# Large and very singular solutions to semilinear elliptic equations

Andrey E. Shishkov(1) and Yevgeniia A. Yevgenieva(2)
(1) RUDN University, Russian Federation
(2) IAMM of NAS of Ukraine, Ukraine

###### Abstract

We consider equation $-\Delta u+f(x,u)=0$ in smooth bounded domain
$\Omega\in\mathbb{R}^{N}$, $N\geqslant 2$, with $f(x,r)>0$ in
$\Omega\times\mathbb{R}^{1}_{+}$ and $f(x,r)=0$ on $\partial\Omega$. We find
the condition on the character of degeneration of $f(x,r)$ near
$\partial\Omega$, which is a criteria of the existence-nonexistence of a very
singular solution with a strong point singularity on $\partial\Omega$.
Moreover we prove that the mentioned condition is a sufficient condition for
the uniqueness of a large solution and conjecture that this condition is also
a necessary condition of the uniqueness.

##  1 Introduction and main results

This paper deals with two problems:

1) the uniqueness of large solutions,

2) the existence of very singular solutions

to a semilinear elliptic equation of the form:

$-\Delta u+f(x,u)=0\quad\text{in }\Omega\in\mathbb{R}^{N},\ N>1,$ (1.1)

where nonlinear absorption term $f(x,s)>0$ $\forall\,x\in\Omega$,
$\forall\,s>0$, degenerates on $\partial\Omega$:

$f(x,s)=0\quad\forall\,x\in\partial\Omega,\qquad
f(x,0)=0\quad\forall\,x\in\overline{\Omega}.$ (1.2)

When $f(s)$ is monotonic the existence of large solution, i.e. a solution of
equation (1.1) satisfying boundary condition:

$\lim_{d(x)\to 0}u(x)=\infty,\quad d(x):=dist(x,\partial\Omega),$ (1.3)

is assosiated with a well known Keller–Osserman [11, 12] condition on the
growth of $f(s)$ as $s\to\infty$. An adaptation of KO-condition to
nonmonotonic $f(s)$ was realized in [15], to general nonlinearities $f(x,s)$ —
in [16]. A generalization of KO-condition for higher order semilinear
equations and inequalities was introduced in [17]. The uniqueness of large
solutions was firstly proved by C. Loewner and L. Nirenberg in [13] for smooth
domain $\Omega$ and $f(s)=s^{p}$, $p=\frac{N+2}{N-2}$. A first general result
about the uniqueness was obtained by C. Bandle and M. Marcus [14] for smooth
bounded domain $\Omega$ and $f(s)=s^{p}$, $p>1$. Asymptotic methods,
introduced in [14], was applied to different classes of nonlinearities
$f(x,s)$ by many authors (see [18] and references therein).

It became clear that the uniqueness of the large solution mostly depends on
the character of degeneration of nonlinearity $f(x,s)$ on the boundary of
$\Omega$. So, in [10] the uniqueness was proved for $C^{2}$–smooth bounded
domain when:

$f(x,s)\geqslant c_{0}d(x)^{\alpha}s^{p}\quad\forall\,x\in\Omega,\
\forall\,s\geqslant 0,\ p>1,\ \alpha>0,\ c_{0}=const>0,$

where $d(x)$ is from (1.3). In [18] authors conjectured that for the
uniqueness the following condition should hold:

$f(x,s)\geqslant
c_{0}\exp\left(-\frac{c_{1}}{d(x)^{\alpha}}\right)s^{p}\quad\forall\,x\in\Omega,\
\forall\,s\geqslant 0,\ 0<\alpha<1,\ c_{1}=const\geqslant 0.$

In this context our first main result is the following. In bounded domain
$\Omega\in\mathbb{R}^{N}$ with $C^{2}$–smooth boundary $\partial\Omega$ we
consider the semilinear equation:

$Lu+H(x)u^{p}:=-\sum_{i,j=1}^{N}\left(a_{ij}(x)u_{x_{i}}\right)_{x_{j}}+H(x)u^{p}=0\quad\text{in
}\Omega,\ p>1,$ (1.4)

where $C^{1,\lambda}$–smooth functions $a_{ij}(\cdot)$ satisfy the ellipticity
condition:

$d_{1}|\xi|^{2}\geqslant\sum_{i,j=1}^{N}a_{ij}(x)\xi_{i}\xi_{j}\geqslant
d_{0}|\xi|^{2}\quad\forall\,\xi\in\mathbb{R}^{N},\
\forall\,x\in\overline{\Omega},\ d_{1}<\infty,\ d_{0}>0,$ (1.5)

and absorption potential $H(\cdot)$ satisfies

$H(x)\geqslant h_{\omega}(d(x))\quad\forall\,x\in\overline{\Omega},\quad
h_{\omega}(s)=:\exp\left(-\frac{\omega(s)}{s}\right)\quad\forall\,s\in(0,\rho_{0}).$
(1.6)

######  Theorem 1.1.

Let potential $H(x)$ satisfies estimate (1.6), where nondecreasing continuous
function $\omega(\cdot)$ satisfies the technical condition:

$s^{\gamma_{1}}\leqslant\omega(s)<\omega_{0}=const<\infty\quad\,s\in(0,\rho_{0}),\
0<\gamma_{1}<1$ (1.7)

and the Dini condition

$\int_{0}^{c}\frac{\omega(s)}{s}ds<\infty;$ (1.8)

Then equation (1.4) admit only one large solution in mentioned domain
$\Omega$.

######  Remark 1.1.

We conjecture that Dini condition (1.8) is also a necessary condition for the
uniqueness of the large solution.

As the indirect confirmation of the validity of the above conjecture we
consider the second main result of this paper about the necessity of Dini
condition (1.8) for the existence of a very singular (v.s.) solution to
equations of structure (1.4). Let us remind that a v.s. solution was firstly
discovered as a nonegative solution of semilinear parabolic equation:

$u_{t}-\Delta u+hu^{p}=0\quad\text{in
}\mathbb{R}^{1}_{+}\times\mathbb{R}^{N},\ 1<p<1+2N^{-1},\ h=const>0,$ (1.9)

satisfying initial condition:

$u(0,x)=\infty\delta(x),\quad\delta(x)\text{ is Dirac measure},$ (1.10)

in the following sense: $u(0,x)=0$ $\forall\,x:|x|\neq 0$, and

$\lim_{t\to 0}\int_{R^{N}}u_{\infty}(t,x)dx=\infty.$

Moreover v.s. solution $u_{\infty}(t,x)$ can be obtained as
$\lim_{k\to\infty}u_{k}(t,x)$, where $u_{k}(t,x)$ is solution of (1.9)
satisfying initial condition $u_{k}(0,x)=k\delta(x)$ (see [19, 20, 21] and
references therein). The next stage was the study of v.s. solutions to problem
(1.9), (1.10) with variable absorption potential $h=h(t,x)\geqslant 0$,
degenerating on initial hyperplane:

$h(0,x)=0\quad\forall\,x\in\mathbb{R}^{N}.$ (1.11)

A new phenomenon was observed in [22]: if $h=h(t)=\exp\left(-\omega
t^{-1}\right)$, $\omega=const>0$, then
$u_{\infty}(t,x)=\lim_{k\to\infty}u_{k}(t,x)$ is not a v.s. solution, but a
large solution, namely, $u_{\infty}(0,x)=\infty$
$\forall\,x\in\mathbb{R}^{N}$. In [23] it was found a sharp condition on the
degeneration of $h(t,x)$ which guarantees the existence of v.s. solution
$u_{\infty}(t,x)$ with strong point singularity:
$h(t)\geqslant\exp\left(-\omega(t)t^{-1}\right)$, where $\omega(\cdot)$ is
continuous nondecreasing function, satisfying the following Dini condition:

$\int_{0}^{c}s^{-1}\omega(s)^{\frac{1}{2}}<\infty.$ (1.12)

Up to now we do not know whether this condition is also necessary condition
for the existence of the v.s. solution to semilinear parabolic equations of
structure (1.9). But the study of v.s. solution in elliptic case, i.e. a v.s.
soltion to equations of structure (1.4), is more advanced in terms of the
corresponding Dini condition (1.8). Namely, in [6] the following result about
sufficiency was proved. Let $\\{u_{k}(x)\\}$ be a sequence of solutions of
equation (1.4), (1.5), (1.6), satisfying boundary condition:

$u_{k}=k\delta_{a}(x),\quad\text{on }\partial\Omega,\ a\in\partial\Omega,\
k=1,2,...$ (1.13)

Let potential $H(x)$ satisfies estimate (1.6), where nonnegative function
$\omega(s)$ satisfies all conditions of Theorem 1.1. Then
$u_{\infty}(x)=\lim_{k\to\infty}u_{k}(x)$ is a v.s. solution of (1.4) with the
strong boundary singularity at $a\in\partial\Omega$ and $\lim_{x\to y}u(x)=0$
$\forall\,y\in\partial\Omega\setminus\\{a\\}$.

Let us consider the following model problem:

$-\Delta u+h_{\omega}(|x^{\prime}|)u^{p}=0\quad\text{in
}\Omega:=\mathbb{R}^{N}_{+}=\\{x\in\mathbb{R}^{N},x_{N}>0\\},$ (1.14)
$u|_{x_{N}=0}=K\delta_{a}(x^{\prime\prime}),\ a\in
L:=\\{x\in\partial\Omega:|x^{\prime}|=0\\},$ (1.15)

where $N\geqslant 2$, $p>1$, $0<K<\infty$, $x^{\prime}=(x_{2},...,x_{N})$,
$x^{\prime\prime}=(x_{1},...,x_{N-1})$, $\delta_{a}(x^{\prime\prime})$ is
Dirac measure on $\partial\Omega$ with $supp\ \delta_{a}=\\{a\\}$;

$h_{\omega}(s)=\exp\left(-\frac{\omega(s)}{s}\right)\quad\forall\,s\geqslant
0.$ (1.16)

Here function $\omega(\cdot)$ satisfies the following conditions:

$\begin{split}\text{(i)}&\quad\omega\in C(0,\infty)\text{ is a positive
nondecreasing function},\\\ \text{(ii)}&\quad
s\to\mu(s):=\frac{\omega(s)}{s}\text{ is monotonically decreasing on
}\mathbb{R}^{1}_{+},\\\ \text{(iii)}&\quad\lim_{s\to
0}\mu(s)=\infty.\end{split}$ (1.17)

Thus $h_{\omega}(|x^{\prime}|)$ is the absorption potential of equation (1.14)
which degenerates on line $L$ from (1.15). If

$P_{0}(x,z)=c_{N}x_{N}|x-z|^{-N},\
c_{N}=\pi^{-\frac{N}{2}}\Gamma\left(\frac{N}{2}\right),$ (1.18)

is the Poisson kernel for $-\Delta$ in $\mathbb{R}^{N}_{+}$, then (see [1])
inequality

$\int_{\\{|x|<R,x_{N}>0\\}}h_{\omega}(|x^{\prime}|)P_{0}(x,a)^{p}x_{N}dx<\infty\quad\forall\,R:0<R<\infty$
(1.19)

guarantees the existence of a unique solution of problem (1.14), (1.15)
dominated by supersolution $KP_{0}(x,a)$. Thus, if condition (1.19) holds,
then for arbitrary monotonically increasing sequence

$\left\\{K_{j}\right\\},\ K_{j}\to\infty\text{ as }j\to\infty$ (1.20)

there exists a monotonically nondecreasing (due to the comparison principle)
sequence of solutions $u_{j}(x)$ of problem (1.14), (1.15) with $K=K_{j}$.
Moreover, since $h_{\omega}(|x^{\prime}|)$ is a positive function in
$\overline{\Omega}\setminus L$, equation (1.14) possesses maximal solution $U$
in $\Omega$, which is a large solution (see [3]):

$\lim_{x_{N}\to 0,|x|<M}U(x)=\infty\quad\forall\,M>0.$ (1.21)

Since $u_{j}(x)\leqslant U$ $\forall\,x\in\Omega$ $\forall\,j\in\mathbb{N}$,
the mentioned sequence converges to some function $u_{\infty}$, which is a
positive solution of (1.14).

######  Theorem 1.2.

Let parameter $p$ in equation (1.14) additionally satisfies

$1<p<1+\frac{2}{N-1}:=p_{0}$ (1.22)

and $\\{u_{j}(x)\\}$ be a sequence of solutions of problem (1.14), (1.15),
corresponding to $K=K_{j}$ from (1.20). Assume that functions $\omega(s)$,
$\mu(s)$ satisfy conditions (1.17) and

$\limsup_{j\to\infty}\mu\left(2^{-j+1}\right)\mu\left(2^{-j}\right)^{-1}<1;\quad\omega(s)\to
0\text{ as }s\to 0.$ (1.23)

Let Dini condition (1.8) does not hold:

$\int_{0}^{1}s^{-1}\omega(s)ds=\infty,$ (1.24)

then $u_{\infty}(x):=\lim_{j\to\infty}u_{j}(x)$ is a solution of (1.14) which
satisfies

$u_{\infty}|_{\partial\Omega\setminus L}=0,\quad
u_{\infty}|_{\partial\Omega\cap L}=\infty.$ (1.25)

######  Remark 1.2.

Let us additionally consider the following problem:

$-\Delta u+h_{\omega}(x_{N})u^{p}=0\quad\text{in }\mathbb{R}^{N}_{+},$ (1.26)
$u|_{x_{N}=0}=K_{j}\delta_{a}(x^{\prime\prime}),\ a\in\mathbb{R}^{N-1},\
K_{j}\to\infty\text{ as }j\to\infty,$ (1.27)

where $h_{\omega}(s)$ is the same as in (1.16). Since
$h_{\omega}(x_{N})\leqslant h_{\omega}(|x^{\prime}|)$ and $h_{\omega}(x_{N})$
degenerates on all hyperplane $\\{x:x_{N}=0\\}$, then due to Theorem 1.2 and
comparison principle solution $u_{\infty}(x):=\lim_{j\to\infty}u_{j}(x)$
satisfies: $u_{\infty}(x^{\prime\prime},0)=\infty$
$\forall\,x^{\prime\prime}\in\mathbb{R}^{N-1}$. Moreover in [6] it was proved
that if $\omega(s)$ from(1.16) satisfies condition (1.8) instead of (1.24)
then

$u_{\infty}(x^{\prime\prime},0)=0\quad\forall\,x^{\prime\prime}\in\mathbb{R}^{N-1}:\
x^{\prime\prime}\neq a.$ (1.28)

Thus, Dini condition (1.8) is necessary and sufficient condition of the
existence of very singular solution $u_{\infty}(x)$ with point singularity.

######  Remark 1.3.

Condition $\omega(s)\to 0$ as $s\to 0$ is technical for our proof of Theorem
1.2 and can be omitted by simple arguments. Let
$\omega(s)\geqslant\omega_{0}=const>0$ $\forall\,s>0$. Then we can find
continuous nondecreasing function $\tilde{\omega}(s)\geqslant 0$:

$\tilde{\omega}(s)>0\ \forall\,s>0,\ \tilde{\omega}(s)\to 0\text{ as }s\to
0,\tilde{\omega}(s)\leqslant\omega_{0}\ \forall\,s>0,$

which satisfies condition (1.24). Let now $\tilde{u}_{j}(x)$ be a sequence of
solutions of problem (1.26), (1.27) with absorption potential
$h_{\tilde{\omega}}(x_{N}):=\exp\left(-\frac{\tilde{\omega}(x_{N})}{x_{N}}\right)$
instead of $h_{\omega}(x_{N})$. Then due to Th.1.2
$\tilde{u}_{\infty}(x^{\prime\prime},0)=\infty$. If now $u_{j}^{(0)}(x)$ be a
sequence of solutions of problem (1.26), (1.27) with
$h_{\omega_{0}}(x_{N}):=\exp\left(-\frac{\omega_{0}}{x_{N}}\right)$ instead of
$h_{\omega}(x_{N})$, then due to comparison principle

$u_{j}(x)\geqslant
u^{(0)}_{j}(x)\geqslant\tilde{u}_{j}(x)\quad\forall\,j\in\mathbb{N},\
\forall\,x\in\overline{\Omega}.$

Therefore $\infty=\tilde{u}_{\infty}(x^{\prime\prime},0)\leqslant
u_{\infty}^{(0)}(x^{\prime\prime},0)\leqslant u_{\infty}(x^{\prime\prime},0)$
and as consequence
$u_{\infty}^{(0)}(x^{\prime\prime},0)=u_{\infty}(x^{\prime\prime},0)=\infty$.
Notice that this last property of instantaneous propagation of strong point
singularity of solution $u_{\infty}^{(0)}(x)$ to all boundary of domain when
$\omega=\omega_{0}>0$, was firstly discovered by M. Marcus, L. Veron [8].

The paper is organized as follows. Section 2 is devoted to the proof of main
auxiliary Theorem 2.1, which demonstrates our variant of the local energy
estimate method in the study of asymptotic behavior of solutions to semilinear
elliptic equations of diffusion-absorption type near the singularity set. In
section 3 the technique, elaborated in section 2, is adapted to the proof of
Theorem 1.1. Finally, in section 4 Theorem 1.2 about necessity of the Dini
condition is proved.

##  2 Local energy estimates near the boundary
singularity set

Let $\Omega\subset\mathbb{R}^{N}_{+}$ be a bounded domain with
$C^{2}$–boundary $\partial\Omega$, such that

$\Gamma_{\overline{R}+\rho_{0}}:=\\{(x^{\prime\prime},0):|x^{\prime\prime}|\leqslant\overline{R}+\rho_{0}\\}\subset\partial\Omega,\quad\Gamma_{\overline{R}+\rho_{0}}\times(0,\rho_{0})\subset\Omega,$
(2.1)

where $\overline{R}>0$, $\rho_{0}>0$. Let $G_{i}$, $i=1,2,...,l$, be bounded
subdomains of hyperplane $\\{x_{N}=0\\}$ with $C^{2}$–boundaries $\partial
G_{i}$ such that

$G_{i}\subset\\{|x^{\prime\prime}|<\overline{R}\\}\quad\forall\,i\leqslant l,$
(2.2) $dist(G_{i},G_{j}):=\inf_{x\in G_{i},y\in
G_{j}}|x-y|>\rho_{0}\quad\forall\,i\neq j,\ \rho_{0}\text{ is from
\eqref{1.14}}.$ (2.3)

In domain $\Omega$ we consider the following problem:

$\displaystyle
Lu+H(x)u^{p}:=-\sum_{i,j=1}^{N}\left(a_{ij}(x)u_{x_{i}}\right)_{x_{j}}+H(x)u^{p}=0,$
(2.4) $\displaystyle u|_{\bar{G}_{i}}=K^{(i)},\ i=1,2,...,l;\quad u=0\text{ on
}\partial\Omega\setminus\biggr{\\{}\bigcup_{i=1}^{l}\bar{G}_{i}\biggr{\\}},$
(2.5)

where $p>1$, $K^{(i)}=const>0$ $\forall\,i\leqslant l$, $C^{1,\lambda}$–smooth
functions $a_{ij}(\cdot)$ satisfy ellipticity condition (1.5), and potential
$H(\cdot)$ satisfies condition (1.6). Introduce now $l$ sequences
$\\{K_{j}^{(i)}\\}$, $i\leqslant l$, $j=1,2,...$:

$K_{j}^{(i)}\to\infty\text{ as }j\to\infty\quad\forall\,i\leqslant l,$ (2.6)

and let $\\{u_{j}\\}$, $j=1,2,...$, be an infinite sequence of solutions of
equation (2.4) satisfying boundary condition

$u_{j}|_{\bar{G}_{i}}=K_{j}^{(i)},\quad u_{j}=0\text{ on
}\partial\Omega\setminus\left\\{\cup_{j=1}^{l}\bar{G}_{i}\right\\}.$ (2.7)

######  Theorem 2.1.

Let functions $h_{\omega}(\cdot)$ and $H(\cdot)$ satisfy relation (1.6) and
let $\omega$ from (1.6) is a nondecreasing continuous function satisfying
technical condition (1.7) and Dini condition (1.8). If $u_{j}$ is a solution
of problem (2.4), (2.7), then $u_{\infty}:=\lim_{j\to\infty}u_{j}$ is a
solution of equation (2.4), satisfying boundary conditions

$\lim_{x\to
y}u_{\infty}(x)=0\quad\forall\,y\in\partial\Omega\setminus\big{\\{}\cup_{i\leqslant
l}\overline{G}_{i}\big{\\}},\quad\lim_{x\to
y}u_{\infty}(x)=\infty\quad\forall\,y\in\cup_{i\leqslant l}\overline{G}_{i}$
(2.8)

###### Proof.

Let us introduce the following families of subdomains of $\Omega$ from (2.1):

$\begin{split}&\Omega_{s}:=\\{x\in\Omega:d(x)>s\\}\quad\forall\,s\in\mathbb{R}_{+}^{1},\\\
&\Omega^{s}:=\\{x\in\Omega:0<d(x)<s\\}\quad\forall\,s\in\mathbb{R}_{+}^{1}\end{split}$
(2.9)

Due to the smoothness of $\partial\Omega$ there exists $\bar{s}>0$ such that
$\partial\Omega^{s}\cap\Omega=\partial\Omega_{s}$ is $C^{2}$–smooth for any
$s:0<s<\bar{s}$. Moreover assume that

$d(x)=x_{N}\quad\forall\,x\in\Gamma_{\bar{R}+\rho_{0}}\times(0,\rho_{0}).$
(2.10)

Let $u$ be a nonnegative solution of equation (1.4) in $\Omega$. Introduce the
following energy function, connected with $u$:

$I(s):=\int_{\Omega_{s}}\left(|\nabla_{x}u|^{2}+h_{\omega}(d(x))u^{p+1}\right)dx,\quad
s>0.$ (2.11)

######  Lemma 2.1.

Function $I(\cdot)$ from (2.11) satisfies:

$I(s)\leqslant
d_{3}\left[\int_{0}^{s}h_{\omega}(r)^{\frac{2}{p+3}}dr\right]^{-\frac{p+3}{p-1}},\quad\forall\,s:0<s<\bar{s},$
(2.12)

where constant $d_{3}<\infty$ does not depend on $u$.

###### Proof.

Multiplying equation (1.4) by $u$ and integrating on $\Omega_{s}$,
$s:0<s<\bar{s}$, we obtain:

$\int_{\Omega_{s}}\left(\sum_{i,j=1}^{N}a_{ij}(x)u_{x_{i}}u_{x_{j}}+H(x)u^{p+1}\right)dx=\int_{\partial\Omega_{s}}\sum_{i,j=1}^{N}a_{ij}(x)u_{x_{i}}u\nu_{j}d\sigma\leqslant\\\
\leqslant\left(\int_{\partial\Omega_{s}}\sum_{i,j=1}^{N}a_{ij}(x)u_{x_{i}}u_{x_{j}}d\sigma\right)^{\frac{1}{2}}\left(\int_{\partial\Omega_{s}}\sum_{i,j=1}^{N}a_{ij}(x)\nu_{i}\nu_{j}u^{2}d\sigma\right)^{\frac{1}{2}},$
(2.13)

where $\nu(x)=(\nu_{1},...,\nu_{N})$ is an outward normal unit vector to
$\partial\Omega$ of the length $1$. Due to (1.5), (1.6), using Holder’s
inequality we have:

$\left(\int_{\partial\Omega_{s}}\sum_{i,j=1}^{N}a_{ij}(x)\nu_{i}\nu_{j}u^{2}d\sigma\right)^{\frac{1}{2}}\leqslant
c(meas\,\partial\Omega_{s})^{\frac{q-1}{2(q+1)}}h_{\omega}(s)^{-\frac{1}{q+1}}\left(\int_{\partial\Omega_{s}}h_{\omega}(s)u^{q+1}d\sigma\right)^{\frac{1}{q+1}}.$

Substituting this estimate into (2.13) and using Young’s inequality we obtain:

$I(s)\leqslant
c_{1}h_{\omega}(s)^{-\frac{1}{p+1}}\left(\int_{\partial\Omega_{s}}\left(|\nabla_{x}u|^{2}+h_{\omega}\left(d(x)\right)u^{p+1}\right)d\sigma\right)^{1-\frac{p-1}{2(p+1)}}.$
(2.14)

It is easy to see that

$\frac{dI(s)}{ds}=-\int_{\partial\Omega_{s}}\left(|\nabla_{x}u|^{2}+h_{\omega}\left(d(x)\right)u^{p+1}\right)d\sigma.$

Substituting this relation into (2.14) we derive the following differential
inequality:

$I(s)\leqslant
c_{2}h_{\omega}(s)^{-\frac{1}{p+1}}\left(-I^{\prime}(s)\right)^{1-\frac{p-1}{2(p+1)}}.$

Solving this inequality we obtain (2.12). ∎

Now we derive the global upper a’priory estimates for solutions $u_{j}$ of
problem (1.4), (2.7) when $j\to\infty$. For an arbitrary small $\delta>0$ we
introduce $C^{1}$–smooth function $\xi_{\delta}(x^{\prime\prime})$ with $supp\
\xi_{\delta}\subset\\{x^{\prime\prime}\in\mathbb{R}^{N-1}:|x^{\prime\prime}|\leqslant\overline{R}+\delta\\}$,
$\delta<2^{-1}\rho_{0}$, such that:

$\displaystyle\xi_{\delta}(x^{\prime\prime})=1\text{ if
}x^{\prime\prime}\in\overline{G}_{i}\quad\forall\,i\leqslant l,$ (2.15)
$\displaystyle\xi_{\delta}(x^{\prime\prime})=0\text{ if
}dist(x^{\prime\prime},\overline{G}_{i}):=\min_{y\in\overline{G}_{i}}|x^{\prime\prime}-y|\geqslant\delta\quad\forall\,i\leqslant
l,$ (2.16) $\displaystyle 0\leqslant\xi_{\delta}(x^{\prime\prime})\leqslant
1\quad\forall\,x^{\prime\prime}\in\mathbb{R}^{N-1}\setminus\cup_{i=1}^{l}\overline{G}_{i}:\min_{i\leqslant
l}dist(x^{\prime\prime},\overline{G}_{i})<\delta.$ (2.17)

It is clear that

$|\nabla\xi_{\delta}|\leqslant
c\delta^{-1}\quad\forall\,\delta:0<\delta<2^{-1}\rho_{0},$ (2.18)

where $c<\infty$ does not depend on $\delta$. Let $u_{j,\delta}$, $j=1,2,...$,
be a solution of equation (1.4) subject to the regularized boundary condition:

$u_{j,\delta}=K_{j}\xi_{\delta}\text{ on }\partial\Omega,\quad
K_{j}:=\max_{i\leqslant l}K_{j}^{(i)}.$ (2.19)

Due to the comparison principle we have:

$u_{j,\delta}(x)\geqslant u_{j}(x)\quad\forall\,x\in\overline{\Omega},\
\forall\,j\in\mathbb{N},\ \forall\,\delta:0<\delta<2^{-1}\rho_{0}.$ (2.20)

Therefore to prove Theorem 2.1 it is sufficient to investigate and estimate
from above solution $u_{j,\delta}$ with arbitrary small $\delta>0$. For the
sake of simplicity of the notations we omit $\delta$ in $u_{j,\delta}$ and
denote $u_{j,\delta}$ by $u_{j}$.

######  Lemma 2.2.

Solution $u_{j}=u_{j,\delta}$ of problem (1.4), (2.19) satisfies the following
estimate:

$\int_{\Omega}\left(|\nabla
u_{j}|^{2}+h_{\omega}\left(d(x)\right)u_{j}^{p+1}\right)dx\leqslant\overline{K}_{j}:=\bar{c}\big{(}K_{j}^{p+1}+\delta^{-1}K_{j}^{2}\big{)},$
(2.21)

where constant $\bar{c}<\infty$ does not depend on $j\in\mathbb{N}$,
$\delta>0$.

###### Proof.

Let us introduce a $C^{2}$–cut–off function $\zeta=\zeta_{\delta}(s)$ such
that $\zeta_{\delta}(s)=1$ if $s\leqslant\delta$, $\zeta_{\delta}(s)=0$ if
$s>2\delta$, $0\leqslant\zeta_{\delta}(s)\leqslant 1$,
$|\nabla\zeta_{\delta}|<c\delta^{-1}$. Multiplying (1.4) by

$v_{j}(x)=u_{j}(x)-K_{j}\xi_{\delta}(x^{\prime\prime})\zeta_{\delta}(d(x)),\quad
K_{j}\text{ is from \eqref{3.11}},$ (2.22)

and integrating it over $\Omega$, due to $v_{j}=0$ on $\partial\Omega$ we
obtain:

$\int_{\Omega}\left(\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}u_{jx_{k}}+H(x)u_{j}^{p+1}\right)dx=\int\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}\left(\xi_{\delta}(x^{\prime\prime})\zeta_{\delta}(d(x))\right)_{x_{k}}K_{j}dx+\\\
+\int_{\Omega}K_{j}H(x)u_{j}^{p}\xi_{\delta}(x^{\prime\prime})\zeta_{\delta}(d(x))dx:=A_{1}+A_{2}.$
(2.23)

Using Young’s inequality and properties (2.15)–(2.18) we get:

$\begin{split}&|A_{1}|\leqslant
2^{-1}\int_{\Omega}\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}u_{jx_{k}}dx+c\delta^{-1}K_{j}^{2},\\\
&|A_{2}|\leqslant
2^{-1}\int_{\Omega}H(x)u_{j}^{p+1}dx+c^{\prime}K_{j}^{p+1},\end{split}$ (2.24)

where constants $c,c^{\prime}<\infty$ do not depend on $\delta$, $j$. In
virtue of (2.24) relation (2.23) easily yields :

$\int_{\Omega}\left(\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}u_{jx_{k}}+H(x)u_{j}^{p+1}\right)dx\leqslant\bar{\bar{c}}(K_{j}^{p+1}+\delta^{-1}K_{j}^{2}),\quad\bar{\bar{c}}=\max(c,c^{\prime}),$
(2.25)

which gives estimate (2.21) due to properties (1.6), (1.5) . ∎

Introduce now the following family of subdomains of domain $\Omega^{s}$ with
arbitrary $s\in(0,\rho_{0})$:

$\Omega^{s}(\tau):=\Omega^{s}\setminus\big{\\{}x=(x^{\prime\prime},x^{N})\in\Omega^{s}:r(x^{\prime\prime}):=\min_{i\leqslant
l}dist(x^{\prime\prime},G_{i})<\tau\big{\\}}\quad\forall\,\tau\in\big{(}0,\frac{\rho_{0}}{2}\big{)},$
(2.26)

where $\rho_{0}>0$ is from (2.10). Introduce also a jet one energy function
for solution $u_{j}$ under consideration:

$J_{j}(s,\tau):=\int_{\Omega^{2s}(\tau)}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)\zeta_{s}(d(x))dx,$ (2.27)

where $\zeta_{s}(\cdot)$ is a function from (2.22): $\zeta_{s}(d)=1$ if
$d\leqslant s$, $\zeta_{s}(d)=0$ if $d>2s$, $0\leqslant\zeta_{s}(d)\leqslant
1$, $|\nabla\zeta_{s}|\leqslant cs^{-1}$.

######  Lemma 2.3.

Energy function $J_{j}(s,\tau)$ from (2.27) satisfies the following
differential inequality

$J_{j}(s,\tau)\leqslant
cs\left(-\frac{d}{d\tau}J_{j}(s,\tau)\right)+Ch_{\omega}(s)^{-\frac{2}{p-1}-\nu}\quad\forall\,\tau\in\big{(}\delta,\frac{\rho_{0}}{2}\big{)},\
\forall\,j\in\mathbb{N},\\\ \forall\,s\in\big{(}0,\frac{\rho_{0}}{2}\big{)},\
\forall\,\nu>0,\ C=C(\nu)\to\infty\text{ as }\nu\to 0,$ (2.28)

where constants $c,C$ do not depend on $j$.

###### Proof.

We multiply equation (1.4) for solution $u_{j}(x)$ by
$u_{j}(x)\zeta_{s}(d(x))$ and integrate it over $\Omega^{2s}(\tau)$,
$\tau>\delta$. As a result we obtain the following relation:

$\overline{J}_{j}(s,\tau):=\int_{\Omega^{2s}(\tau)}\left(\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}u_{jx_{k}}+H(x)u_{j}^{p+1}\right)\zeta_{s}(d(x))dx=R_{1}+R_{2}:=\\\
:=\int_{\Gamma^{2s}(\tau)}\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}u_{j}\nu_{k}(x)\zeta_{s}(d(x))d\sigma-\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}\zeta_{s}(d(x))_{x_{k}}u_{j}dx,$
(2.29)

where $\Gamma^{2s}(\tau):=\bigcup_{i\leqslant l}\Gamma_{i}^{2s}(\tau)$,
$\Gamma_{i}^{2s}(\tau):=\\{x=(x^{\prime\prime},x_{N}):x_{N}<2s,dist(x^{\prime\prime},G_{i})=\tau\\}$.
Notice that due to (2.10) and (2.2), (2.3) we have:
$\Gamma_{i}^{2s}(\tau)\cap\Gamma_{j}^{2s}(\tau)=\emptyset$ $\forall\,i\neq j$,
$\forall\,\tau<2^{-1}\rho_{0}$. Now using Hölder inequality we estimate
$R_{1}$ from above:

$|R_{1}|\leqslant\left(\int_{\Gamma^{2s}(\tau)}\sum_{i,k=1}^{N}a_{ik}u_{jx_{i}}u_{jx_{k}}\zeta_{s}(d(x))d\sigma\right)^{\frac{1}{2}}\left(\int_{\Gamma^{2s}(\tau)}\sum_{i,k=1}^{N}a_{ik}\nu_{i}\nu_{k}u_{j}^{2}\zeta_{s}(d(x))d\sigma\right)^{\frac{1}{2}}=\\\
=(R_{1}^{(1)})^{\frac{1}{2}}(R_{1}^{(2)})^{\frac{1}{2}}.$ (2.30)

Using (1.5) we estimate $R_{1}^{(2)}$:

$R_{1}^{(2)}\leqslant
d_{1}\int_{\Gamma^{2s}(\tau)}u_{j}^{2}\zeta_{s}(d(x))d\sigma=d_{1}\left(\int_{\Gamma^{2s}(\tau)\setminus\Gamma^{s}(\tau)}u_{j}^{2}\zeta_{s}(d(x))d\sigma+\int_{\Gamma^{s}(\tau)}u_{j}^{2}d\sigma\right)=\\\
=d_{1}\left(R_{1,1}^{(2)}+R_{1,2}^{(2)}\right).$

Since $u_{j}(x^{\prime\prime},0)=0$
$\forall\,x^{\prime\prime}\in\Gamma^{s}(\tau):\delta<\tau<\rho_{0}$, we derive
by Poincare’s inequality:

$R_{1,2}^{(2)}=\int_{\Gamma^{s}(\tau)}u_{j}^{2}d\sigma\leqslant
d_{2}s^{2}\int_{\Gamma^{s}(\tau)}\left|\frac{\partial u_{j}}{\partial
x_{N}}\right|^{2}d\sigma\leqslant d_{2}s^{2}\int_{\Gamma^{s}(\tau)}|\nabla
u_{j}|^{2}d\sigma\quad\forall\,\tau:\delta<\tau<\frac{\rho_{0}}{2}.$ (2.31)

We estimate term $R_{1,1}^{(2)}$ by the standard trace interpolation
inequality (see e.g. [7]):

$\begin{split}&\int_{\Gamma_{i,x_{N}}(\tau)}u_{j}(x^{\prime\prime},x^{N})^{2}d\sigma^{\prime\prime}\leqslant
c_{1}\left(\int_{\tau<|x^{\prime\prime}|<\rho_{0}}|\nabla_{x^{\prime\prime}}u_{j}(x^{\prime\prime},x_{N})|^{2}dx^{\prime\prime}\right)^{\frac{1}{2}}\left(\int_{\tau<|x^{\prime\prime}|<\rho}u_{j}(x^{\prime\prime},x_{N})^{2}dx^{\prime\prime}\right)^{\frac{1}{2}}\\\
&+c_{2}\int_{\tau<|x^{\prime\prime}|<\rho_{0}}u_{j}(x^{\prime\prime},x_{N})^{2}dx^{\prime\prime}\quad\forall\,\tau:\delta<\tau<\frac{\rho_{0}}{2},\
\forall\,x_{N}\in(s,2s),\ \forall\,i\leqslant l,\end{split}$ (2.32)

where
$\Gamma_{i,x_{N}}(\tau):=\\{x=(x^{\prime\prime},x_{N}):dist(x^{\prime\prime},G_{i})=\tau,x_{N}=const\\}$,
constants $c_{1},c_{2}$ do not depend on $\tau,s$. Integrating the last
inequality on $x_{N}$ over $(s,2s)$ and summing the obtained inequalities from
$i=1$ up to $i=l$, we obtain after simple computations:

$R_{1,1}^{(2)}\leqslant
c_{1}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}|\nabla_{x^{\prime\prime}}u_{j}|^{2}dx\right)^{\frac{1}{2}}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}u_{j}(x)^{2}dx\right)^{\frac{1}{2}}+c_{2}\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}u_{j}(x)^{2}dx=\\\
=:c_{1}(R_{1,1,1}^{(2)})^{\frac{1}{2}}(R_{1,1,2}^{(2)})^{\frac{1}{2}}+c_{2}R_{1,1,2}^{(2)}\quad\forall\,\tau\in(\delta,2^{-1}\rho_{0}),\
\forall\,s\in(0,2^{-1}\rho_{0}).$ (2.33)

Using Hölder inequality we get:

$R_{1,1,2}^{(2)}\leqslant
c_{3}s^{\frac{p-1}{p+1}}h_{\omega}(s)^{-\frac{2}{p+1}}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}h_{\omega}(d(x))u_{j}(x)^{p+1}dx\right)^{\frac{2}{p+1}}.$
(2.34)

It follows from (2.33), (2.34) that

$\begin{split}&R_{1,1}^{(2)}\leqslant
c_{4}s^{\frac{p-1}{p+1}}h_{\omega}(s)^{-\frac{2}{p+1}}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}h_{\omega}(d(x))u_{j}(x)^{p+1}dx\right)^{\frac{2}{p+1}}+\\\
&+c_{5}s^{\frac{p-1}{2(p+1)}}h_{\omega}(s)^{-\frac{1}{p+1}}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}|\nabla_{x}u_{j}|^{2}dx\right)^{\frac{1}{2}}\left(\int_{\Omega^{2s}(\tau)}h_{\omega}(d(x))u_{j}(x)^{p+1}dx\right)^{\frac{1}{p+1}}\leqslant\\\
&\leqslant
c_{4}s^{\frac{p-1}{p+1}}h_{\omega}(s)^{-\frac{2}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{p+1}}+c_{5}s^{\frac{p-1}{2(p+1)}}h_{\omega}(s)^{-\frac{1}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{2(p+1)}},\end{split}$
(2.35)

where
$I_{j}(s)=\int_{\Omega_{s}}\left(|\nabla_{x}u_{j}|^{2}+h_{\omega}(d(x))u_{j}(x)^{p+1}\right)dx$.
Plugging estimates (2.31) and (2.35) into (2.30) and using Young’s inequality
we obtain:

$\begin{split}&|R_{1}|\leqslant c_{6}\left(\int_{\Gamma^{2s}(\tau)}|\nabla
u_{j}|^{2}\zeta_{s}d\sigma\right)^{\frac{1}{2}}\biggr{[}s^{2}\int_{\Gamma^{s}(\tau)}|\nabla
u_{j}|^{2}d\sigma+s^{\frac{p-1}{p+1}}h_{\omega}(s)^{-\frac{2}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{p+1}}+\\\
&+s^{\frac{p-1}{2(p+1)}}h_{\omega}(s)^{-\frac{1}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{2(p+1)}}\biggr{]}^{\frac{1}{2}}\leqslant
c_{7}\biggr{[}s\int_{\Gamma^{2s}(\tau)}|\nabla u_{j}|^{2}\zeta_{s}d\sigma+\\\
&+s^{-1+\frac{p-1}{2(p+1)}}h_{\omega}(s)^{-\frac{1}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{2(p+1)}}+s^{-1+\frac{p-1}{p+1}}h_{\omega}(s)^{-\frac{2}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{p+1}}\biggr{]}.\end{split}$
(2.36)

Lastly we estimate $R_{2}$. Using Hölder inequality and property (1.5) we get:

$|R_{2}|\leqslant
cs^{-1}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}|\nabla
u_{j}|^{2}dx\right)^{\frac{1}{2}}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}u_{j}^{2}dx\right):=cs^{-1}(R_{2}^{(1)})^{\frac{1}{2}}(R_{2}^{(2)})^{\frac{1}{2}}.$
(2.37)

Term $R_{2}^{(2)}$ coincides with $R_{1,1,2}^{(2)}$ and it can be estimated as
in (2.34). Therefore using Young’s inequality we derive from (2.37):

$|R_{2}|\leqslant
cs^{-\left(1-\frac{p-1}{2(p+1)}\right)}h_{\omega}(s)^{-\frac{1}{p+1}}\left(\int_{\Omega^{2s}(\tau)\setminus\Omega^{s}(\tau)}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\right)^{1-\frac{p-1}{2(p+1)}}.$
(2.38)

Thus, due to estimates (2.36) and (2.38) it follows from (2.29) that

$J_{j}(s,\tau)\leqslant d_{0}^{-1}\overline{J}_{j}(s,\tau)\leqslant
cs\int_{\Gamma^{2s}(\tau)}|\nabla_{x}u_{j}|^{2}\zeta_{s}(d(x))d\sigma+c_{1}s^{-\frac{2}{p+1}}h_{\omega}(s)^{-\frac{2}{p+1}}\times\\\
\times\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{p+1}}+c_{2}s^{-\frac{p+3}{2(p+1)}}h_{\omega}(s)^{-\frac{1}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{2(p+1)}}.$
(2.39)

It is easy to check that

$\int_{\Gamma^{2s}(\tau)}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)\zeta_{s}(d(x))d\sigma\leqslant-\bar{c}\frac{d}{d\tau}J_{j}(s,\tau),$
(2.40)

where $\bar{c}=const>0$ does not depend on $\tau$, $s$, $j$. Substituting
(2.40) into (2.39) we obtain:

$\begin{split}&J_{j}(s,\tau)\leqslant\bar{\bar{c}}s\left(-\frac{d}{d\tau}J_{j}(s,\tau)\right)+c_{1}F_{j}(s)\quad\forall\,\tau\in\left(\delta,\frac{\rho_{0}}{2}\right),\
\forall\,s\in\left(0,\frac{\rho_{0}}{2}\right),\\\
&F_{j}(s):=\frac{\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{p+1}}}{s^{\frac{2}{p+1}}h_{\omega}(s)^{\frac{2}{p+1}}}+\frac{\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{2(p+1)}}}{s^{\frac{p+3}{2(p+1)}}h_{\omega}(s)^{\frac{1}{p+1}}}\end{split}$
(2.41)

It only remains to estimate $F_{j}(s)$ from above. Due to lemma 2.1 we have
the following uniform with respect to $j\in\mathbb{N}$ upper estimate for
energy functions $I_{j}$:

$I_{j}(s)\leqslant
d_{3}\left[\int_{0}^{s}h_{\omega}(r)^{\frac{2}{p+3}}dr\right]^{-\frac{p+3}{p-1}}\quad\forall\,s:0<s<\bar{s}.$
(2.42)

Since $\omega(\cdot)$ is a nondecreasing function it is easy to check (see
lemma 2.4 from [6]) that

$\int_{0}^{s}\exp\left(-\frac{a\omega(t)}{t}\right)dt\geqslant\frac{s^{2}}{2s+a\omega(s)}\exp\left(-\frac{a\omega(s)}{s}\right)\quad\forall\,a>0.$
(2.43)

Therefore due to (2.43) it follows from (2.42) that

$I_{j}(s)\leqslant
d_{3}\left(\frac{2s+\frac{2}{p+3}\omega(s)}{s^{2}}\right)^{\frac{p+3}{p-1}}\exp\left(\frac{2}{p-1}\frac{\omega(s)}{s}\right):=d_{3}\Phi_{1}(s)h_{\omega}(s)^{-\frac{2}{p-1}}.$
(2.44)

Substituting estimate (2.44) in the definition of function $F_{j}(s)$ we
obtain:

$F_{j}(s)\leqslant
d_{4}\left(\left(\frac{\Phi_{1}(s)}{s}\right)^{\frac{2}{p+1}}+\left(\frac{\Phi_{1}(s)}{s}\right)^{\frac{p+3}{2(p+1)}}\right)h_{\omega}(s)^{-\frac{2}{p-1}}.$
(2.45)

Due to condition (1.7) for function $\omega(s)$ we have:

$h_{\omega}(s)^{-\frac{2}{p-1}}\geqslant\exp\left(\frac{2}{p-1}s^{-(1-\gamma_{1})}\right)\qquad\forall\,s\in(0,\rho_{0}),$

which yields the upper estimate for $F_{j}(s)$:

$F_{j}(s)\leqslant
C(\nu)h_{\omega}(s)^{-\frac{2}{p-1}-\nu}\quad\forall\,\nu>0,\
C(\nu)\to\infty\text{ as }\nu\to 0.$ (2.46)

∎

Now we get down to the main stage of the proof of the theorem, which consists
in a careful analysis of the vanishing properties of energy functions
$J_{j}(s,\tau)$, satisfying inequalities (2.25) for all $j\in\mathbb{N}$.
Notice that due to global estimate (2.21) function $J_{j}(s,\tau)$ satisfies
the following ”initial” condition:

$J_{j}(s,\delta)\leqslant\overline{K}_{j}:=\bar{c}(K_{j}^{p+1}+\delta^{-1}K_{j}^{2})\quad\forall\,j\in\mathbb{N},$
(2.47)

where $\delta>0$ and $K_{j}$ are from boundary condition (2.19). Let us fix
$j$ large enough and $\nu>0$ small enough. Then define $s_{j}>0$ by the
following relation:

$C(\nu)h_{\omega}(s_{j})^{-\frac{2}{p-1}-\nu}=\overline{K}_{j}^{\theta}\quad\forall\,j\in\mathbb{N},\
C(\nu)\text{ is from \eqref{3.33}},$ (2.48)

where $0<\theta<1$ will be defined later on. It follows from (2.28), (2.47)
that $J_{j}(s_{j},\tau)$ satisfies the following differential inequality:

$\begin{split}&J_{j}(s_{j},\tau)\leqslant\tilde{c}s_{j}\left(-\frac{d}{d\tau}J_{j}(s_{j},\tau)\right)+\overline{K}_{j}^{\theta}\quad\forall\,\tau:\delta<\tau<2^{-1}\rho_{0},\\\
&J_{j}(s_{j},\delta)\leqslant\overline{K}_{j}.\end{split}$ (2.49)

Let us define now value $\tau_{j}$ by the equality:

$J_{j}(s_{j},\delta+\tau_{j})=2\overline{K}_{j}^{\theta},$ (2.50)

where $\theta$ has been introduced in (2.48). In order to find an upper
estimate for $\tau_{j}$, we observe that

$J_{j}(s_{j},\tau)>2\overline{K}_{j}^{\theta}\quad\forall\,\tau\in(\delta,\delta+\tau_{j}).$

Therefore (2.49) yields:

$J_{j}(s_{j},\tau)\leqslant
2\tilde{c}s_{j}\left(-\frac{d}{d\tau}J_{j}(s_{j},\tau)\right)\quad\forall\,\tau\in(\delta,\delta+\tau_{j}).$
(2.51)

Solving this differential inequality and taking into account the initial
condition in (2.49), we deduce:

$J_{j}(s_{j},\tau)\leqslant\overline{K}_{j}\exp\left(-\frac{\tau-\delta}{2\tilde{c}s_{j}}\right)\quad\forall\,\tau\in(\delta,\delta+\tau_{j}).$
(2.52)

By (2.50) and (2.52) we get:
$2\overline{K}_{j}^{\theta}\leqslant\overline{K}_{j}\exp\left(-\frac{\tau_{j}}{2\tilde{c}s_{j}}\right)$,
where $\tilde{c}$ is constant from (2.49). Consequently, $\tau_{j}$ satisfies:

$0<\tau_{j}\leqslant 2\tilde{c}s_{j}(-\ln 2+(1-\theta)\ln\overline{K}_{j}).$
(2.53)

Next, we notice that due to definitions (2.11), (2.27) we have:

$\int_{\Omega^{\rho_{0}}(\delta+\tau_{j})}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\leqslant
I_{j}(s_{j})+J_{j}(s_{j},\delta+\tau_{j}),\text{ if
}\delta+\tau_{j}<2^{-1}\rho_{0}.$ (2.54)

Due to estimate (2.44) and condition (1.7) on $\omega(s)$, it follows
analogously to (2.46):

$I_{j}(s_{j})\leqslant
C_{1}(\nu)h_{\omega}(s_{j})^{-\frac{2}{p-1}-\nu},\quad\forall\,\nu>0,\
C_{1}(\nu)\to\infty\text{ as }\nu\to 0.$ (2.55)

Using now definition (2.48) of $s_{j}$ and (2.50) of $\tau_{j}$, we deduce
from (2.54) and (2.55):

$\int_{\Omega^{\rho_{0}}(\delta+\tau_{j})}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\leqslant\left(2+\frac{C_{1}(\nu)}{C(\nu)}\right)\overline{K}_{j}^{\theta}.$
(2.56)

Now we fix sequence $\\{{K}_{i}\\}$ and sequence $\\{\overline{K}_{i}\\}$
defined by (2.47). It is easy to see that to prove Theorem 2.1 it is
sufficient to consider only one specific sequence $\\{{K}_{i}\\}$,
$K_{i}\to\infty$ as $i\to\infty$. As such a sequence we consider

$\overline{K}_{i}:=\exp\exp i,\quad i\in\mathbb{N}.$ (2.57)

Now we have to fix parameter $\theta$ from definition (2.48) of $s_{j}$,
namely, we have to guarantee the validity of the following inequality:

$\left(2+\frac{C_{1}(\nu)}{C(\nu)}\right)\overline{K}_{j}^{\theta}\leqslant\overline{K}_{j-1}$
(2.58)

for all sufficiently large $j$. Due to (2.57) inequality (2.58) is equivalent
to:

$\ln\left(2+\frac{C_{1}(\nu)}{C(\nu)}\right)+\theta\exp
j\leqslant\exp(j-1)=e^{-1}\exp j.$ (2.59)

It is easy to see that (2.59) is satisfied by

$\theta=(2e)^{-1}\quad\forall\,j\geqslant j_{0}:=1+\ln
2+\ln\ln\left(2+\frac{C_{1}(\nu)}{C(\nu)}\right).$ (2.60)

With such $\theta$ inequality (2.56) yields to:

$\int_{\Omega^{\rho_{0}}(\delta+\tau_{j})}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\leqslant\overline{K}_{j-1}.$
(2.61)

Now we can obtain an explicit upper estimates of $\tau_{j}$, $s_{j}$, defined
by (2.48), (2.50). Firstly, (2.48) yields to:

$\begin{split}&C(\nu)\exp\left(\big{(}\frac{2}{p-1}+\nu\big{)}\frac{\omega(s_{j})}{s_{j}}\right)=\overline{K}_{j}^{\theta}\
\Rightarrow\
\frac{\theta}{2}\ln\overline{K}_{j}\leqslant\left(\frac{2}{p-1}+\nu\right)\frac{\omega(s_{j})}{s_{j}}\leqslant\theta\ln\overline{K_{j}}\\\
&\forall\,j\geqslant j^{\prime}=j^{\prime}(\nu)=\ln\ln
C(\nu)+\ln\theta^{-1}+\ln 2.\end{split}$ (2.62)

It follows from (2.62), (1.7) and (2.57) that:

$s_{j}\leqslant
2\left(\frac{2}{p-1}+\nu\right)\theta^{-1}(\ln\overline{K}_{j})^{-1}\omega(s_{j})\leqslant
2\left(\frac{2}{p-1}+\nu\right)\theta^{-1}\omega_{0}\exp(-j).$ (2.63)

Due to monotonicity of $\omega(\cdot)$ this estimate yields:

$\omega(s_{j})\leqslant\omega(C_{3}\exp(-j)),\quad
C_{3}=2\left(\frac{2}{p-1}+\nu\right)\theta^{-1}\omega_{0}.$ (2.64)

As to $\tau_{j}$, we derive from (2.53) and (2.63)

$\tau_{j}\leqslant 2\tilde{c}s_{j}(1-\theta)\ln\overline{K}_{j}\leqslant
4\tilde{c}(1-\theta)\left(\frac{2}{p-1}+\nu\right)\theta^{-1}\omega(s_{j})\leqslant
C_{2}\omega(s_{j}),$ (2.65)

where $C_{2}=4\theta^{-1}(1-\theta)\tilde{c}\left(\frac{2}{p-1}+\nu\right)$.
Substituting (2.64) into (2.65) we obtain:

$\tau_{j}\leqslant C_{2}\omega(C_{3}\exp(-j)).$ (2.66)

So, estimates (2.61), (2.63), (2.65) are results of the first circle of the
computation and a starting point for the second circle. We define value
$s_{j-1}$ similarly to (2.48):

$C(\nu)h_{\omega}(s_{j-1})^{-\frac{2}{p-1}-\nu}=\overline{K}_{j-1}^{\theta},\quad\theta=(2e)^{-1},\
C(\nu)\text{ is from \eqref{3.33}}.$ (2.67)

Then energy function $J_{j}(s_{j-1},\tau)$ satisfies the following
differential inequality:

$J_{j}(s_{j-1},\tau)\leqslant\tilde{c}s_{j-1}\left(-\frac{d}{d\tau}J_{j}(s_{j-1},\tau)\right)+\overline{K}_{j-1}^{\theta}\quad\forall\,\tau\in(\delta+\tau_{j},2^{-1}\rho_{0})$
(2.68)

instead of (2.49), and the following ”initial” condition, which is consequence
of inequality (2.61), derived in first round of computations:

$J_{j}(s_{j-1},\delta+\tau_{j})\leqslant\overline{K}_{j-1}.$ (2.69)

Next we define $\tau_{j-1}$ by the analog of (2.50):

$J_{j}(s_{j-1},\delta+\tau_{j}+\tau_{j-1})=2\overline{K}_{j-1}^{\theta}.$
(2.70)

Analogously to (2.51), using (2.68), (2.69), (2.70) we derive the following
relation:

$J_{j}(s_{j-1},\tau)\leqslant
2\tilde{c}s_{j-1}\left(-\frac{d}{d\tau}J_{j}(s_{j-1},\tau)\right)\quad\forall\,\tau\in(\delta+\tau_{j},\delta+\tau_{j}+\tau_{j-1}).$
(2.71)

If we solve it using ”initial” condition (2.69), we obtain:

$J_{j}(s_{j-1},\tau)\leqslant\overline{K}_{j-1}\exp\left(-\frac{\tau-\delta-\tau_{j}}{2\tilde{c}s_{j-1}}\right)\quad\forall\,\tau\in(\delta+\tau_{j},\delta+\tau_{j}+\tau_{j-1}).$
(2.72)

Definition (2.70) of $\tau_{j-1}$ and estimate (2.72) lead to the explicit
estimate of $\tau_{j-1}$:

$\tau_{j-1}\leqslant 2\tilde{c}s_{j-1}\left(-\ln
2+(1-\theta)\ln\overline{K}_{j-1}\right),$

and, finally, to the following analogs of estimates (2.63), (2.66):

$s_{j-1}\leqslant C_{3}\exp(-j+1),\quad\tau_{j-1}\leqslant
C_{2}\omega(C_{3}\exp(-j+1)).$ (2.73)

Inequality (2.44) yields analog of (2.55):

$I_{j}(s_{j-1})\leqslant\frac{C_{1}(\nu)}{C(\nu)}\left(C(\nu)h_{\omega}(s_{j-1})^{-\frac{2}{p-1}-\nu}\right).$
(2.74)

Summing estimates (2.70) and (2.74), using definition (2.67) of $s_{j-1}$ and
keeping in mind the validity of property (2.58) for all $j\geqslant j_{0}$, we
get:

$\int_{\Omega^{\rho_{0}}(\delta+\tau_{j}+\tau_{j-1})}\left(|\nabla_{x}u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\leqslant\left(2+\frac{C_{1}(\nu)}{C(\nu)}\right)\overline{K}_{j-1}^{\theta}\leqslant\overline{K}_{j-2},\text{
if }j>j_{0}+1.$ (2.75)

This estimate is the final step of the second circle of computations and a
starting point for the next circle. Realizing $i$ such circles, we obtain the
following analog of estimates (2.75), (2.73):

$\int_{\Omega^{\rho_{0}}(\delta+\sum_{k=0}^{i-1}\tau_{j-k})}\left(|\nabla_{x}u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\leqslant\overline{K}_{j-i}.$
(2.76) $\tau_{j-k}\leqslant C_{2}\omega(C_{3}\exp(-j+k)),\quad
s_{j-k}\leqslant C_{3}\exp(-j+k),\quad\forall\,k\leqslant i-1.$ (2.77)

There are two restrictions on value $i$. First of them follows from (2.59),
(2.60):

$j-i\geqslant j_{0}:=1+\ln 2+\ln\ln(2+C_{1}(\nu)C(\nu)^{-1}).$ (2.78)

The second restriction follows from analogous of (2.51), (2.71), namely,
estimates of the interval, where differential inequality for energy function
$J_{j}(s_{j-1},\tau)$ has to be satisfied:

$\delta+\sum_{k=0}^{i-1}\tau_{j-k}\leqslant 2^{-1}\rho_{0}.$ (2.79)

Due to estimate (2.77) and monotonicity of $\omega(\cdot)$ we have:

$\sum_{k=0}^{i-1}\tau_{j-k}\leqslant
C_{2}\sum_{k=0}^{i-1}\omega(C_{3}\exp(-j+k))\leqslant
C_{2}C_{3}^{-1}\int_{C_{3}\exp(-j)}^{C_{3}\exp(-j+i)}r^{-1}\omega(r)dr\leqslant\\\
\leqslant
C_{2}C_{3}^{-1}\int_{0}^{C_{3}\exp(-(j-i))}r^{-1}\omega(r)dr=:C_{2}C_{3}^{-1}\Phi(j-i),$
(2.80)

where $\Phi(s)\to 0$ as $s\to\infty$ due to Dini condition (1.8). Therefore
for arbitrary $\delta>0$, $\rho_{0}>2\delta$ there exists finite
$j^{(0)}=j^{(0)}(\delta,\rho_{0})$ such that
$\delta+C_{2}C_{1}^{-1}\Phi(j^{(0)})\leqslant 2^{-1}\rho_{0}$ and, as
consequence, condition (2.79) is satisfied for $j-i\geqslant j^{(0)}$. Thus,
due to (2.80) estimate (2.76) leads to:

$\int_{\Omega^{\rho_{0}}(\delta+C_{2}C_{3}^{-1}\Phi(j-i))}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\leqslant\overline{K}_{j-i}\quad\forall\,j:j-i\geqslant\bar{j},$
(2.81)

where $\bar{j}:=\max\\{j_{0},j^{(0)},j^{\prime}\\}$, $j^{\prime}$ is from
(2.62), $u_{j}(x)=u_{j,\delta}(x)$. Let us notice that for arbitrary small
$\delta>0$ we can find finite number $j^{(1)}=j^{(1)}(\delta)$ such that:

$j^{(1)}=\min(j\in\mathbb{N}:\delta\geqslant K_{j}^{-(p-1)}).$ (2.82)

From definition (2.47) of $\overline{K}_{j}$ it follows:

$\overline{K}_{j}\leqslant 2\bar{c}K_{j}^{p+1}\quad\forall\,j\geqslant
j^{(1)},$ (2.83)

which means that $\overline{K}_{j}$ does not depend on $\delta$ for all
$j\geqslant j^{(1)}$. Now due to condition (1.8) we can find finite
$j^{(2)}=j^{(2)}(\delta)\geqslant j^{(1)}(\delta)$ such that

$C_{2}C_{3}^{-1}\Phi(j-i)\leqslant\delta\quad\forall\,j:j-i>j^{(2)}$

and, as a consequence of (2.81):

$\int_{\Omega^{\rho_{0}}(2\delta)}\left(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1}\right)dx\leqslant\overline{K}_{j^{\prime\prime}}\quad\forall\,j>j^{\prime\prime}:=\max\\{\bar{j},j^{(2)}\\}.$
(2.84)

This estimate yields to the following uniform with respect to $j\in\mathbb{N}$
a’priory estimate:

$\|u_{j,\delta}\|_{H^{1}(\Omega^{\rho_{0}}(2\delta),\partial\Omega^{\rho_{0}}(2\delta)\cap\partial\Omega)}\leqslant
C=C(\delta)<\infty\quad\forall\,j\in\mathbb{N},$ (2.85)

where for arbitrary set $S\subset\partial\Omega$ by $H^{1}(\Omega,S)$ we
denote, as usually, the closure in the norm of $H^{1}(\Omega)$ of set
$C^{1}(\Omega,S):=\\{f\in C^{1}(\Omega):f|_{S}=0\\}$. Since
$h_{\omega}(d(x))\geqslant 0$ in $\overline{\Omega}$, all functions
$u_{j,\delta}$ are subsolutions of the corresponding linear elliptic equation.
Therefore, by Harnack inequality, for subsolutions of linear elliptic
equations (see, e.g., [7])

$\Big{(}\sup_{\Omega^{\rho_{1}}(3\delta)}u_{j,\delta}\Big{)}^{2}\leqslant
c(\delta,\rho_{0})\int_{\Omega^{\rho_{0}}(2\delta)}|u_{j,\delta}(x)|^{2}dx\quad\forall\,j\in\mathbb{N},\
\forall\,\delta>0,\ \rho_{1}=\frac{\rho_{0}}{2}.$ (2.86)

From (2.86) and (2.84) it follows:

$\sup_{\Omega^{\rho_{1}}(3\delta)}u_{j,\delta}\leqslant
c_{1}(\delta)\quad\forall\,j\in\mathbb{N},\ \forall\,\delta>0.$ (2.87)

In virtue of (2.20) the last inequality yields:

$\sup_{\Omega^{\rho_{1}}(3\delta)}u_{j}\leqslant
c_{1}(\delta)\quad\forall\,j\in\mathbb{N},\ \forall\,\delta>0,$ (2.88)

where $u_{j}$ is a solution of problem (1.4), (2.7). It is easy to see that
$u_{j}(x)$ is a solution of the following problem:

$\displaystyle-Lu_{j}=g_{j}(x):=H(x)u_{j}(x)^{p}\quad\text{in
}\Omega^{\rho_{1}}(3\delta)$ (2.89) $\displaystyle
u_{j}|_{\partial\Omega^{\rho_{1}}(3\delta)\cap\partial\Omega}=0\quad\forall\,j\in\mathbb{N},$
(2.90)

where, thanks to (2.88),
$\|g_{j}\|_{L^{q}(\Omega^{\rho_{1}}(3\delta))}\leqslant c_{2}(\delta)$
$\forall\,j\in\mathbb{N}$, $\forall\,q>1$. Therefore due to the classical
$L^{q}$ a’priory estimates for solutions of linear elliptic problems (see, for
example, [7]):

$\|u_{j}\|_{W^{2,q}(\Omega^{\rho_{2}}(4\delta))}\leqslant
c_{3}(\delta)\quad\forall\,j\in\mathbb{N},\ \forall\,q>1,\
\rho_{2}=\frac{\rho_{1}}{2}.$ (2.91)

Due to the comparison principle sequence $\\{u_{j}\\}$, $j=1,2,...$, is
monotonically nondecreasing in $\Omega$ and therefore $u_{j}(x)\to
u_{\infty}(x)$ pointwise for all $x\in\overline{\Omega}$. Then in virtue of
uniform estimate (2.91) and the compact embedding of space
$W^{2,q}(\Omega^{\rho_{2}}(4\delta))$ into
$C^{1,\lambda}(\overline{\Omega}^{\rho_{2}}(4\delta))$,
$0<\lambda<1-\frac{N}{q}$, it follows that

$\|u_{j}-u_{\infty}\|_{C^{1,\lambda}(\overline{\Omega}^{\rho_{2}}(4\delta))}\to
0\text{ as }j\to\infty.$ (2.92)

Since $\delta$ is an arbitrary positive number and $u_{j}=0$ on
$\partial\Omega\setminus\\{\cup_{i\leqslant l}\overline{G}_{i}\\}$ it follows
from (2.92) that $u_{\infty}(x)=0$
$\forall\,x\in\partial\Omega\setminus\\{\cup_{i\leqslant
l}\overline{G}_{i}\\}$. Theorem 2.1 is proved. ∎

##  3 About uniqueness of large solution

Here we prove Theorem 1.1. The proof consists in checking that the equation
under consideration has so called strong barrier property. This property was
introduced in [9, 10], where the sufficiency of this property for the
uniqueness of the large solution for the equation under consideration was
proved.

######  Definition 3.1.

(see [10], Def. 2.6) Let $z\in\partial\Omega$. We say that equation (1.4)
possesses a strong barrier at point $z$ if there exists number $R_{z}>0$ such
that for every $r\in(0,R_{z})$ there exists positive supersolution
$u=u_{r,z}\in C(\overline{\Omega}\cap B_{r}(z))$ of equation (1.4) in
$\Omega\cap B_{r}(z)$ with

$\lim_{y\to x,\ y\in\Omega\cap B_{r}(z)}u_{r,z}(y)=\infty\text{ for all
}x\in\Omega\cap\partial B_{r}(z).$ (3.1)

If an equation has mentioned property for arbitrary point $z\in\partial\Omega$
then we say that the equation under consideration has the strong barrier
property.

Without loss of generality we suppose that $\Omega\in\mathbb{R}_{+}^{N}$,
$z=0\in\partial\Omega$, $B_{R}(0)\cap\Omega=\\{x=(x^{\prime\prime},x_{N})\in
B_{R}(0):x_{N}>0\\}:=\Omega_{R}$ and $d(x):=dist(x,\partial\Omega)=x_{N}$
$\forall\,x\in\Omega_{R}$. Introduce also weight function

$\rho(x):=dist(x,\partial
B_{R}(0))=R-|x|=R-(|x^{\prime\prime}|^{2}+x_{N}^{2})^{\frac{1}{2}}.$ (3.2)

and surface $\Gamma_{R}:=\\{x\in\Omega_{R}:\rho(x)=d(x)\\}$. It is easy to
check that $\Gamma_{R}$ is a paraboloid:

$\Gamma_{R}=\left\\{x\in\Omega_{R}:x_{N}=\frac{R^{2}-|x^{\prime\prime}|^{2}}{2R}\quad\forall
x^{\prime\prime}:|x^{\prime\prime}|\leqslant R\right\\}.$ (3.3)

Now we construct solution $u$ from Definition 3.1. Namely, let
$\\{u_{j}(x)\\}$ be an increasing sequence of solutions of equation (1.4) in
domain $\Omega_{R}$ satisfying boundary conditions:

$u|_{\partial\Omega_{R}\cap\Omega}=K_{j},\quad u|_{\partial\Omega\cap
B_{R}(0)}=0\quad K_{j}\to\infty\text{ as }j\to\infty.$ (3.4)

We prove that as desired solution $u$ we can take
$u_{\infty}=\lim_{j\to\infty}u_{j}(x)$. Firstly we introduce sequence
$\\{u_{j,\delta}(x)\\}$ of solutions of equation (1.4), satisfying the
following ”regularized” boundary conditions:

$\begin{split}&u|_{\partial\Omega_{R}\cap\Omega}=K_{j},\quad
u|_{\partial\Omega\cap B_{R}(0)}=K_{j}\xi_{\delta}(|x^{\prime\prime}|),\\\
&\text{where }\xi_{\delta}(s)=\begin{cases}0,&\forall\,s\leqslant R-\delta,\\\
1-\delta^{-1}(R-s),&\forall\,s>R-\delta\end{cases}\end{split}$ (3.5)

and $\delta>0$ may be arbitrary small. Due to the comparison principle
$u_{j,\delta}(x)\geqslant u_{j}(x)$ $\forall\,j\in\mathbb{N}$,
$\forall\,\delta>0$ and consequently $u_{\infty,\delta}(x)\geqslant
u_{\infty}(x)$ $\forall\,x\in\overline{\Omega}_{R}$. Main part of our analysis
consists in the proof that

$u_{\infty,\delta}(x)=0\quad\forall\,x=(x^{\prime\prime},x_{N}):x_{N}=0,\
|x^{\prime\prime}|<R-c\delta,$ (3.6)

where $c=const<\infty$ does not depend on $\delta$. This proof is some
adaptation of the proof of Theorem 2.1. Introduce analogous to (2.9) families
of subdomains:

$\displaystyle\Omega_{R,s}=\\{x\in\Omega_{R}:\bar{\rho}(x)>s\\}\quad\forall\,s:0<s<\frac{R}{2},$
(3.7)
$\displaystyle\Omega_{R}^{s}=\\{x\in\Omega_{R}:0<\bar{\rho}(x)<s\\}\quad\forall\,s:0<s<\frac{R}{2},$
(3.8)

where $\bar{\rho}(x):=\min\\{\rho(x),d(x)\\}$, $d(x)=x_{N}$, $\rho(\cdot)$ is
from (3.2). If $u$ is an arbitrary nonegative solution of equation (1.4) in
$\Omega_{R}$ we introduce the following energy function:

$I(s):=\int_{\Omega_{R,s}}(|\nabla_{x}u|^{2}+h_{\omega}(d(x))u^{p+1})dx\quad\forall\,s\in(0,2^{-1}R).$
(3.9)

######  Lemma 3.1.

Function $I(\cdot)$ from (3.9) satisfies:

$I(s)\leqslant
c_{1}\left[\int_{0}^{s}h_{\omega}(r)^{\frac{2}{p+3}}dr\right]^{-\frac{p+3}{p-1}}\quad\forall\,s\in(0,2^{-1}R),$
(3.10)

where constant $c_{1}<\infty$ does not depend on $u$.

The proof repeats the proof of lemma 2.1 with small nonessential changes and
we omit it.

######  Lemma 3.2.

Solution $u_{j}(x):=u_{j,\delta}$ of problem (1.4), (3.5) in domain
$\Omega_{R}$ satisfies the following a priory estimate:

$\int_{\Omega_{R}}(|\nabla
u_{j}|^{2}+h_{\omega}(d(x))u_{j}^{p+1})dx\leqslant\overline{K}_{j}:=c_{2}(K_{j}^{p+1}+\delta^{-1}K_{j}^{2}),$
(3.11)

where $c_{2}<\infty$ does not depend on $j\in\mathbb{N}$.

###### Proof.

It is easy to see that function

$v_{j}(x):=u_{j}(x)-K_{j}\xi_{\delta}(|x|)=0\text{ on }\partial\Omega_{R}.$

Therefore multiplying equation (1.4) by $v_{j}$ and integrating by parts we
deduce easily the following analog of relation (2.23):

$\int_{\Omega_{R}}\left(\sum_{i,k=1}^{N}a_{ik}u_{jx_{i}}u_{jx_{k}}+H(x)u_{j}^{p+1}\right)dx=K_{j}\int_{\Omega_{R}}\sum_{i,k=1}^{N}a_{ik}u_{jx_{i}}\xi_{\delta}(|x|)_{x_{k}}dx+\\\
+K_{j}\int_{\Omega_{R}}H(x)u_{j}^{q}\xi_{\delta}(|x|)dx.$ (3.12)

Further proof coincides with the proof of lemma 2.2. ∎

Introduce now analogous to (2.26) family of subdomains of $\Omega^{s}_{R}$:

$\Omega^{s}_{R}(\tau):=\Omega^{s}_{R}\cap\big{\\{}(x^{\prime\prime},x_{N}):|x^{\prime\prime}|<R-\tau,x_{N}<s\big{\\}}\\\
\forall\,\tau\in(\delta,R),\ \forall\,s\in(0,s_{\delta}),\
s_{\delta}:=\delta-\frac{\delta^{2}}{2R}=\frac{R^{2}-(R-\delta)^{2}}{2R}\text{
(see \eqref{4.3})}$ (3.13)

and analogous to (2.27) energy function

$J_{j}(s,\tau):=\int_{\Omega^{2s}(\tau)}\left(|\nabla
u_{j}|^{2}+h_{\omega}(x_{N})u_{j}^{p+1}\right)\zeta_{s}(x_{N})dx,$ (3.14)

where $\zeta_{s}(\cdot)$ is a function from (2.22), (2.27).

######  Lemma 3.3.

Energy function $J_{j}(s,\tau)$ satisfies the following relation:

$\begin{split}&J_{j}(s,\tau)\leqslant
c_{3}s\left(-\frac{d}{d\tau}J_{j}(s,\tau)\right)+Ch_{\omega}(s)^{-\frac{2}{p-1}-\nu}\quad\forall\,\tau\in(\delta,R),\
\forall\,j\in\mathbb{N},\\\ &\forall\,s\in(0,2^{-1}s_{\delta}),\
\forall\,\nu>0,\ C=C(\nu)\to\infty\text{ as }\nu\to 0,\end{split}$ (3.15)

where $c_{3},C(\nu)$ do not depend on $j$; $s_{\delta}$ is from (3.13).

###### Proof.

Multiplying equation (1.4) by $u_{j}(x)\zeta_{s}(x_{N})$ and integrating in
$\Omega^{2s}_{R}(\tau)$, $\tau\in(\delta,R)$, $2s\leqslant
s_{\delta}=\delta-\frac{\delta^{2}}{2R}$, we obtain the following analog of
(2.29):

$\overline{J}_{j}(s,\tau):=\int_{\Omega^{2s}_{R}(\tau)}\left(\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}u_{jx_{k}}+H(x)u_{j}^{p+1}\right)\zeta_{s}(x_{N})dx=\\\
=\int_{\partial^{\prime\prime}\Omega^{2s}_{R}(\tau)}\sum_{i,k=1}^{N}a_{ik}(x)u_{jx_{i}}u_{j}\nu_{k}(x)\zeta_{s}(x_{N})d\sigma-\\\
-\int_{\Omega^{2s}_{R}(\tau)\setminus\Omega^{s}_{R}(\tau)}\sum_{i,k=1}^{N}a_{ik}u_{jx_{i}}\zeta_{s}(x_{N})_{x_{k}}u_{j}dx:=R_{1}+R_{2}\quad\forall\,\tau\in(\delta,R),\
s<2^{-1}s_{\delta},$ (3.16)

where
$\partial^{\prime\prime}\Omega^{2s}_{R}(\tau)=\\{(x^{\prime\prime},x_{N}):|x^{\prime\prime}|=R-\tau,0<x_{N}<2s\\}$.
Remark now the following important property of our subdomains:

$\Omega^{2s}_{R}(\tau)\setminus\Omega^{s}_{R}(\tau)\subset\Omega_{s}(\tau)\setminus\Omega_{2s}(\tau)\quad\forall\,\tau>\delta,\
\forall\,s:0<s<2^{-1}s_{\delta}$ (3.17)

and, as consequence,

$\int_{\Omega^{2s}_{R}(\tau)\setminus\Omega^{s}_{R}(\tau)}(|\nabla
u_{j}|^{2}+h_{\omega}(x_{N})u_{j}^{p+1})dx\leqslant
I_{j}(s)-I_{j}(2s)\quad\forall\,\tau>\delta,\ \forall\,s<2^{-1}s_{\delta}.$
(3.18)

Estimating term $R_{1}$ in (3.16) by the same way as in (2.30)–(2.36), we
obtain due to properties (3.17), (3.18):

$\begin{split}&|R_{1}|\leqslant
c_{4}s\int_{\partial^{\prime\prime}\Omega^{2s}_{R}(\tau)}|\nabla
u_{j}|^{2}\zeta_{s}(x_{N})d\sigma+s^{-1+\frac{p-1}{2(p+1)}}h_{\omega}(s)^{-\frac{1}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{2(p+1)}}+\\\
&+s^{-1+\frac{p-1}{p+1}}h_{\omega}(s)^{-\frac{2}{p+1}}\left(I_{j}(s)-I_{j}(2s)\right)^{1-\frac{p-1}{p+1}}\quad\forall\,\tau\in(\delta,R),\
\forall\,s<\frac{\delta}{2}-\frac{\delta^{2}}{4R}.\end{split}$ (3.19)

Analogously to (2.38) we have:

$|R_{2}|\leqslant
cs^{-\left(1-\frac{p-1}{2(p+1)}\right)}h_{\omega}(s)^{-\frac{1}{p+1}}(I_{j}(s)-I_{j}(2s))^{1-\frac{p-1}{2(p+1)}}\quad\forall\,s<2^{-1}s_{\delta}.$
(3.20)

Keeping in mind the analog of (2.40):

$\int_{\partial^{\prime\prime}\Omega^{2s}(\tau)}\left(|\nabla
u_{j}|^{2}+h_{\omega}(x_{N})u_{j}^{p+1}\right)\zeta_{s}(x_{N})d\sigma\leqslant-
c_{0}\frac{d}{d\tau}J_{j}(s,\tau),$ (3.21)

in virtue of estimates (3.19), (3.20) we deduce from (3.16) inequality (2.41)
for all $\tau\in(\delta,R)$ and $s\in(0,2^{-1}s_{\delta})$. Estimating now
$F_{j}(s)$ by the same way as in (2.43)–(2.45), we obtain (2.46) and,
consequently, necessary relation (3.15). ∎

Further proof of Theorem 1.1 repeats with obvious changes corresponding part
of the proof of Theorem 2.1. As a result of the estimates and the
constructions which are similar to (2.47)–(2.84) we obtain the following
analog of uniform estimate (2.84):

$\int_{\Omega^{s_{\delta}}(2\delta)}(|\nabla
u_{j}|^{2}+h_{\omega}(x_{N})u_{j}^{p+1})dx\leqslant\overline{K}_{j^{\prime\prime}}\quad\forall\,j>j^{\prime\prime}=j^{\prime\prime}(\delta),$
(3.22)

where $j^{\prime\prime}(\delta)\to\infty$ as $\delta\to 0$. Since
$u_{j}(x):=u_{j,\delta}(x)=0$
$\forall\,x=(x^{\prime\prime},x_{N}):x_{N}=0,|x^{\prime\prime}|<R-\delta$, we
deduce from (3.22) the following property by the same arguments as in
(2.85)–(2.92):

$\|u_{j,\delta}-u_{\infty,\delta}\|_{C^{1,\lambda}(\overline{\Omega}^{2^{-1}s_{\delta}}(4\delta))}\to
0\text{ as }j\to\infty,$ (3.23)

and, consequently, property (3.6) with $c=4$. Now since
$u_{\infty,\delta}(x)\geqslant u_{\infty}(x)$
$\forall\,x\in\overline{\Omega}_{R}$ $\forall\,\delta>0$ (3.23) yields
$u_{\infty}(x)=0$
$\forall\,x=(x^{\prime\prime},x_{N}):x_{N}=0,|x^{\prime\prime}|<R$. Moreover
$u_{\infty}(x)=\infty$ $\forall\,x=(x^{\prime\prime},x_{N}):|x|=R,x_{N}>0$.
Thus $u_{\infty}$ is desired strong barrier for equation (1.4). Theorem 1.1 is
proved.

##  4 Necessity of Dini condition for non-propagation of the point singularity

In this section we prove Theorem 1.2. First of all we construct a family of
subsolutions $v_{j}(x)$ of the problem under consideration, connected with
solutions $u_{j}(x)$, $j=1,2,...$. Namely, introduce the family of subdomains
$\Omega_{j}$ of domain $\Omega$ from (1.14):

$\Omega_{j}:=\biggr{\\{}x\in\Omega:|x^{\prime}|^{2}:=\sum_{i=2}^{N}x_{i}^{2}<r_{j}^{2},x_{1}\in\mathbb{R}^{1}\biggr{\\}},\
r_{j}=2^{-j},\ j=1,2,...,$ (4.1)

and numbers

$a_{j}=\exp\left(-\mu(r_{j})\right)=\exp\left(-\frac{\omega(r_{j})}{r_{j}}\right),A_{j}=\left(a_{j}r_{j}^{2}\right)^{\frac{1}{p-1}}\quad\forall\,j\in\mathbb{N}.$
(4.2)

Consider now a family of auxiliary problems:

$-\Delta v+a_{j}v^{p}=0\text{ in }\Omega_{j},$ (4.3) $v=0\text{ on
}\partial\Omega_{j}\setminus\\{x:x_{N}=0\\},$ (4.4)
$v(x^{\prime\prime},0)=K_{j}\delta_{a}(x^{\prime\prime}).$ (4.5)

Due to condition (1.22) on $p$, inequality (1.19) is satisfied for equation
(4.3) and therefore problem (4.3)–(4.5) has solution $v_{j}$, $j=1,2,...$.
Since $u_{j}(x)\geqslant v_{j}(x)$ on $\partial\Omega_{j}$ then

$u_{j}(x)\geqslant v_{j}(x)\text{ in }\Omega_{j}\quad\forall\,j\in\mathbb{N}.$
(4.6)

Next we estimate $v_{j}(x)$ from below. Let us perform the rescaling of
problem (4.3)–(4.5). Introducing new variables and unknown function:

$y=r_{j}^{-1}x,\ w_{j}(y)=A_{j}v_{j}(r_{j}y),\ y\in
G:=\\{y\in\mathbb{R}^{N}:|y^{\prime}|<1,y_{N}>0\\},$ (4.7)

where $A_{j}$ is from (4.2), we obtain with respect to $w_{j}(y)$ the
following problem:

$-\Delta w_{j}+w_{j}^{p}=0\text{ in }G,$ (4.8) $w_{j}(y)=0\text{ on }\partial
G\setminus\\{y:y_{N}=0\\},$ (4.9)
$w_{j}(y_{1},...,y_{N-1},0):=w_{j}(y^{\prime\prime},0)=K_{j}A_{j}r_{j}^{-(N-1)}\delta_{ar_{j}^{-1}}(y^{\prime\prime}).$
(4.10)

Without loss of generality we suppose that $a=0$ and specify now the choice of
the sequence $\\{K_{j}\\}$:

$K_{j}:=A_{j}^{-1}r_{j}^{N-1}\ \forall\,j\in\mathbb{N}.$ (4.11)

Then $w_{j}(y)=w(y)$ $\forall\,j\in\mathbb{N}$, where $w(y)$ is a solution of
the problem:

$-\Delta_{y}w+w^{p}=0\text{ in }G$ (4.12) $w(y)=0\text{ on }\partial
G\setminus\\{y:y_{N}=0\\}$ (4.13)
$w(y^{\prime\prime},0)=\delta_{0}(y^{\prime\prime}).$ (4.14)

It is obvious (due to comparison principle) that $w(\cdot)$ satisfies
estimate:

$0\leqslant w(y)\leqslant
P_{0}(y,0)=\frac{c_{N}y_{N}}{(y_{1}^{2}+y_{2}^{2}+...+y_{N}^{2})^{\frac{N}{2}}}\quad\forall\,y\in
G,$ (4.15)

where $P_{0}(\cdot,\cdot)$ is the Poisson kernel from (1.18). Therefore,
function $w^{(y_{1})}(y^{\prime}):=w(y_{1},y^{\prime})$ has the following
properties:

$\|w^{(y_{1})}(\cdot)\|_{L^{\infty}\left(B_{1,+}^{N-1}\right)}<\infty\
\forall\,y_{1}\neq 0,\\\
\varphi(y_{1}):=\varphi_{w}(y_{1}):=\|w^{(y_{1})}(\cdot)\|_{L^{\infty}\left(B_{1,+}^{N-1}\right)}\to
0\text{ as }|y_{1}|\to\infty,$ (4.16)

where
$B_{1,+}^{N-1}=\left\\{y^{\prime}=(y_{2},...,y_{N})\in\mathbb{R}^{N-1}:|y^{\prime}|<1,\
y_{N}>0\right\\}$. Let us fix an arbitrary value $y_{1}^{(0)}>0$ and consider
our solution $w$ as a solution of equation (4.12) in infinite cylindrical
domain $G\cap\left\\{y:y_{1}>y_{1}^{(0)}\right\\}$ satisfying boundary
condition:

$w(y)=0\text{ on }\partial G\cap\left\\{y:y_{1}>y_{1}^{(0)}\right\\}.$ (4.17)

Due to lemma 3.1 from [5], for solution $w$ there exists a number $\alpha>0$
such that

$\lim_{y_{1}\to\infty}\exp\left(\sqrt{\lambda_{1}}\left(y_{1}-y_{1}^{(0)}\right)\right)w(y)=\alpha\psi_{1}(y^{\prime}),\quad\max_{y^{\prime}\in
B_{1,+}^{N-1}}\psi_{1}(y^{\prime})=\psi_{1}(\tilde{y}^{\prime})=1$ (4.18)

uniformly in $B_{1,+}^{N-1}$. Here $\lambda_{1}>0$ is the first eigenvalue and
$\psi_{1}$ is the corresponding normalized eigenfunction of $-\Delta$ in
$B_{1,+}^{N-1}$, constant $\alpha=\alpha(y_{1}^{(0)})$ satisfies estimate:

$0<\alpha\leqslant c\quad\sup_{y^{\prime}\in
B_{1,+}^{N-1}}w^{(y_{1}^{(0)})}(y^{\prime})=c\varphi(y_{1}^{(0)}),$ (4.19)

where $c<\infty$ doesn’t depend on solution $w$, function $\varphi(\cdot)$ is
from (4.16).

######  Remark 4.1.

Lemma 3.1 is formulated in [5] for cylindrical domain $G=\mathbb{R}^{1}\times
B_{1}^{N-1}$. But its proof is based on results of §2 of the paper [4], which
are true for much more general class of cylindrical domains, particularly, for
$G=\mathbb{R}^{1}\times B_{1,+}^{N-1}$.

Thus, due to (4.16), (4.11), (4.7) it follows from (4.18) the existence of
constant $\beta<\infty$ such that

$\frac{\alpha}{2A_{j}}\psi_{1}\left(r_{j}^{-1}x^{\prime}\right)\exp\biggr{(}-\sqrt{\lambda_{1}}\frac{(x_{1}-x_{1}^{(0)})}{r_{j}}\biggr{)}\leqslant
v_{j}(x)\leqslant\\\
\leqslant\frac{2\alpha}{A_{j}}\psi_{1}\left(r_{j}^{-1}x^{\prime}\right)\exp\biggr{(}-\sqrt{\lambda_{1}}\frac{(x_{1}-x_{1}^{(0)})}{r_{j}}\biggr{)}\quad\forall\,x\in\Omega_{j}:x_{1}>x_{1}^{(j)}:=\beta
r_{j},\ \forall\,j\in\mathbb{N}.$ (4.20)

Due to (4.6) inequality (4.20) yields the first rough estimate from below of
solution $u_{j}$ under consideration:

$u_{j}(x)\geqslant v_{j}(x)\geqslant
B_{j}\psi_{1}\left(r_{j}^{-1}x^{\prime}\right)\exp\left(-\sqrt{\lambda_{1}}\frac{x_{1}}{r_{j}}\right)\quad\forall\,x\in\Omega_{j}:x_{1}>\beta
r_{j},$ (4.21)

where
$B_{j}=(2A_{j})^{-1}\alpha(y_{1}^{(0)})\exp\left(\sqrt{\lambda_{1}}y_{1}^{(0)}\right)$.
Let us define number $\tau_{j}>0$ by the following relation:

$B_{j}\exp\left(-\sqrt{\lambda_{1}}\frac{\tau_{j}}{r_{j}}\right)=A_{j-1}^{-1}=\left(a_{j-1}r_{j-1}^{2}\right)^{-\frac{1}{p-1}},$
(4.22)

which yields by simple computations:

$\frac{\tau_{j}}{r_{j}}=\frac{\mu(r_{j})-\mu(r_{j-1})}{\sqrt{\lambda_{1}}(p-1)}+c_{1},\quad
c_{1}=y_{1}^{(0)}+\frac{\ln\alpha}{\sqrt{\lambda_{1}}}-\frac{3-p}{(p-1)\sqrt{\lambda_{1}}}.$
(4.23)

Since due to condition (1.23), $(\mu(r_{j})-\mu(r_{j-1}))\to\infty$ as $j\to
0$, there exists number $j^{\prime}$ such that

$\frac{r_{j}\left(\mu(r_{j})-\mu(r_{j-1})\right)}{\sqrt{\lambda_{1}}(p-1)}\leqslant\tau_{j}\leqslant\frac{2r_{j}\left(\mu(r_{j})-\mu(r_{j-1})\right)}{\sqrt{\lambda_{1}}(p-1)}\quad\forall\,j>j^{\prime}$
(4.24)

and, particularly,

$\tau_{j}>\beta r_{j}\quad\forall\,j\geqslant j^{\prime}.$ (4.25)

As consequence, estimate (4.21) yields:

$u_{j}(\tau_{j},x^{\prime})\geqslant v_{j}(\tau_{j},x^{\prime})\geqslant
A_{j-1}^{-1}\psi_{1}\left(r_{j}^{-1}x^{\prime}\right).$ (4.26)

Let us fix arbitrary large $j\in\mathbb{N}$ in (4.26) and consider
$\left\\{v_{j-k}(x)\right\\}$, $k=1,2,...,j-j^{\prime}$ ($j^{\prime}$ is from
(4.25)), as solutions of the following boundary value problems

$-\Delta v_{j-k}+a_{j-k}v_{j-k}^{p}=0\text{ in
}\Omega_{j-k}\cap\\{x_{1}>\tau_{j}+...+\tau_{j-k+1}\\},$ (4.27)
$v_{j-k}=0\text{ on
}\partial\Omega_{j-k}\cap\biggr{\\{}x_{1}>\sum_{i=0}^{k-1}\tau_{j-i}\biggr{\\}},$
(4.28)
$v_{j-k}\biggr{(}\sum_{i=0}^{k-1}\tau_{j-i},x^{\prime}\biggr{)}=\gamma_{j-k}(x^{\prime}):=\begin{cases}A_{j-k}^{-1}\psi_{1}\left(\frac{x^{\prime}}{r_{j-k+1}}\right)&\text{if
}|x^{\prime}|<r_{j-k+1},\\\ 0&\text{if
}r_{j-k+1}\leqslant|x^{\prime}|\leqslant r_{j-k}.\end{cases}$ (4.29)

where $\Omega_{j-k}=\left\\{x\in\Omega:|x^{\prime}|<r_{j-k}\right\\}$,
$A_{j-k}=\left(a_{j-k}r_{j-k}^{2}\right)^{\frac{1}{p-1}}$,
$a_{j-k}=\exp(-\mu(r_{j-k}))$. To define sequence $\\{\tau_{j-k}\\}$,
$k=1,2,...$, we need to convert problem (4.27)-(4.29) to the new variables:

$y=r_{j-k}^{-1}x,\ w_{j-k}(y):=A_{j-k}v_{j-k}(r_{j-k}y).$ (4.30)

It is easy to see that all these functions $w_{j-k}(y)$ can be obtained as a
shift of unique function $w(y)$:

$w_{j-k}(y_{1},y^{\prime}):=w\biggr{(}y_{1}-\sum_{i=0}^{k-1}\tau_{j-i}r_{j-k}^{-1},y^{\prime}\biggr{)},$
(4.31)

where $w(y)$ is a solution of the problem:

$-\Delta_{y}w+w^{p}=0\text{ in }G\cap\\{y_{1}>0\\},$ (4.32) $w=0\text{ on
}\partial\Omega_{j-k}\cap\left\\{y_{1}>0\right\\},$ (4.33)
$w(0,y^{\prime}):=\begin{cases}\psi_{1}\left(2y^{\prime}\right)&\text{if
}|y^{\prime}|<2^{-1},\\\ 0&\text{if }2^{-1}\leqslant|y^{\prime}|\leqslant
1.\end{cases}$ (4.34)

Due to lemma 3.1 from [5] for this function $w$ there exists number
$\alpha_{1}>0$ such that the following property takes place:

$\lim_{y_{1}\to\infty}\exp\left(\sqrt{\lambda_{1}}y_{1}\right)w(y)=\alpha_{1}\psi_{1}(y^{\prime}),$
(4.35)

where constant $\alpha_{1}>0$ satisfies estimate

$\alpha_{1}\leqslant c\sup_{y^{\prime}\in
B_{1,+}^{N-1}}\psi_{1}(2y^{\prime})=c$

with the same as in (4.19) constant $c$. As in (4.20), property (4.35) yields
the existence of constant $\beta_{1}<\infty$ such that

$\frac{\alpha_{1}}{2A_{j-k}}\psi_{1}(r_{j-k}^{-1}x^{\prime})\exp\left(\frac{-\sqrt{\lambda_{1}}(x_{1}-h_{j,k})}{r_{j-k}}\right)\leqslant
v_{j-k}(x)\leqslant\frac{2\alpha_{1}}{A_{j-k}}\psi_{1}(r_{j-k}^{-1}x^{\prime})\times\\\
\times\exp\left(\frac{-\sqrt{\lambda_{1}}(x_{1}-h_{j,k})}{r_{j-k}}\right)\quad\forall\,x\in\Omega_{j-k}:x_{1}\geqslant
h_{j,k}+r_{j-k}\beta_{1};h_{j,k}:=\sum_{i=0}^{k-1}\tau_{j-i}.$ (4.36)

Now define value $\tau_{j-k}$ by relation:

$\frac{\alpha_{1}}{2A_{j-k}}\exp\left(-\sqrt{\lambda_{1}}\frac{\tau_{j-k}}{r_{j-k}}\right)=A_{j-k-1}^{-1},\quad
k=1,2,....$ (4.37)

Then using nonnegativity of $u_{j}$ in $\Omega$, properties (4.26) and initial
condition (4.29) with $k=1$, we get:

$u_{j}(\tau_{j},x^{\prime})\geqslant
v_{j-1}(\tau_{j},x^{\prime})\quad\forall\,x^{\prime}:|x^{\prime}|<r_{j}.$

Using additionally that $v_{j-1}(\tau_{j},x^{\prime})=0$ if
$r_{j}<|x^{\prime}|<r_{j-1}$ we obtain that

$u_{j}(\tau_{j},x^{\prime})\geqslant
v_{j-1}(\tau_{j},x^{\prime})\quad\forall\,x^{\prime}:|x^{\prime}|<r_{j-1}.$
(4.38)

Now using the comparison principle for solution $u_{j}$ and subsolution
$v_{j-1}$ of equation (1.14) in domain $\Omega_{j-1}\cap\\{x_{1}>\tau_{j}\\}$
we obtain

$u_{j}(x)\geqslant
v_{j-1}(x)\quad\forall\,x\in\Omega_{j-1}\cap\\{x_{1}>\tau_{j}\\}.$ (4.39)

Now we prove our main intermediate inequality:

$u_{j}(x)\geqslant
v_{j-k}(x)\quad\forall\,x\in\Omega_{j-k}\cap\biggr{\\{}x_{1}\geqslant\sum_{i=0}^{k-1}\tau_{j-i}\biggr{\\}}\
\forall\,k:1\leqslant k<j-j^{\prime},$ (4.40)

where $j^{\prime}$ is from (4.24), (4.25). Let us suppose that (4.40) is true
for some $k\geqslant 1$. Then we have to prove that

$u_{j}(x)\geqslant
v_{j-k-1}(x)\quad\forall\,x\in\Omega_{j-k-1}\cap\Big{\\{}x_{1}\geqslant\sum_{i=0}^{k}\tau_{j-i}\Big{\\}}.$
(4.41)

To prove (4.41) it is sufficient, due to comparison principle, to show that

$u_{j}\Big{(}\sum_{i=0}^{k}\tau_{j-i},x^{\prime}\Big{)}\geqslant
v_{j-k-1}\Big{(}\sum_{i=0}^{k}\tau_{j-i},x^{\prime}\Big{)}.$ (4.42)

From initial condition (4.29) for function $v_{j-k-1}(x)$ we have:

$v_{j-k-1}\Big{(}\sum_{i=0}^{k}\tau_{j-i},x^{\prime}\Big{)}=\begin{cases}A_{j-k-1}^{-1}\psi_{1}\left(\frac{x^{\prime}}{r_{j-k}}\right),&\mbox{if
}|x^{\prime}|\leqslant r_{j-k}\\\ 0,&\mbox{if }r_{j-k}<|x^{\prime}|\leqslant
r_{j-k-1}.\end{cases}$ (4.43)

Additionally, estimate (4.36) yields:

$v_{j-k}\Big{(}\sum_{i=0}^{k}\tau_{j-i},x^{\prime}\Big{)}\geqslant\frac{\alpha_{1}}{2A_{j-k}}\psi_{1}(r_{j-k}^{-1}x^{\prime})\exp\Big{(}-\frac{\sqrt{\lambda_{1}}\tau_{j-k}}{r_{j-k}}\Big{)}\text{
if }\tau_{j-k}\geqslant\beta_{1}r_{j-k}.$ (4.44)

In virtue of definition (4.37) of $\tau_{j-k}$ the last inequality leads to:

$v_{j-k}\Big{(}\sum_{i=0}^{k}\tau_{j-i},x^{\prime}\Big{)}\geqslant
A_{j-k-1}^{-1}\psi_{1}(r_{j-k}^{-1}x^{\prime})\quad\forall\,x^{\prime}:|x^{\prime}|\leqslant
r_{j-k}.$ (4.45)

Since $u_{j}\left(\sum_{i=0}^{k}\tau_{j-i},x^{\prime}\right)\geqslant
0=v_{j-k-1}\left(\sum_{i=0}^{k}\tau_{j-i},x^{\prime}\right)$ if
$r_{j-k}\leqslant|x^{\prime}|\leqslant r_{j-k-1}$, relations (4.43) and (4.45)
lead to (4.42). It is necessary now to investigate the properties of sequence
$\\{\tau_{j-k}\\}$, $k=1,2,...$ more precisely. Using standard computations
definition (4.37) yields:

$\frac{\tau_{j-k}}{r_{j-k}}=\frac{\mu(r_{j-k})-\mu(r_{j-k-1})}{\sqrt{\lambda_{1}}(p-1)}+c_{2},\quad
c_{2}=\frac{\ln\alpha_{1}}{\sqrt{\lambda_{1}}}-\frac{3-p}{(p-1)\sqrt{\lambda_{1}}}.$

Due to condition (1.23) on the function $\mu(\cdot)$ there exists constant
$j^{\prime\prime}<\infty$, which does not depend on $j$, such that the
following relations hold:

$\frac{r_{j-k}\left(\mu(r_{j-k})-\mu(r_{j-k-1})\right)}{\sqrt{\lambda_{1}}(p-1)}\leqslant\tau_{j-k}\leqslant\frac{2\left(\mu(r_{j-k})-\mu(r_{j-k-1})\right)r_{j-k}}{\sqrt{\lambda_{1}}(p-1)},\quad\tau_{j-k}>\beta_{1}r_{j-k}\\\
\forall\,k<j-j^{\prime\prime}.$ (4.46)

Moreover condition (1.23) yields existence of constant ${\ae}<1$ such that

$\mu(r_{j-k-1})\mu(r_{j-k})^{-1}<{\ae}<1\quad\forall\,k:j-k>j^{\prime\prime}.$
(4.47)

Therefore it follows from (4.46) that

$\frac{2\mu(r_{i})}{\sqrt{\lambda_{1}}(p-1)}\geqslant\frac{\tau_{i}}{r_{i}}\geqslant\frac{(1-{\ae})\mu(r_{i})}{\sqrt{\lambda_{1}}(p-1)}\quad\forall\,i>j^{(1)}:=\max(j^{\prime},j^{\prime\prime}).$
(4.48)

Consequently due to definition (1.17) of $\mu(\cdot)$:

$\sum_{k=0}^{j-i}\tau_{i+k}=\sum_{k=0}^{j-i}\tau_{j-k}\geqslant{\ae}_{1}\sum_{k=0}^{j-i}\omega(r_{j-k})\geqslant{\ae}_{1}\sum_{k=0}^{j-i}\int_{r_{j-k}}^{r_{j-k-1}}\frac{\omega(s)}{s}ds={\ae}_{1}\int_{r_{j}}^{r_{i-1}}\frac{\omega(s)}{s}ds\\\
\forall\,j,i:j>i>j^{(1)},\
{\ae}_{1}:=\frac{1-{\ae}}{\sqrt{\lambda_{1}}(p-1)}.$ (4.49)

Additionally, left inequality in (4.48) and assumption (1.23) on function
$\omega(\cdot)$ yields $\tau_{i}\to 0$ as $i\to\infty$. Therefore for
arbitrary fixed number $g>0$ there exists number $i_{g}$ such that
$\tau_{i}<g$ $\forall\,i>i_{g}$. Then in virtue of condition (1.24) and
inequalities (4.49) there exists number $\bar{j}=\bar{j}(i,g)<\infty$ such
that

$g<\sum_{k=0}^{\overline{j}-i}\tau_{\overline{j}-k}=\sum_{k=0}^{\overline{j}-i}\tau_{i+k},\
\sum_{k=1}^{\overline{j}-i}\tau_{i+k}\leqslant
g\quad\forall\,i>j^{(2)}:=\max\left\\{j^{(1)},i_{g}\right\\}.$ (4.50)

Notice that $\overline{j}(i,g)-i\to\infty$ as $i\to\infty$. Let us rewrite
proved relation (4.40) in the following equivalent form:

$u_{j}(x)\geqslant
v_{i}(x)\quad\forall\,x=(x_{1},x^{\prime})\in\Omega_{i}\cap\Big{\\{}x_{1}\geqslant\sum_{k=1}^{j-i}\tau_{i+k}\Big{\\}}\
\forall\,j>i>j^{\prime}.$ (4.51)

Let us define a sequence of points $\\{x^{(i)}\\}$:

$\begin{split}&x^{(i)}=\left(x^{(i)}_{1},x^{{}^{\prime}(i)}\right):x^{\prime(i)}=r_{i+1}\,\tilde{y}^{\prime},\
\tilde{y}^{\prime}\text{ is from \eqref{2.19}},\\\
&x_{1}^{(i)}=\sum_{k=0}^{\overline{j}-i}\tau_{i+k}=\sum_{k=0}^{\overline{j}-i}\tau_{\overline{j}-k}\quad\forall\,i>j^{(2)},\
\overline{j}=\overline{j}(i,g)\text{ is from \eqref{2.50}}.\end{split}$

We deduce from (4.29) after simple transformation:

$v_{i}\Big{(}\sum_{k=1}^{j-i}\tau_{i+k},x^{\prime}\Big{)}=A_{i}^{-1}\psi_{1}(r_{i+1}^{-1}x^{\prime})\quad\forall\,x^{\prime}:|x^{\prime}|<r_{i+1}.$

Therefore using (4.50) and the definition of point $x^{(i)}$ we get in virtue
of (4.18):

$v_{i}\left(g+\lambda_{i}\tau_{i},x^{\prime(i)}\right)=A_{i}^{-1}\psi_{1}(r_{i+1}^{-1}x^{\prime(i)})=A_{i}^{-1}\quad\forall\,i>j^{(2)},\
0<\lambda_{i}<1.$ (4.52)

Let us define sequence $X^{(i)}=(X_{1}^{(i)},x^{\prime(i)})$,
$X_{1}^{(i)}=g+\lambda_{i}\tau_{i}$. Then since $A_{i}^{-1}\to\infty$ as
$i\to\infty$ we deduce from (4.51) and (4.52):

$u_{\infty}(X^{(i)})\geqslant u_{\overline{j}}(X^{(i)})\geqslant
v_{i}(X^{(i)})=A_{i}^{-1}\to\infty\text{ as }i\to\infty,$ (4.53)

where $\overline{j}=\overline{j}(i,g)$ is from (4.50). Since $X_{1}^{(i)}\to
g$ and $x^{\prime(i)}\to 0$ as $i\to\infty$, relation (4.53) yields that
$u_{\infty}(g,0,...,0)=\infty$. Since $(g,0,...,0)$ is an arbitrary point from
$L\cap\\{\mathbb{R}_{+}^{1}\\}=L_{+}$ then
$u_{\infty}|_{L\cap\\{\mathbb{R}_{+}^{1}\\}}=\infty$. Now we return to model
problem (4.12)–(4.14) and consider its solution $w$ as a solution of equation
(4.12) in infinite cylindrical domain $G\cap\\{y:y_{1}<y_{1}^{(0)}<0\\}$,
satisfying boundary condition: $w(y)=0$ on $\partial
G\cap\\{y:y_{1}<y_{1}^{(0)}<0\\}$. If we repeat all above considerations using
this solution $w$ in the mentioned cylindrical domain, we obtain that
$u_{\infty}|_{L\cap\\{\mathbb{R}_{-}^{1}\\}}=\infty$. Theorem 1.2 is proved.

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