepsilon}[\lambda_{\varepsilon}(m-1)-m+2]\Phi_{\varepsilon}
    (\lambda_{\varepsilon}^2
               \Phi_{\varepsilon}^2+\Phi_{\varepsilon}^{'2})^{\frac{m-2}{2}}=\\
 =(\overline {a}_0-\varepsilon)\Phi_{\varepsilon}|\Phi_{\varepsilon}|^{m-2}-
 \overline {\mu}
\frac{1}{\Phi_{\varepsilon}}(\lambda_{\varepsilon}^2\Phi_{\varepsilon}^2+
\Phi_{\varepsilon}^{'2})^{\frac{m}{2}},
\quad \omega\in (-\frac{\omega_0+\varepsilon}{2},\frac{\omega_0+\varepsilon}{2}),  
\tag{$4.11_\varepsilon$} \\
\Phi(-\frac{\omega_0+\varepsilon}{2})=\Phi(\frac{\omega_0+\varepsilon}{2})=0,
\tag{$4.12_\varepsilon$} \\
{\int_{0}^{+\infty}}\frac{[(m-1)y^2+\lambda_{\varepsilon}^2](y^2+
\lambda_{\varepsilon}^2)^{\frac{m-4}{2}} }
        {(m-1+\overline {\mu})(y^2+\lambda_{\varepsilon}^2)^{\frac{m}{2}}+
 \lambda_{\varepsilon}(2-m)(y^2+\lambda_{\varepsilon}^2)^{\frac{m-2}{2}}
 +\varepsilon - \overline {a}_0}\,dy
 =\frac{\omega_0+\varepsilon}{2}.\tag{$4.13_\varepsilon$}
  \endgather    
$$
  The $(4.11_\varepsilon)$-$(4.13_\varepsilon)$ is obtained from
the (4.11)-(4.13) by change in the last $\omega_0$ by
$\omega_0+\varepsilon$ and $\overline {a}_0$ by $\overline {a}_0-\varepsilon $.
From the monotonicity properties of  
$\overline {\lambda}(\omega_0,\overline {a}_0)$
established in \S 3 (see (3.30)), we obtain
$$0<\lambda _{\varepsilon}<\overline {\lambda}, \quad \lim_{\varepsilon
\to +0}\lambda _{\varepsilon}=\overline {\lambda}.\tag4.14 $$ 

Now we establish lower bounds for the functions $q_{n-1}$  and
$\Phi_{\varepsilon}(\omega)$. 

\proclaim{Corollary 4.2}  From Lemma 3.1 it follows that
$$\sqrt {q_{n-1}}^{(m-1)\lambda _{\varepsilon}-m}\ge \varkappa _0=
\min {\biggl \{1;
\biggl (\cos\frac{\omega_0}{2}}\biggr)^{(m-1)\lambda _{\varepsilon}-m}\biggr \}.\tag4.15$$
\endproclaim

\proclaim{Lemma 4.3} There exists $\varepsilon^*>0$ such that
$$\Phi_{\varepsilon}\biggl (\frac{\omega_0}{2}\biggr )\ge \frac{\varepsilon}{\omega_0+\varepsilon} \qquad 
\forall \varepsilon \in (0,\varepsilon ^*). \tag4.16$$
\endproclaim

\demo{Proof} 
We turn to the $(\overline *):$ $P_m(\overline {\lambda})>0$. Since
$P_m(\overline {\lambda})$ is a polynomial, by continuity, 
there exists a $\delta ^*$-neighborhood of $\overline {\lambda}$,
in which $(\overline *)$ is satisfied as before, i.e. there
exists $\delta ^*>0$ such that $P_m(\lambda )>0$ for 
$\forall \lambda$ such that $|\lambda - \overline {\lambda}|<\delta ^*$. 
We choose the number $\delta ^*>0$
in the such way; then 
$$P_m(\overline {\lambda} -\delta)>0 \quad \forall \delta\in(0,\delta ^*)\,.
\tag4.17$$

Recall that $\overline {\lambda}$ solves the (4.13). By
(4.14), now  for every $\delta\in(0,\delta ^*)$ we can put
$$\lambda _{\varepsilon}=\overline {\lambda}-\delta$$
and solve  $(4.13_{\varepsilon})$ together with this 
$\lambda _{\varepsilon}$ with respect to $\varepsilon ;$ let $\varepsilon (\delta )>0$ be
obtained solution. Since (4.14) is true,
$$\lim_{\delta \to +0}\varepsilon (\delta)=+0.$$
Thus we have the sequence of problems 
$(4.11_\varepsilon)$-$(4.13_\varepsilon)$ with respect to 
$$(\lambda _{\varepsilon},\Phi_{\varepsilon}(\omega))\quad 
\forall \varepsilon \; \bigl|\; 0<\varepsilon < \min
(\varepsilon (\delta);\pi -\omega_0)=\varepsilon ^*(\delta),\quad
\forall \delta\in (0,\delta ^*).\tag4.18$$ 
We consider $\Phi_{\varepsilon}(\omega)$ with $\forall \varepsilon$
from (4.18).  In the same way as (3.18) we verify that
$$\Phi''_{\varepsilon}(\omega)<0 \quad \forall \omega\in \biggl [-\frac{\omega_0+\varepsilon}{2},
\frac{\omega_0+\varepsilon}{2}\biggr ].$$
But this inequality means that the function
$\Phi_{\varepsilon}(\omega)$ is convex on $[-\frac{\omega_0+\varepsilon}{2},
\frac{\omega_0+\varepsilon}{2}]$, i.e. 
$$\multline
\Phi_{\varepsilon}(\alpha_1\omega_1+\alpha_2\omega_2)\ge \alpha_1\Phi_{\varepsilon}(\omega_1)+
\alpha_2\Phi_{\varepsilon}(\omega_2)\quad \forall \omega_1, \omega_2\in \biggl [-\frac{\omega_0+
\varepsilon}{2},\frac{\omega_0+\varepsilon}{2}\biggr ];\\ 
\alpha_1\ge 0,\;\alpha_2\ge 0\;\bigl |\alpha_1+\alpha_2=1.
\endmultline\tag4.19$$
We put
$\alpha_1=\frac{\omega_0}{\varepsilon +\omega_0}$, 
$\alpha_2=\frac{\varepsilon}{\varepsilon +\omega_0}$, 
$\omega_1=\frac{\varepsilon +\omega_0}{2}$, $\omega_2=0$. Then
$(4.12_\varepsilon)$ we obtain
$$\Phi_{\varepsilon}\biggl (\frac{\omega_0}{2}\biggr )\ge \frac{\varepsilon}{\omega_0+
\varepsilon}\Phi_{\varepsilon}(0)=\frac{\varepsilon}{\omega_0+
\varepsilon}, $$
and this lemma is proved. \hfill Q.E.D.
\enddemo

\proclaim{Corollary 4.4} 
$$\frac{\varepsilon}{\omega_0+\varepsilon}\le \Phi_{\varepsilon}(\omega)\le 1
\quad \forall \omega\in [-\omega_0/2,\omega_0/2];\quad \forall \varepsilon
\in (0,\varepsilon ^*).\tag4.20$$
\endproclaim

\demo{Proof of the Theorem 4.1}
Let $(\lambda _{\varepsilon},\Phi_{\varepsilon}(\omega))$ be the solution of
$(4.11_\varepsilon)$-$(4.13_\varepsilon)$, with $\varepsilon
\in (0,\varepsilon ^*)$, where $\varepsilon ^*$ is defined by (4.18).
We set
$$w_{\varepsilon}(r,\omega)=(r\sqrt {q_{n-1}})^{\lambda _{\varepsilon}}
\Phi_{\varepsilon}(\omega),\quad \omega\in [-\omega_0/2,\omega_0/2].$$
Elementary calculations show that
$$\gather 
w_{\varepsilon}\bigl |_{\Gamma_0}\ge 0, \tag4.21 \\
w_{\varepsilon}\bigl |_{\Omega_d}\ge \frac{\varepsilon}{\omega_0+\varepsilon}
\biggl(d\cos\frac{\omega_0}{2}\biggr )^{\overline \lambda } \tag4.22\endgather
$$
 (by virtue of (4.14), Lemma 3.1 and (4.20)). Then for (4.8)-(4.9) we obtain  
$$\multline
{\goth M}_0w_{\varepsilon}(r,\omega)=(r\sqrt {q_{n-1}})^{(m-1)\lambda _
{\varepsilon}-m}\biggl \{
-\frac{d}{d \omega}[(\lambda _{\varepsilon}^2
          \Phi_{\varepsilon}^2+{\Phi'}_{\varepsilon}^2)^{\frac{m-2}{2}}
          {\Phi}'_{\varepsilon}]-\\
   - \lambda_{\varepsilon}[\lambda_{\varepsilon}(m-1)-m+2]\Phi_{\varepsilon}
    (\lambda_{\varepsilon}^2
               \Phi_{\varepsilon}^2+{\Phi'}_{\varepsilon}^2)^{\frac{m-2}{2}}+
\overline {a}_0\Phi_{\varepsilon}^{m-1}-\\
- \overline {\mu}
\frac{1}{\Phi_{\varepsilon}}(\lambda_{\varepsilon}^2\Phi_{\varepsilon}^2+
{\Phi'}_{\varepsilon}^2)^{\frac{m}{2}}\biggr \}=\varepsilon (r\sqrt {q_{n-1}})^{(m-1)
\lambda _{\varepsilon}-m}\Phi_{\varepsilon}^{m-1}
\endmultline$$
by virtue of $(4.11_\varepsilon)$.
From (4.14) and Lemma 3.1 we have
$$\sqrt {q_{n-1}}^{(m-1)\lambda _{\varepsilon}-m}\ge \overline
{\varkappa } _0=\min {\biggl \{1;
\biggl (\cos\frac{\omega_0}{2}}\biggr)^{(m-1)\overline {\lambda }-m}\biggr \}.\tag4.23$$
Taking into account (4.20), we get
$${\goth M}_0w_{\varepsilon}(r,\omega)\ge \frac{\varepsilon ^m\overline
{\varkappa } _0}{(\omega_0+\varepsilon)^{m-1}}r^{(m-1)\lambda _
{\varepsilon}-m}.\tag4.24$$

Now we use the weak comparison principle for (4.7)-(4.9). By
definition of weak solution 
$$\gather 
Q(v,\phi)\equiv \int_{G}\biggl \{|\nabla v|^{m-2}v_{x_i}\phi_{x_i}+
\frac{\overline {a}_0}{(x_{n-1}^2+x_n^2)^\frac{m}{2}}v|v|^{m-2}\phi-\\
-\overline {\mu }v^{-1}|\nabla v|^m\phi -F(x)\phi \biggr \}dx=0 \quad
\forall \phi(x)\in {\goth N}^1_{m, 0, 0}(G). 
\tag 4.25
\endgather$$
Now let $\phi(x)\in {\goth N}^1_{m, 0, 0}(G)$ be
such that
$$\phi (x)\ge 0\quad \forall x\in \overline {G}; \qquad
\phi (x)=0\quad \forall x\in G\setminus G_0^d.$$
Then  $\forall A>0$,
$$\multline
Q(Aw_{\varepsilon},\phi)=\int_{G_0^d} \phi (x)\biggl ({\goth M}_0(Aw_{\varepsilon})
-F(x) \biggr )dx\\
=\int_{G_0^d} \phi (x)\biggl (A^{m-1}{\goth M}_0
w_{\varepsilon}(x)-F(x) \biggr )dx \\
\geq \int_{G_0^d} \phi (x)\biggl \{\biggl (\frac{A}{\varepsilon
+\omega_0}\biggr )^{m-1} \overline {\varkappa}_0\varepsilon ^m
|x|^{(m-1)\lambda _{\varepsilon}-m}-k_1t^{1-m}|x|^{\beta}\biggr \}dx\\
\geq  
\biggl \{\biggl (\frac{A}{\varepsilon
+\omega_0}\biggr )^{m-1} \overline {\varkappa}_0\varepsilon
^m-k_1\biggl (\frac{m-1+q}{m-1}\biggr )^{m-1}\biggr \}
\int_{G_0^d} \phi (x)|x|^{(m-1)\lambda _{\varepsilon}-m}dx\ge 0,
\endmultline\tag4.26
$$
if $A>0$ is chosen sufficiently large,
$$A\ge \frac{(m-1+q)(\varepsilon +\omega_0)}{\varepsilon (m-1)}\biggl (\frac{k_1}
{\overline {\varkappa}_0\varepsilon}\biggr )^{\frac{1}{m-1}}.\tag4.27$$

To obtain the first inequality of (4.26) we used (4.2), (4.9), and (4.24).
And for the second inequality, we used (4.3), (4.6), (4.10), and (4.14). 

Furthermore, by the theorem 2.1, $v(x)\biggl |_{\Omega_d}\le c_0d^{\alpha}$;
therefore by (4.22),
$$Aw_{\varepsilon}\biggl |_{\Omega_d}\ge \frac{A\varepsilon}{\omega_0+\varepsilon}
\biggl (d\cos\frac{\omega_0}{2}\biggr )^{\overline \lambda }\ge v(x)\biggl |_{\Omega_d},
 \tag4.28$$
if  $A>0$ is chosen such that
$$A\ge \frac{c_0(\varepsilon +\omega_0)}{\varepsilon \biggl (
\cos\frac{\omega_0}{2}\biggr )^{\overline \lambda }}d^{\alpha - \overline \lambda }.\tag4.29$$

Thus, if $A>0$ is chosen according to  (4.27), (4.29), then from
(4.25), (4.26), (4.28), (4.21) and (4.7) we obtain:
$$ \gather
Q(Aw_{\varepsilon},\phi)\ge 0, \quad Q(v,\phi)=0 \quad \text
{in}\quad G^d_0;\\
Aw_{\varepsilon}\biggl |_{\partial G_0^d}\ge v\biggl |_{\partial G_0^d}.
\endgather$$
It is easy to verify that  rest of the conditions of the weak
comparison principle are fulfilled. By this principle we obtain
$$v(x)\le Aw_{\varepsilon}(x),\quad \forall x\in \overline {G_0^d}.$$
Similarly we can prove that
$$v(x)\ge -Aw_{\varepsilon}(x),\quad \forall x\in \overline {G_0^d}.$$
Thus, finally, we have
$$|v(x)|\le Aw_{\varepsilon}(x)\le A|x|^{\lambda _{\varepsilon}},\quad 
\forall x\in \overline {G_0^d}.\tag4.30$$
Resubstituting the old variables, by (4.6), (4.10) we obtain from (4.30)
the required bound (4.5). Therefore, Theorem 4.1 is proved.
\enddemo



\head References \endhead
        
\item{1.} {\it Babuska J., Kellogg R.B.} Numerical solution of the neutron 
diffusion equation in the presence of corners and interfaces, in: "Numerical 
reactor calculations", Intern. atomic energy agency, Vienna, 1972, pp. 
473-486.

\item{2.} {\it Benilan Ph., Br\'ezis H.} Some variational problems of the Thomas -
Fermi type, in: "Variotional inequalities" {\it (Cottle, Gianessi, Lions, 
Eds.)}, Wiley, NY, 1980, pp. 53-73.

\item{3.} {\it Cirmi G.R., Porzio M.M.} $L^\infty -$solutions for
some nonlinear degenerate elliptic and parabolic equations., Ann. mat. pura 
ed appl. (IV). 1995. Vol. {\bf CLXIX}. P. 67-86.

\item{4.} {\it Dobrowolski M.} Numerical approximation of elliptic interface 
and corner problems, Habilitationsschrift, Universit\"at Bonn, 1981 (see also:
Z.A.A.M. 1984. V. {\bf 64}, pp. 270-271.)

\item{5.} {\it Dobrowolski M.} On quasilinear elliptic equations in domains 
with conical boundary points, J. reine und angew. Math. 1989.
Vol. {\bf 394}. P. 186-195.

\item{6.} {\it Dr\'abek P., Kufner A., Nicolosi F.} Quasilinear elliptic 
equations with degenerations and singularities. Walter de Gruyter. Berlin, NY,
1997.

\item{7.} {\it Gilbarg D., ’rudinger N.} Elliptic partial differential
equations of second order, 2nd ed. Revised third printing. Springer, 1998.

\item{8.} {\it Jackson J.D.} Classical Electrodynamics, John Wiley \& 
Sons, NY, 1975.

\item{9.} {\it Kinderlehrer D., Stampacchia G.,} An introduction
to variational inequalities and their applications. 1980.
Academic Press. N-Y, London.

\item{10.} {\it Ladyzhenskaya O.A., Ural'tseva N.N.} Linear and
quasilinear elliptic equations, 2nd ed. Moscow. 1973 (Russian).

\item{11.} {\it Sommerfeld A.} Asymptotische Integration der 
Differentialgleichung der Thomas - Fermischen atoms, Z. f\"ur 
Physik. 1932. V. {\bf 78}. P. 283-308.


\item{12.} {\it Strong G. {\rm \&} Fix J.} 
An analysis of the finite element method, Prentice-Hall, 
Englewood Cliffs, 1973.

\item{13.} {\it Tolksdorf P.} On the Dirichlet problem for quasilinear
elliptic equations in domains with conical boundary points, Comm. Part. 
Different. Equns. 1983. Vol. {\bf 8}, N7. P. 773-817.

\item{14.} {\it Borsuk M.V.} Estimates of solutions to the Dirichlet problem
for second-order nondivergent elliptic   equations in a neighbourhood
of a conic point of the boundary , Differ. Equations. 1994. V.{\bf
30}, No. 1. P. 94-99.

\item{15.} {\it Borsuk M.V.} Estimates of generalized solutions to the 
Dirichlet problem for second-order quasilinear  elliptic equations in a
domain with a conic point on the boundary , Differ. Equations. 1995. V.{\bf 31}
, No. 6. P. 936-941.

\item{16.} {\it Borsuk M.V., Dobrowolski M.} On the behavior of
solutions of the Dirichlet problem for a class of degenerate
elliptic equations in the neighborhood of conical boundary
points, Nonlinear boundary value problems. Kiev (to appear).

\item{17.} {\it Borsuk M.V., Plesha M.I.} Estimates of generalized solutions 
of the Dirichlet problem for quasilinear elliptic second-order equations in a 
domain with conical boundary point, Ukrainer  Math. journal. 1998.
{\bf V. 50}, No. 10. P. 1299-1309 (Ukrainian).

\item{18.} {\it Borsuk M., Portnyagin D.} Barriers on cones for
degenerate quasilinear elliptic operators, Electron. J. Diff.
Eqns., Vol. 1998 (1998), {\bf No. 11}, pp. 1-8.

\filbreak


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