\begin{theorem}\label{main theorem}
Let $\Omega\subset\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.
If $\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \ then the Bergman metric of $\Omega$ is well-defined in a nonempty open subset of $\Omega$, denoted by $\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\Omega^*$.
\end{theorem}

For any bounded domain $D\subset \mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.
Cheng and Yau \cite{CY80} proved that every bounded pseudoconvex domain in $\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.
Later, Mok and Yau \cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.

A natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,
Cheng \cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.
Cheng's conjecture was confirmed in dimension two by Fu–Wong \cite{FW97} and Nemirovski--Shafikov \cite{NS06}, and was resolved in all dimensions by Huang–Xiao \cite{HX16} based on earlier work of many authors.
Subsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \cite{HL23}, Ebenfelt–Xiao–Xu \cite{EXX22,EXX24}, and references therein.
Related variations of Cheng’s conjecture were also discussed by S. Li in his papers \cite{L1, L2, L3}.

In a more recent development, Savale and Xiao \cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\mathbb{C}^2$.
They proved that a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\mathbb{C}^2$.
A prior result by Fu–Wong \cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\mathbb{C}^2$.

Despite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.
We will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \ref{def1-9-24}) is not Einstein.

\begin{definition}\label{def1-9-24}
Let $\Omega$ be a possibly unbounded domain in $\mathbb{C}^{n}$  with $n>1$ and let $p\in\partial\Omega$.
We say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\rho_{1},\dots,\rho_{m}\colon U\to\mathbb{R}$ with $m>1$  such that
$\Omega\cap U=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\}$  and
$\{\partial\rho_{1}|_p,\dots,\partial\rho_{m}|_p\}$  are linearly independent over $\mathbb{C}$.
\end{definition}

One of the main tools used in the proof of Theorem \ref{main theorem} is the rescaling argument, which has been used to work on many related
problems. In particular, in connection with our present work, we
mention the papers by Wong \cite{W77},
Kim \cite{Kim}, Kim-Yu \cite{KY}, Krantz-Yu \cite{KYu}
and Boas-Straube-Yu \cite{BSY}, where the rescaling method has been used to study the boundary limit of various quantities associated with the Bergman metric. Indeed, our current work has benefited from their studies. A recent application of the rescaling method can also be found in Huang-Zhu \cite{HZ}, where it is employed in solving a CR transversality problem. Another recent application of the rescaling method was used in working on the pinched properties of a K\"ahler metric is a recent paper of Bracci-Gauthier-Zimmer \cite{BGZ}.

We next recall the definition of strongly pseudoconvex polyhedral boundary points for a domain $\Omega\subset{\mathbb C}^n$.
\begin{definition}\label{def1-9-24}
Let $\Omega$ be a possibly unbounded domain in $\mathbb{C}^{n}$ with $n>1$ and let $p\in\partial\Omega$.
We say that $p$ is a strongly pseudoconvex polyhedral boundary point if there exists a neighborhood $U$ of $p$ in $\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\rho_{1},\dots,\rho_{m}\colon U\to\mathbb{R}$ with $m>1$ such that
$\Omega\cap U=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\}$ and
$\{\partial\rho_{1}|_p,\dots,\partial\rho_{m}|_p\}$ are linearly independent over $\mathbb{C}$.
\end{definition}

\begin{proposition}\label{prop1-4-3}
Let $\Omega$ be a possibly unbounded pseudoconvex domain in $\mathbb C^n$ which is stronlgy pseudoconvex polyhedral at some boundary point $p\in\partial\Omega$. Let $U$ be a neighborhood of $p$ such that $U\cap\Omega$ is connected, and on which the Bergman metric $g_{\Omega}$ is well defined. Then its Bergman metric $g_\Omega$ is K\"ahler-Einstein on $U\cap\Omega$ if and only if its Bergman canonical invariant $J_{\Omega}\equiv (n+1)^n\frac{\pi^n}{n!}$ on $U\cap\Omega$.
\end{proposition}
\begin{proof}
By Corollary \ref{coro1-3-30}, for any smooth boundary point $q\in U\cap\partial\Omega$ near which $\partial\Omega$ is strongly pseudoconvex, one has
 \begin{equation}\label{e2-3-31}
 \lim_{z\rightarrow q}J_{\Omega}(z)=\frac{(n+1)^n\pi^n}{n!}.
 \end{equation}
 The Bergman metric $g_{\Omega}$ {is} K\"ahler-Einstein if its Ricci curvature $R_{\Omega}=c g_{\Omega}$ for some constant $c$. By Corollary \ref{coro1-3-30}, one has $c=-1$. Consequently, the K\"ahler-Einstein assumption implies that $\log J_{\Omega}$ is a pluriharmonic function on $U\cap\Omega$. Now, for any attached holomorphic disk $\phi:\Delta\rightarrow\Omega$ where $\phi$ is holomorphic in $\Delta:=\{t\in {\mathbb C}: |t|<1\}$, continuous up to $\overline {\Delta}$, and $\phi(\partial\Delta)$ is contained in the smooth part of $ U\cap \partial\Omega$, we have that $\log
 J_{\Omega}(\phi(t))$ is harmonic. Since it is constant on the strongly pseudoconvex part of the boundary by \eqref{e2-3-31}, it assumes the value
\[
\log\frac{(n+1)^n\pi^n}{n!}
\]
everywhere on $\Delta$. Now, since $\partial \Omega$ is strongly pseudoconvex near $q$, the union of such disks fills up an open subset of $\partial\Omega$ near $q$. Since $\log
 J_{\Omega}$ is well defined in $U\cap\Omega$ on which it is real analytic, we conclude that $\log J_{\Omega}\equiv
 \log\frac{(n+1)^n\pi^n}{n!}$ over $U\cap\Omega$ as $U\cap\Omega$ is connected by definition.
Conversely, if $J_\Omega(z)$ takes a constant value near $p$, then the Bergman metric is obviously K\"ahler-Einstien. Thus, we have the conclusion of the proposition.
\end{proof}
\begin{remark}\label{remark1-10-21}

Note that the zero set of the Bergman kernel function, denoted by $E$, is a complex analytic variety in $\Omega$.
Thus, $J_\Omega$ is a well-defined real-analytic function on $\Omega\setminus E$. Since $\Omega\setminus E$ is connected, $J_\Omega$ is constant if and only if it is constant on some nonempty open subset of $\Omega$. In particular, when $\Omega$ contains a $C^2$-smooth strongly pseudoconvex boundary point, the Bergman metric of the domain $\Omega$ is Kähler–Einstein wherever it is well-defined if and only if $J_\Omega=c$ is a constant on a certain open subset of $\Omega\setminus E$ .
 In this case, $c=\frac{(n+1)^n\pi^n}{n!}$, and the Bergman space $A^2(\Omega)$ separates holomorphic directions at any point in $\Omega\setminus E$ and thus the Bergman metric is well-defined in $\Omega\setminus E$.
\end{remark}
\section{Stability of the Bergman kernels}\label{sec2}

\begin{proof}[Proof of Theorem \ref{main theorem}]
Let $\Omega$ be a pseudoconvex domain which is strongly pseudoconvex polyhedral at a boundary point $p\in\partial\Omega$ as defined in Definition~\ref{def1-9-24}. After shrinking $U$ if necessary, we may assume that
\begin{enumerate}
\item[(i)] there are $C^{2}$-smooth strongly plurisubharmonic functions $\{\rho_{j}\}_{j=1}^{m}$ with $m>1$ on $U$ such that
\begin{equation}\label{9-24-a2}
 U\cap\Omega=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\};
\end{equation}
\item[(ii)] the vectors $\partial\rho_{1}(q),\dots,\partial\rho_{m}(q)$ are linearly independent over $\mathbb C$ for $q\in U\cap \overline\Omega$.
\end{enumerate}
By Lemma \ref{25-9-23a1}, after a suitable change of coordinates on a small neighborhood $V\Subset U$ of $p$, we may assume that $p=0$ and
\begin{equation}\label{9-24-a1}
V\cap\Omega=\{(z_1,\cdots, z_n)\in V: {\rm Im~ }z_1>\Phi_1(z, \overline z), \cdots, {\rm Im}~ z_m>\Phi_m(z, \overline z)\}
\end{equation}
where $$\Phi_1(z, \overline z)=|z|^2+R_1(z), \Phi_j(z, \overline z)=\sum a^j_{\alpha\overline\beta} z_{\alpha}\overline z_{\beta}+R_j(z), \quad 2\leq j\leq m$$
with each remainder $R_j=\mathcal O(|z|^3)$. Write $$U_0=\{z\in\mathbb C^n: |z_j|<\varepsilon_0, 1\leq j\leq n\}$$ with $\varepsilon_0\ll 1$ such that $U_0\Subset V$ and on $U_0$ one has
$$|R_j(z)|\leq \frac{A_0}{2}|z|^2, \forall z\in U_0, 1\leq j\leq m,$$
where $A_0$ is defined in (\ref{10-27-a3}).
We first construct a bounded continuous plurisubharmonic function $\psi$ in $\Omega$ where $\psi$ is strictly plurisubharmonic near $p=0$ as follows:
\begin{lemma}\label{le3-28-1}
After shrinking $U_0$ if necessary, there exists a plurisubharmonic function $\psi:\Omega\rightarrow (-\infty, 0)$ such that $$\psi (z)>-c_0, ~\left(\frac{\partial^2\psi(z)}{\partial z_j\partial\overline z_k}\right)\geq c ~I_{n}, ~~z\in U_0\cap\Omega$$for some constants $c_0>0,~c>0$.
\end{lemma}
\begin{proof}
Let $(V, z)$ be the coordinates given as in (\ref{9-24-a1}) and set $\varphi=\frac{A_0}4|z|^2-y_1$. Then $\varphi$ is strictly plurisubharmonic on $V$ and satisfies $\varphi(0)=0, \varphi(z)<0$ when $z\in U_0\cap\overline\Omega\setminus \{0\}$. Take $r>0$ such that $\overline {\mathbb B^n(0, r)}\Subset U_0$. Set $M=\max\{\varphi(z): z\in \partial \mathbb B^n(0, r)\cap\overline \Omega\}$. Then $M<0$. Now we define $\psi$ as follows:
\begin{equation}
 \psi=\begin{cases}
 \max\{\varphi(z), M\}, &z\in \mathbb B^n(0, r)\cap\Omega\\
 M, &z\in\Omega\setminus \mathbb B^n(0, r).
 \end{cases}
\end{equation}
Then $\psi$ is a bounded and continuous plurisubharmonic function on $\Omega$ with $$\psi(0)=0, ~~\psi(z)<0, \forall z\in\overline\Omega\setminus\{0\}.$$
Furthermore, $\psi$ is equal to $\varphi$ near $0$ with $$\left(\frac{\partial^2\psi}{\partial z_i\partial\overline z_j}\right)= \frac{A_0}4 I_n $$
in some neighborhood $U$ of $0$ in $\mathbb C^n$. Moreover, $\psi>-c_0$ on $U$ for some positive constant $c_0$.
\end{proof}

\begin{corollary}
Let $\Omega\subset\mathbb{C}^{n}$, with $n>1$,  be a bounded pseudoconvex domain.  If $\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point,then its  Bergman metric cannot be Einstein.
\end{corollary}

One of the main tools used in the proof of Theorem \ref{main theorem} is the rescaling argument, which has been  used  to work on many related
problems. In particular, in connection with our present work, we
mention   the papers by Wong \cite{W77},
Kim \cite{Kim},  Kim-Yu \cite{KY}, Krantz-Yu \cite{KYu}
and Boas-Straube-Yu \cite{BSY},  where the rescaling method has been used to study the boundary limit of  various quantities associated with  the Bergman metric. Indeed, our current work  has benefited   from  their studies.  A recent application of the rescaling method can also be found in Huang-Zhu \cite{HZ}, where it is employed in solving a CR transversality problem. Another recent  application of the rescaling method was used in working on the pinched  properties of a K\"ahler metric is a recent  paper of Bracci-Gauthier-Zimmer \cite{BGZ}.

The  ideas of our proof of the main theorem can be stated briefly as follows: First, we   show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball. Next we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point  tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.
To obtain such a sequence, we assign weight $2$ to one of the complex normal directions, weight $1.5$  to the other normal directions, and weight $1$  to the remaining CR directions. The main part of the paper is then devoted to showing that the Bergman invariant function of this product domain coincides with that of  $\Omega$.
A direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned  product domain, leading to a contradiction.  In this respect, our proof departs from earlier approaches to the Cheng conjecture and its generalizations, where the contradiction is derived via spherical CR geometry and the Qi-Keng Lu uniformization theorem \cite{HX20}.

The ideas of the proof of Theorem~\ref{main theorem} are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”

The ideas of the proof, in establishing the main theorem, are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”

Question:
Let \(\Omega\subset \mathbb C^n\) with \(n>1\) be a pseudoconvex domain, possibly unbounded. Assume that \(\Omega\) has a strongly pseudoconvex polyhedral boundary point \(p\in \partial\Omega\), meaning that there exist a neighborhood \(U\) of \(p\) in \(\mathbb C^n\) and \(C^2\)-smooth strongly plurisubharmonic functions \(\rho_1,\dots,\rho_m:U\to\mathbb R\) with \(m>1\) such that
\[
\Omega\cap U=\{z\in U: \rho_1(z)<0,\dots,\rho_m(z)<0\},
\]
and the complex differentials \(\{\partial\rho_1|_p,\dots,\partial\rho_m|_p\}\) are linearly independent over \(\mathbb C\). Under these assumptions, which statement about the Bergman metric of \(\Omega\) holds?

There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric cannot be Einstein on any open subset of \(\Omega^*\).

Choices:
B. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and if the Bergman metric is Einstein on some open subset of \(\Omega^*\), then \(\Omega\) is biholomorphic to the unit ball.
C. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined.
D. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \(\Omega^*\).
E. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman invariant function on \(\Omega^*\) is constant and equal to that of the unit ball.

conclusion-from-contradiction replaced by global classification

constant-ball invariant asserted unconditionally instead of under Einstein hypothesis

The stem gives hypotheses and relevant definitions but does not explicitly or implicitly reveal the correct conclusion. No option is singled out by wording in the prompt.

The item is largely a theorem-recall/application question: the correct choice appears to match the intended theorem conclusion closely. However, it is not a pure tautology because the distractors include stronger, weaker, and altered statements.

Some reasoning is needed to distinguish the exact supported conclusion from a weaker true statement and stronger false variants, but the problem mainly tests recognition of the theorem rather than deeper generative derivation.

The distractors are plausible and mathematically targeted: one is weaker-but-true, one is an overstrong rigidity claim, one reverses the quantifier structure, and one asserts an unjustified ball-like invariant. These reflect realistic failure modes.

A solid MCQ with good distractors and no answer leakage, but it remains mostly a theorem-recall item rather than a strongly generative reasoning question.


Question:
Let \(\Omega\subset \mathbb C^n\) with \(n>1\) be a pseudoconvex domain, possibly unbounded. Assume that \(\Omega\) has a strongly pseudoconvex polyhedral boundary point \(p\in \partial\Omega\), meaning that there exist a neighborhood \(U\) of \(p\) in \(\mathbb C^n\) and \(C^2\)-smooth strongly plurisubharmonic functions \(\rho_1,\dots,\rho_m:U\to\mathbb R\) with \(m>1\) such that
\[
\Omega\cap U=\{z\in U: \rho_1(z)<0,\dots,\rho_m(z)<0\},
\]
and the complex differentials \(\{\partial\rho_1|_p,\dots,\partial\rho_m|_p\}\) are linearly independent over \(\mathbb C\). Under these assumptions, which statement about the Bergman metric of \(\Omega\) holds?

Choices:
B. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and if the Bergman metric is Einstein on some open subset of \(\Omega^*\), then \(\Omega\) is biholomorphic to the unit ball.
C. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined.
D. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \(\Omega^*\).
E. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman invariant function on \(\Omega^*\) is constant and equal to that of the unit ball.

The stem does not explicitly state the conclusion or otherwise reveal choice A. It gives only the hypotheses and asks for the valid consequence.

This is close to a theorem-recall item: the stem lists the hypotheses of a specific result and asks for its conclusion. However, it is not a pure verbatim restatement because the options vary in logical strength and quantification.

Some reasoning is needed to distinguish the exact conclusion from weaker true statements and stronger false variants, especially around quantifiers and rigidity claims. Still, the item mainly tests recognition of the theorem rather than substantial derivation.

The distractors are mathematically plausible and well targeted: one is weaker but true, one overstates the conclusion via ball-rigidity, one weakens the quantifier structure, and one asserts an unjustified invariant conclusion. These reflect realistic failure modes.

A solid MCQ with strong distractors and no answer leakage, but it leans heavily on theorem recall and is only moderately generative rather than deeply reasoning-based.


\label{thm-main}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	If $Q(A)$ is totally positive, then
	\begin{enumerate}[\rm(1)]
		\item $A$ is totally positive;
		\item $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive;
		\item the row generating function $\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\ge0$.
	\end{enumerate}

\begin{prob}\label{prob-TP}
	Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
	under which conditions are both $A$ and its reversal $\rev{A}:=[a_{n,n-k}]_{n,k\ge0}$ totally positive?
\end{prob}

\hspace*{\parindent}
Following Karlin \cite{Kar68},
a finite or infinite matrix $A=[a_{n,k}]_{n,k\ge0}$ is {\it totally positive}
if all its minors are nonnegative.
Such matrices have been widely studied in both pure and applied mathematics \cite{Bre95,Bre96,Kar68,Pin10}.
In recent five years there has been great interest concerning total positivity of combinatorial matrices \cite{CDD+21,CS24,LLY+22,LPW23,LWW23,MMW22,MW22,PSZ23,Sok23,Zhu21,Zhu24}.
However, there are still many combinatorial matrices that are experimentally totally positive but without proof.
For instance, a longstanding conjecture of Brenti \cite[Conjecture 6.10]{Bre96} asked the total positivity of the Eulerian triangle
$$
[A(n,k)]_{n,k\ge0}=
\left[
 \begin{array}{*{6}c}
 1 \\
 1 & 1 \\
 1 & 4 & 1 \\
 1 & 11 & 11 & 1 \\
 1 & 26 & 66 & 26 & 1 \\
 \vdots &&&&& \ddots
 \end{array}
\right],
$$
where the {\it Eulerian number} $A(n,k)$ counts the number of permutations of $[n+1]$ with $k$ descents,
and satisfies the recurrence
$$
A(n,k)=(n-k+1)A(n-1,k-1)+(k+1)A(n-1,k)
$$
with initial conditions $A(0,k)=\del_{0k}$.
Although the total positivity of the Eulerian triangle is hard to prove,
the first author {\it et al.} \cite{CDD+21},
by using a planar network approach,
showed the total positivity of the reversal of the Stirling triangle of the second kind
$$
\overleftarrow{S}:=[\overleftarrow{S}(n,k)]_{n,k\ge0}=
\left[
 \begin{array}{*{6}c}
 1 \\
 1 & 1 \\
 1 & 3 & 1 \\
 1 & 6 & 7 & 1 \\
 1 & 10 & 25 & 15 & 1 \\
 \vdots &&&&& \ddots
 \end{array}
\right],
$$
which enjoys a similar recurrence to the Eulerian triangle, i.e.,
$$
\overleftarrow{S}(n,k)=(n-k+1)\overleftarrow{S}(n-1,k-1)+\overleftarrow{S}(n-1,k)
$$
with initial conditions $\overleftarrow{S}(0,k)=\del_{0k}$.
On the other hand, it is known \cite{Bre95,Zhu14,CLW15rec} that the Stirling triangle of the second kind is totally positive.
So it is natural to ask the following question.

\begin{prob}\label{prob-TP}
 Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
 under which conditions are both $A$ and its reversal $\overleftarrow{A}:=[a_{n,n-k}]_{n,k\ge0}$ totally positive?
\end{prob}

One particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\dots$.
The reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\boldsymbol{\al}=(a_n)_{n\ge0}$:
$$
T_{\infty}(\boldsymbol{\al}):=[a_{i-j}]_{i,j\ge0}=
\left[
 \begin{array}{*{5}c}
 a_0 \\
 a_1 & a_0 \\
 a_2 & a_1 & a_0 \\
 a_3 & a_2 & a_1 & a_0 \\
 \vdots &&&& \ddots
 \end{array}
\right].
$$
An infinite nonnegative sequence $\boldsymbol{\al}$ is called a {\it P\'olya frequency sequence} (or shortly, {\it PF sequence}),
if its Toeplitz matrix $T_{\infty}(\boldsymbol{\al})$ is totally positive.
We identify a finite sequence $a_0,a_1,\dots,a_n$ with the infinite sequence $a_0,a_1,\dots,a_n,0,0,\dots$.
A fundamental characterization for PF sequences is due to Schoenberg and Edrei,
which states that the sequence $\boldsymbol{\al}=(a_n)_{n\ge0}$ is PF if and only if its generating function
\begin{equation}\label{eq-PF}
 \sum_{n\ge0} a_n x^n=C x^k e^{\gamma x} \frac{\prod_{j\ge0} (1+\al_j x)}{\prod_{j\ge0} (1-\be_j x)},
\end{equation}
where $C>0$, $k\in\mathbb{N}$, $\al_j,\be_j,\gamma\ge0$
and $\sum_{j\ge0}(\al_j+\be_j)<\infty$.
In particular, Aissen, Schoenberg and Whitney states that
a finite sequence of nonnegative numbers is PF if and only if its generating function is {\it real-rooted}, that is,
it is a constant polynomial or all of its zeros are real numbers (see \cite[p. 399, 412]{Kar68} for instance).
In this sense, we refer the generating functions of PF sequences as {\it PF formal power series}.
Hence, it gives rise to the another problem.

\begin{prob}\label{prob-RZ}
 Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
 under which conditions is the row generating function $\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\ge0$,
 or equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?
\end{prob}

The objective of this paper is to give a common answer to Problems \ref{prob-TP} and \ref{prob-RZ}.
An efficient method of proving total positivity of matrices is to prove the total positivity of their (right) production matrices \cite{DFR05},
which has been frequently used in literature \cite{CLW15rio,CLW15rec,PZ16,LLY+22,CS24,Zhu14,Zhu24}.
However, this method does not work when proving total positivity of matrix reversals.
So we turn to consider the total positivity of left production matrices,
and discover that it implies the total positivity of the matrix itself, its reversal, and the Toeplitz matrix of each row, simultaneously.

Let $A$ be a lower triangular matix with nonzero diagonals,
then $A$ is invertible.
Following Liang {\it et al.} \cite{LPW23}, call
\begin{equation}\label{eq-leftprod}
Q(A):=A\cdot
\left[
 \begin{array}{cc}
 1 & O \\
 O & A^{-1}
 \end{array}
\right]
\end{equation}
the {\it left production matrix} of $A$.
This implies that $Q(A)$ is also lower triangular with nonzero diagonals.
Moreover, we have
\begin{equation}\label{eq-A=Q}
A=Q(A)
\left[
 \begin{array}{cc}
 I_1 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_2 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_3 & \\
 & Q(A)
 \end{array}
\right]
\dots.
\end{equation}
Note that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \eqref{eq-A=Q}.
In such cases, we also call $Q(A)$ the left production matrix of $A$.

One particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\dots$.
The reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\bal=(a_n)_{n\ge0}$:
$$
T_{\infty}(\bal):=[a_{i-j}]_{i,j\ge0}=
\left[
 \begin{array}{*{5}c}
 a_0 \\
 a_1 & a_0 \\
 a_2 & a_1 & a_0 \\
 a_3 & a_2 & a_1 & a_0 \\
 \vdots &&&& \ddots
 \end{array}
\right].
$$
An infinite nonnegative sequence $\bal$ is called a \defn{P\'olya frequency sequence} (or shortly, \defn{PF sequence}),
if its Toeplitz matrix $T_{\infty}(\bal)$ is totally positive.
We identify a finite sequence $a_0,a_1,\dots,a_n$ with the infinite sequence $a_0,a_1,\dots,a_n,0,0,\dots$.
A fundamental characterization for PF sequences is due to Schoenberg and Edrei,
which states that the sequence $\bal=(a_n)_{n\ge0}$ is PF if and only if its generating function
\begin{equation}\label{eq-PF}
 \sum_{n\ge0} a_n x^n=C x^k e^{\ga x} \frac{\prod_{j\ge0} (1+\al_j x)}{\prod_{j\ge0} (1-\be_j x)},
\end{equation}
where $C>0$, $k\in\mathbb{N}$, $\al_j,\be_j,\ga\ge0$
and $\sum_{j\ge0}(\al_j+\be_j)<\infty$.
In particular, Aissen, Schoenberg and Whitney states that
a finite sequence of nonnegative numbers is PF if and only if its generating function is \defn{real-rooted}, that is,
it is a constant polynomial or all of its zeros are real numbers (see \cite[p. 399, 412]{Kar68} for instance).
In this sense, we refer the generating functions of PF sequences as \defn{PF formal power series}.
Hence, it gives rise to the another problem.

\begin{prob}\label{prob-RZ}
 Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
 under which conditions is the row generating function $\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\ge0$,
 or equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?
\end{prob}

Let $A$ be a lower triangular matix with nonzero diagonals,
then $A$ is invertible.
Following Liang {\it et al.} \cite{LPW23}, call
\begin{equation}\label{eq-leftprod}
Q(A):=A\cdot
\left[
 \begin{array}{cc}
 1 & O \\
 O & A^{-1}
 \end{array}
\right]
\end{equation}
the \defn{left production matrix} of $A$.
This implies that $Q(A)$ is also lower triangular with nonzero diagonals.
Moreover, we have
\begin{equation}\label{eq-A=Q}
A=Q(A)
\left[
 \begin{array}{cc}
 I_1 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_2 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_3 & \\
 & Q(A)
 \end{array}
\right]
\dots.
\end{equation}
Note that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \eqref{eq-A=Q}.
In such cases, we also call $Q(A)$ the left production matrix of $A$.

Recently, Mao {\it et al.} \cite{MMW22} showed Theorem \ref{thm-main} (1) whenever $A$ is a proper Riordan array,
by using matrix decompositions and induction.
However, sometimes Theorem \ref{thm-main} (2) seems more useful.
In fact, it is possible that the total positivity of a given matrix $A$ is not easy to obtain
whenever neither its left nor right production matrix is totally positive (for instance, the matrix $\rev{S}$),
but we can turn to consider its reversal $\rev{A}$ which may have a totally positive left production matrix.
In this paper, we will prove the three results in Theorem \ref{thm-main} simultaneously,
by constructing a unified planar network that represent $A$, $\rev{A}$, and Toeplitz matrices of rows of $A$, respectively,
when selecting different sets of sources and sinks.
Furthermore, we will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$.

\begin{thm}\label{thm-T}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 Define
 \setlength{\arraycolsep}{3pt}
 $$
 M_{n,r}:=
 \left[
 \begin{array}{ccc}
 I_0 \\
 & Q_n(A) \\
 & & I_r
 \end{array}
 \right]
 \left[
 \begin{array}{ccc}
 I_1 \\
 & Q_n(A) \\
 & & I_{r-1}
 \end{array}
 \right]
 \dots
 \left[
 \begin{array}{ccc}
 I_r \\
 & Q_n(A) \\
 & & I_0
 \end{array}
 \right],
 $$
 where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
 Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
 Then for each $r\ge0$, we have
 $$
 T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
 $$
 where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
 and $^T$ stands for transpose.
\end{thm}

\begin{thm1}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 If $Q(A)$ is totally positive, then
 \begin{enumerate}[\rm(1)]
 \item $A$ is totally positive;
 \item $\rev{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive.
 \end{enumerate}
\end{thm1}

\begin{thm2}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 Define
 \setlength{\arraycolsep}{3pt}
 \begin{equation}\label{eq-M}
 M_{n,r}:=
 \left[
 \begin{array}{ccc}
 I_0 \\
 & Q_n(A) \\
 & & I_r
 \end{array}
 \right]
 \left[
 \begin{array}{ccc}
 I_1 \\
 & Q_n(A) \\
 & & I_{r-1}
 \end{array}
 \right]
 \dots
 \left[
 \begin{array}{ccc}
 I_r \\
 & Q_n(A) \\
 & & I_0
 \end{array}
 \right],
 \end{equation}
 where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
 Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
 Then for each $r\ge0$, we have
 \begin{equation}\label{eq-Toep}
 T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
 \end{equation}
 where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
 and $^T$ stands for transpose.
\end{thm2}

\begin{thm1}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 If $Q(A)$ is totally positive, then
 \begin{enumerate}[\rm(1)]
 \item[\rm(3)] the row generating function $\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\ge0$.
 \end{enumerate}
\end{thm1}

\begin{equation}\label{eq-A=Q}
A=Q(A)
\left[
	\begin{array}{cc}
		I_1 &  \\
			& Q(A)
	\end{array}
\right]
\left[
	\begin{array}{cc}
		I_2 &  \\
			& Q(A)
	\end{array}
\right]
\left[
	\begin{array}{cc}
		I_3 &  \\
			& Q(A)
	\end{array}
\right]
\dots.
\end{equation}

\begin{thm}\label{thm-main}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	If $Q(A)$ is totally positive, then
	\begin{enumerate}[\rm(1)]
		\item $A$ is totally positive;
		\item $\rev{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive;
		\item the row generating function $\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\ge0$.
	\end{enumerate}
\end{thm}

Recently, Mao {\it et al.} \cite{MMW22} showed Theorem \ref{thm-main} (1) whenever $A$ is a proper Riordan array,
by using matrix decompositions and induction.
However, sometimes Theorem \ref{thm-main} (2) seems more useful.
In fact, it is possible that the total positivity of a given matrix $A$ is not easy to obtain
whenever neither its left nor right production matrix is totally positive (for instance, the matrix $\overleftarrow{S}$),
but we can turn to consider its reversal $\overleftarrow{A}$ which may have a totally positive left production matrix.
In this paper, we will prove the three results in Theorem \ref{thm-main} simultaneously,
by constructing a unified planar network that represent $A$, $\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively,
when selecting different sets of sources and sinks.
Furthermore, we will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$.

For a given a matrix $A$, denote by
$$
A\submat{i_0, \ldots, i_r}{j_0, \ldots, j_r}
\quad \text{and} \quad
A\minor{i_0, \ldots, i_r}{j_0, \ldots, j_r}
$$
the submatrix and the minor of $A$ determined by
the rows indexed $i_0 < \cdots < i_r$ and
the columns indexed $j_0 < \cdots < j_r$, respectively.
Let
$$
A_r:=A	\left[
	\begin{array}{c}
		0,1,\dots,r \\
		0,1,\dots,r
	\end{array}
	\right]

$$
be the $r$th leading principal submatrix (of order $r+1$) of $A$.

\begin{thm}\label{thm-T}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	Define
	\setlength{\arraycolsep}{3pt}
	$$
	M_{n,r}:=
	\left[
		\begin{array}{ccc}
			I_0 \\
				& Q_n(A) \\
				&     & I_r
		\end{array}
	\right]
	\left[
		\begin{array}{ccc}
			I_1 \\
				& Q_n(A) \\
				&     & I_{r-1}
		\end{array}
	\right]
	\dots
	\left[
		\begin{array}{ccc}
			I_r \\
				& Q_n(A) \\
				&     & I_0
		\end{array}
	\right],
	$$
	where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
	Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
	Then for each $r\ge0$, we have
	$$
	T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
	$$
	where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
	and $^T$ stands for transpose.
\end{thm}

This paper is organized as follows.
In section 2, we introduce the Lindstr\"om-Gessel-Viennot lemma
and planar network characterizations of totally positive matrices.
In section 3, we give proofs of Theorem \ref{thm-main} and Theorem \ref{thm-T}
by constructing a unified planar network.
As applications, we present sufficient conditions for total positivity and real-rootedness properties in the exponential Riordan arrays and the iteration matrices (section 4),
and the $n$-recursive matrices (section 5), repectively.
Examples include matrices and polynomials formed by the Stirling numbers of both kinds (of type A and type B), the Lah numbers, the idempotent numbers, the Delannoy numbers, and the derangement numbers of type A and type B.

They propose to prove the three statements in Theorem~\ref{thm-main} "simultaneously, by constructing a unified planar network that represent $A$, $\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively, when selecting different sets of sources and sinks." They further state that they will "identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$." Concretely, Section 2 introduces the Lindstr\"om-Gessel-Viennot lemma and planar-network characterizations of totally positive matrices, and Section 3 gives the proofs of Theorem~\ref{thm-main} (and Theorem~\ref{thm-T}) "by constructing a unified planar network."

They propose to prove the three statements in the main theorem simultaneously, by constructing a unified planar network that represent $A$, $\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively, when selecting different sets of sources and sinks. They further state that they will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$. Concretely, they first introduce the Lindstr\"om-Gessel-Viennot lemma and planar-network characterizations of totally positive matrices, and then they give the proofs of the main theorem and also establish the following theorem.

\begin{thm}\label{thm-T}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	Define
	\setlength{\arraycolsep}{3pt}
	$$
	M_{n,r}:=
	\left[
		\begin{array}{ccc}
			I_0 \\
				& Q_n(A) \\
				&     & I_r
		\end{array}
	\right]
	\left[
		\begin{array}{ccc}
			I_1 \\
				& Q_n(A) \\
				&     & I_{r-1}
		\end{array}
	\right]
	\dots
	\left[
		\begin{array}{ccc}
			I_r \\
				& Q_n(A) \\
				&     & I_0
		\end{array}
	\right],
	$$
	where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
	Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
	Then for each $r\ge0$, we have
	$$
	T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
	$$
	where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
	and $^T$ stands for transpose.
\end{thm}

They do this by constructing a unified planar network.

Question:
Let $A=[a_{n,k}]_{n,k\ge 0}$ be a lower triangular matrix, and define its left production matrix by
$$
Q(A):=A\begin{bmatrix}1&O\\ O&A^{-1}\end{bmatrix},
$$
where $O$ denotes zero blocks. A matrix is called totally positive if all of its minors are nonnegative, and the reversal of $A$ is the lower triangular matrix $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge 0}$. Which statement holds for every such matrix $A$ for which $Q(A)$ is totally positive?

$A$ is totally positive, $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive, and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ is real-rooted.

Choices:
B. $A$ is totally positive and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ is real-rooted, but the reversal $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ need not be totally positive.
C. $A$ is totally positive and $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive.
D. $A$ is totally positive, $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive, and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ has only nonnegative coefficients.
E. If $A$ is totally positive, $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive, and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ is real-rooted, then $Q(A)$ is totally positive.

unified planar-network transfer to the reversal

dropped the real-rootedness conclusion for row generating polynomials

real-rootedness of row generating polynomials replaced by coefficient nonnegativity

reversed the implication from consequences back to total positivity of the left production matrix

The stem gives only the hypothesis on the left production matrix and asks for the valid conclusion. It does not explicitly state the three target conclusions, though mentioning the reversal signals that it may matter.

The item is essentially a theorem-recall question: the correct option restates the full theorem-level conclusion from the stated hypothesis. It does not substantially transform the result into a new reasoning task.

There is some pressure to compare a full conclusion against weaker, truncated, or partially false variants, but success mainly depends on recognizing the exact theorem rather than generating a nontrivial argument.

The distractors are mathematically plausible and distinct: one is a weaker true statement, others reflect common confusions about finite truncations, PF/real-rootedness implications, and whether reversal total positivity follows.

A mathematically polished MCQ with strong distractors and little answer leakage, but it functions mostly as direct theorem recognition rather than a genuinely generative reasoning problem.


\begin{theorem}
\label{T2}
Assume \eqref{(2)}-\eqref{(6)} hold. Then,
\begin{itemize}
\item[(i)] If $0<q<1-\theta$, there exists a distributional solution $u\in W_{0}^{1,(r_{j})}(\Omega)$ to problem \eqref{(1)}, such that
\begin{align}
\label{(17)}
r_{j}=\frac{N(\overline{p}-\theta)}{\overline{p}(N-\theta)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(ii)] If $1-\theta<q\leq1$, there exists a distributional solution $u\in W_{0}^{1,(\eta_{j})}(\Omega)$ to problem \eqref{(1)}, such that
\begin{align}
\label{(017)}
r_{j}<\frac{N(\overline{p}-1+q)}{\overline{p}(N+q-1)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(iii)] If $q>1$, there exists a distributional solution $u\in W_{0}^{1,(p_{j})}(\Omega)$ to problem \eqref{(1)}.
\end{itemize}
\end{theorem}

Anisotropic elliptic equations describe physical and biological phenomena where
diffusion or conductivity varies with direction, such as flow in heterogeneous
porous media \cite{R18}, epidemic propagation across irregular landscapes
\cite{R19}, and thermistor models exhibiting directional heat–electric
responses \cite{HHH1,HHH2,HHH3}. Singular elliptic equations also arise in many
applications, featuring blow-up behaviour, boundary singularities
\cite{CD,ZD}. These problems often require
techniques beyond the classical isotropic theory \cite{V}, especially when
combined with nonstandard growth conditions \cite{R1,R2}.

Let \(\Omega \subset \mathbb{R}^{N}\) be a bounded open set, where \(N \geq 2\), and let
\((p_{j}) = (p_{1}, \ldots, p_{N}) \in \mathbb{R}^{N}\) satisfy
\[
1 \leq p^{-} = \min_{j \in J}\{p_{j}\} \leq p^{+} = \max_{j \in J}\{p_{j}\} < \infty .
\]

The anisotropic Sobolev spaces \(W^{1,(p_{j})}(\Omega)\) and \(W_{0}^{1,(p_{j})}(\Omega)\) are defined respectively by
\[
\begin{aligned}
W^{1,(p_{j})}(\Omega)
&= \big\{ v \in W^{1,1}(\Omega) :\; D_{j}v \in L^{p_{j}}(\Omega)\ \text{for all } j \in J \big\},\\[2mm]
W_{0}^{1,(p_{j})}(\Omega)
&= \big\{ v \in W_{0}^{1,1}(\Omega) :\; D_{j}v \in L^{p_{j}}(\Omega)\ \text{for all } j \in J \big\}.
\end{aligned}
\]

Equivalently, \(W_{0}^{1,(p_{j})}(\Omega)\) is the closure of \(C_{0}^{\infty}(\Omega)\) with respect to the norm
\[
\|v\|_{W_{0}^{1,(p_{j})}(\Omega)}
= \sum_{j \in J} \|D_{j}v\|_{L^{p_{j}}(\Omega)}.
\]

Endowed with this norm, \(W_{0}^{1,(p_{j})}(\Omega)\) is a separable and reflexive Banach space.
Its dual space is denoted by \(W^{-1,(p'_{j})}(\Omega)\), where for each \(j \in J\), \(p_{j}'\) is the conjugate exponent of \(p_{j}\), that is,
\[
\frac{1}{p_{j}} + \frac{1}{p_{j}'} = 1.
\]

This article's main goal is to investigate the following problem.
\begin{equation}
\label{(1)}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+b(x)\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$ ($N\geq 3$), $f$, $q$, $\theta$, and $p_{j}$ are satisfies  the following conditions
\begin{align}
&\label{(2)}
q>0,\quad 0<\theta<1,\\
&\label{(3)}
f\geq0,\;\; f\not\equiv0,\quad f\in L^{1}(\Omega),\\
&\label{(4)}
2\leq p_{1}\leq p_{2}\leq ... \leq p_{N-1}\leq p_{N}\quad \mbox{and}\quad 2\leq \overline{p}:=N\left(\sum\limits_{j\in J}\frac{1}{p_{j}}\right)^{-1}<N.
\end{align}
For some positive numbers $\alpha,\beta,\mu$, and $\nu$, we assume in this study that $a,b$ are measurable functions satisfying the following conditions.
\begin{align}
&\label{(5)}
 0<\alpha\leq a(x)\leq \beta,\\
&\label{(6)}
0<\mu\leq b(x)\leq \nu.
\end{align}
The isotropic case of problem \eqref{(1)}, where $p_j$ is constant for all $j \in J$, has been extensively studied by several authors, with notable contributions in \cite{olivia, boca92, boca97, dalill}. Also, a problem like \eqref{(1)} without the singular term was studied in more general context in \cite{FR1,FA1}.

A further motivation for this work comes from the isotropic analogue of problem \eqref{(1)}, studied in \cite{R0}. There, the authors considered
\begin{equation}
\label{P3}
\left\{\begin{array}{ll}\displaystyle-\mbox{div}\left(\left[a(x)+u^{q}\right]\nabla u\right)+b(x)\frac{\vert \nabla u\vert^{2}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
with $q>0$, $0<\theta<1$, and $f\in L^{1}(\Omega)$. They proved the existence of nonnegative solutions and derived precise summability results depending on the balance between the exponent of the  absorption term $q$ and the singular term's exponent $\theta$.
The present work investigates the same type of phenomenon in an \textit{anisotropic setting}, where the diffusion operator involves directional exponents $(p_{j})$ and the harmonic mean $\overline{p}$ replaces the Laplacian structure.

The problem \eqref{P3} in the case of $q = 0$ has been addressed by several authors, including \cite{A1, A2, A3, A4, G1, G2}.

 In the context of anisotropic elliptic problems like \eqref{(1)} with no absorption term (i.e., when $q=0$, $a\equiv 1$, and $b\equiv 0)$, Di Castro in \cite{R2} addressed the following problem:
\begin{equation*}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)=f& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation*}
where $p_j$ increases with respect to $j$ such that $2 - \frac{1}{N} \leq \overline{p} < N$, and $f \in L^1(\Omega)$. The author demonstrated that this problem admits a  solution $u \in W_0^{1, (s_j)}(\Omega)$, where
\begin{align}
\label{HH}
1\leq s_{j}<\frac{N(\overline{p}-1)}{\overline{p}(N-1)}p_{j},\quad\forall\; j\in J.
\end{align}
and for all  $\Phi \in C_0^1(\Omega),$
\begin{equation}\label{HH1}
\sum\limits_{j\in J}\int_{\Omega}\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi\,dx =\int_{\Omega}f\Phi\,dx .
\end{equation}

Further, this problem has been studied under different assumptions on the data in \cite{Stamp, BO1, BO2, OR1, BO3}, particularly when $p_j$ is constant for all $j \in J$.

 In the case where $q = 0$ and with lower order terms with sign conditions, Di Castro in \cite{R1} explored the following  general anisotropic problem
\begin{equation}
\label{P2}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}g_{j}(x,u,\nabla u)=f& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
where $f \in L^1(\Omega)$ and $g_j: \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ is a Carath\'eodory function satisfying the following conditions
\begin{itemize}
\item[(H1)] $g_{j}(x,s,\xi)\mbox{sgn}(s)\geq0$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$;
\item[(H2)] $\vert g_{j}(x,s,\xi)\vert\leq h(\vert s\vert)\vert\xi_{j}\vert^{p_{j}}$, for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$, where $h:\mathbb{R}\rightarrow \mathbb{R}^{+}$ is a continuous, nondecreasing function such that $h(t)>\gamma>0$ for $\vert t\vert$ sufficiently large;
\item[(H3)] There exists $\nu>0$ such that $\vert g_{j}(x,s,\xi)\vert\geq \nu \vert \xi_{j}\vert^{p_{j}}$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$.
\end{itemize}
The author investigated the existence and regularity of solutions to this problem, yielding the following results
\begin{itemize}
\item[$\bullet$] If conditions (H1) and (H2) hold, and if $p_j$ satisfies the inequality
$$\frac{\overline{p}(N-1)}{N(\overline{p}-1)}<p_{j}<\frac{\overline{p}(N-1)}{N-\overline{p}},\quad \forall\; j\in J,$$
then there exists at least one  solution $u \in W_0^{1, (s_j)}(\Omega)$, where $s_j$ is defined in \eqref{HH} and \eqref{HH1} holds.
\item[$\bullet$] If conditions (H1), (H2), and (H3) hold, then a weak solution $u$ exists in $W_0^{1, (p_j)}(\Omega)$.
\end{itemize}

On the other hand, when \(b = 0\) and \(q \neq 0\), we have recently addressed, see \cite{FH}, a corresponding problem involving a singular right-hand side of the type \(\displaystyle\frac{f}{u^{l}}\), with \(l > 0\).

We end this  brief history of the problem by recalling the paper \cite{bacii}, in their work, the authors establish the existence of weak solutions for a broad class of anisotropic Dirichlet problems of the type
\[
\mathcal{A}u + \Phi(x,u,\nabla u) = \Psi(u,\nabla u) + \mathcal{L}u + f \quad \text{in } \Omega,
\]
where \(\Omega \subset \mathbb{R}^N\) \((N \ge 2)\) is a bounded domain and \(f \in L^1(\Omega)\) is arbitrary.
The anisotropic principal operator is
\[
\mathcal{A}u = -\sum_{j=1}^N \partial_j\left( \vert\partial_j u\vert^{p_j - 2}\partial_j u \right),
\]
and the lower-order term \(\Phi\) is given by
\[
\Phi(x,u,\nabla u) = \left( 1 + \sum_{j=1}^N \mathfrak{a}_j \vert\partial_j u\vert^{p_j} \right)|u|^{m-2}u,
\]
with parameters \(m,p_j>1\), \(\mathfrak{a}_j \ge 0\), and \(\sum_{k=1}^N \frac{1}{p_k} > 1\).
A key novelty of the study is the inclusion of a possibly singular gradient-dependent nonlinearity
\[
\Psi(u,\nabla u) = \sum_{j=1}^N |u|^{\theta_j - 2}u , \vert\partial_j u\vert^{q_j},
\]
where \(\theta_j>0\) and \(0 \le q_j < p_j\).
The lower-order operator $\mathcal{L}$ belongs to the general class $\mathcal{BC}$ introduced in \cite{baci}.
The class $\mathcal{BC}$ consists of \emph{bounded} operators
\[
\mathcal{L} : W_0^{1,(p_j)}(\Omega) \rightarrow W^{-1,(p_j')}(\Omega)
\]
that satisfy the following structural conditions:

\begin{itemize}
    \item[(P$_1$)]
    The operator $\mathcal{A} - \mathcal{L}$ is coercive on $W_0^{1,(p_j)}(\Omega)$, namely,
    \[
    \frac{\langle \mathcal{A}u - \mathcal{L}u,\, u \rangle}{\|u\|_{W_0^{1,(p_j)}(\Omega)}} \to \infty
    \qquad \text{as } \|u\|_{W_0^{1,(p_j)}(\Omega)} \to \infty .
    \]

    \item[(P$_2$)]
    If $u_l \rightharpoonup u$ and $v_l \rightharpoonup v$ weakly in $W_0^{1,(p_j)}(\Omega)$, then
    \[
    \lim_{l \to \infty} \langle \mathcal{L}u_l,\, v_l \rangle
    = \langle \mathcal{L}u,\, v \rangle .
    \]
\end{itemize}

These assumptions allow the authors to include a broad family of lower-order terms while preserving the coercivity and compactness properties needed for their existence theory.

The authors obtain existence results under two different regimes:
\begin{enumerate}
\item when \(\theta_j > 1\) for every \(j = 1,\dots, N\);

\item when \(\theta_j \le 1\) for at least one index \(j\).
\end{enumerate}
   In the latter case, by assuming \(f \ge 0\) almost everywhere in \(\Omega\), they further prove the existence of nonnegative weak solutions.

 In our study, we face three main difficulties. The first comes from the anisotropic structure of the operator, which requires estimates adapted to different growth conditions in each direction. The second difficulty is the presence of the singular lower-order term depending on the gradient $\frac{\vert D_{j} u\vert^{p_{j}}}{\vert u\vert^{\theta}}$, which complicates the analysis, especially near the points where the unknown $u$ may approach zero. The third difficulty is related to the additional term
 $$
\displaystyle-\sum\limits_{j\in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)$$
whose nonlinear form interacts with the previous terms and makes the compactness arguments more delicate.
\begin{definition}
\label{Definition3.1}
Let the condition in \eqref{(3)} hold. A function $u\in W_{0}^{1,1}(\Omega)$ is a distributional solution to problem \eqref{(1)} if
\begin{align}
\label{(13)}
& u>0\quad\quad \mbox{ a.e. in}\; \Omega,\\
\label{(14)}
& \sum\limits_{j\in J}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u\in (L^{\sigma_{j}}(\Omega))^N,\quad \mbox{for all}\; \sigma_{j}<p_{j}',\\
\label{(15)}
& \sum\limits_{j\in J} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\in L^{1}(\Omega),
\end{align}
and for all  $\Phi \in W_0^{1,d_j}(\Omega),\; d_j>p_j$
\begin{align}
\label{(16)}
&\displaystyle\sum\limits_{j\in J}\int_{\Omega}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi +\sum\limits_{j\in J}\int_{\Omega} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\Phi =\int_{\Omega}f\Phi\,dx .
\end{align}
\end{definition}
\par Our main results are as follows

A further motivation for this work comes from the isotropic analogue of problem \eqref{(1)}, studied in \cite{R0}. There, the authors considered
\begin{equation}
\label{P3}
\left\{\begin{array}{ll}\displaystyle-\mbox{div}\left(\left[a(x)+u^{q}\right]\nabla u\right)+b(x)\frac{\vert \nabla u\vert^{2}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
with $q>0$, $0<\theta<1$, and $f\in L^{1}(\Omega)$. They proved the existence of nonnegative solutions and derived precise summability results depending on the balance between the exponent of the absorption term $q$ and the singular term's exponent $\theta$.
The present work investigates the same type of phenomenon in an \textit{anisotropic setting}, where the diffusion operator involves directional exponents $(p_{j})$ and the harmonic mean $\overline{p}$ replaces the Laplacian structure.

In the context of anisotropic elliptic problems like \eqref{(1)} with no absorption term (i.e., when $q=0$, $a\equiv 1$, and $b\equiv 0)$, Di Castro in \cite{R2} addressed the following problem:
\begin{equation*}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)=f& \hbox{in}\;\Omega, \\
 u =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation*}
where $p_j$ increases with respect to $j$ such that $2 - \frac{1}{N} \leq \overline{p} < N$, and $f \in L^1(\Omega)$. The author demonstrated that this problem admits a solution $u \in W_0^{1, (s_j)}(\Omega)$, where
\begin{align}
\label{HH}
1\leq s_{j}<\frac{N(\overline{p}-1)}{\overline{p}(N-1)}p_{j},\quad\forall\; j\in J.
\end{align}
and for all $\Phi \in C_0^1(\Omega),$
\begin{equation}\label{HH1}
\sum\limits_{j\in J}\int_{\Omega}\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi\,dx =\int_{\Omega}f\Phi\,dx .
\end{equation}

In the case where $q = 0$ and with lower order terms with sign conditions, Di Castro in \cite{R1} explored the following general anisotropic problem
\begin{equation}
\label{P2}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}g_{j}(x,u,\nabla u)=f& \hbox{in}\;\Omega, \\
 u =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
where $f \in L^1(\Omega)$ and $g_j: \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ is a Carath\'eodory function satisfying the following conditions
\begin{itemize}
\item[(H1)] $g_{j}(x,s,\xi)\mbox{sgn}(s)\geq0$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$;
\item[(H2)] $\vert g_{j}(x,s,\xi)\vert\leq h(\vert s\vert)\vert\xi_{j}\vert^{p_{j}}$, for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$, where $h:\mathbb{R}\rightarrow \mathbb{R}^{+}$ is a continuous, nondecreasing function such that $h(t)>\gamma>0$ for $\vert t\vert$ sufficiently large;
\item[(H3)] There exists $\nu>0$ such that $\vert g_{j}(x,s,\xi)\vert\geq \nu \vert \xi_{j}\vert^{p_{j}}$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$.
\end{itemize}
The author investigated the existence and regularity of solutions to this problem, yielding the following results
\begin{itemize}
\item[$\bullet$] If conditions (H1) and (H2) hold, and if $p_j$ satisfies the inequality
$$\frac{\overline{p}(N-1)}{N(\overline{p}-1)}<p_{j}<\frac{\overline{p}(N-1)}{N-\overline{p}},\quad \forall\; j\in J,$$
then there exists at least one solution $u \in W_0^{1, (s_j)}(\Omega)$, where $s_j$ is defined in \eqref{HH} and \eqref{HH1} holds.
\item[$\bullet$] If conditions (H1), (H2), and (H3) hold, then a weak solution $u$ exists in $W_0^{1, (p_j)}(\Omega)$.
\end{itemize}

\item when \(\theta_j \le 1\) for at least one index \(j\).
\end{enumerate}
 In the latter case, by assuming \(f \ge 0\) almost everywhere in \(\Omega\), they further prove the existence of nonnegative weak solutions.

In our study, we face three main difficulties. The first comes from the anisotropic structure of the operator, which requires estimates adapted to different growth conditions in each direction. The second difficulty is the presence of the singular lower-order term depending on the gradient $\frac{\vert D_{j} u\vert^{p_{j}}}{\vert u\vert^{\theta}}$, which complicates the analysis, especially near the points where the unknown $u$ may approach zero. The third difficulty is related to the additional term
 $$
\displaystyle-\sum\limits_{j\in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)$$
whose nonlinear form interacts with the previous terms and makes the compactness arguments more delicate.
\begin{definition}
\label{Definition3.1}
Let the condition in \eqref{(3)} hold. A function $u\in W_{0}^{1,1}(\Omega)$ is a distributional solution to problem \eqref{(1)} if
\begin{align}
\label{(13)}
& u>0\quad\quad \mbox{ a.e. in}\; \Omega,\\
\label{(14)}
& \sum\limits_{j\in J}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u\in (L^{\sigma_{j}}(\Omega))^N,\quad \mbox{for all}\; \sigma_{j}<p_{j}',\\
\label{(15)}
& \sum\limits_{j\in J} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\in L^{1}(\Omega),
\end{align}
and for all $\Phi \in W_0^{1,d_j}(\Omega),\; d_j>p_j$
\begin{align}
\label{(16)}
&\displaystyle\sum\limits_{j\in J}\int_{\Omega}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi +\sum\limits_{j\in J}\int_{\Omega} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\Phi =\int_{\Omega}f\Phi\,dx .
\end{align}
\end{definition}
\par Our main results are as follows

\item when \(\theta_j \le 1\) for at least one index \(j\).
\end{enumerate}
 In the latter case, by assuming \(f \ge 0\) almost everywhere in \(\Omega\), they further prove the existence of nonnegative weak solutions.

We first note that hypothesis \eqref{(4)} already ensures that
\[
1 < r_{j} \le p_{j} \qquad \text{for every } j \in J.
\]
Moreover, by combining assumptions \eqref{(2)} and \eqref{(4)}, we obtain the stronger condition
\[
r_{j} > p_{j} - 1, \qquad \forall\, j \in J.
\]

The following lemma will be of central importance in the remainder of the proof, and it will be especially needed in the passage to the limit.
\begin{lem}\label{lmq}
Assume that the assumptions \eqref{(2)}-\eqref{(6)} hold true.
Then, we have
\begin{equation}\label{ugest}
\int_{\Omega} u_{n}^{q\sigma_j}\left\vert D_{j} u_{n}\right\vert^{\sigma_j(p_{j}-1)}\;dx \leq C\quad\quad \forall \sigma_j<p_j^\prime.
\end{equation}
\end{lem}
\begin{proof}
 Taking $ \varphi(u_n)=1-\frac{1}{(1-u_n)^\lambda},$ with $\lambda>1,$ as test function in the weak formulation \eqref{(21)}, we get
\begin{equation*}
(\lambda-1)\sum_{j\in J}\int_{\Omega} \frac{[a(x)+u_n^q]\vert D_j u_n\vert^{p_{j}}}{(1+u_n)^\lambda}+\sum_{j\in J}\int_{\Omega}b(x)\frac{u_{n}\vert D_{j}u_{n}\vert^{p_{j}}}{\left( u_{n}+\frac{1}{n}\right)^{\theta+1}}\varphi =\int_\Omega f_n \varphi(u_n).
\end{equation*}
We recall that there exists a positive constant $\mathcal{C}$ such that
$$(a(x)+u_n^q) \geq \mathcal{C}(1+u_n)^q,\quad \quad \forall q>1.$$
Dropping the positive term and using the fact that $\vert \varphi(u_n)\vert\leq 1,$ we obtain
\begin{equation}\label{est1}
\sum_{j\in J}\int_{\Omega} \frac{\vert D_j u_n\vert^{p_j}}{(1+u_n)^{\lambda-q}}\leq\frac{1}{\mathcal{C}(\lambda-1)}\int_\Omega \vert f_n\vert .
\end{equation}
For $1 < \sigma_{j} < p'_{j} = \frac{p_{j}}{p_{j} - 1}$, an application of H\"older's inequality together with estimate \eqref{est1} yields, for every $j \in J$,
\begin{align*}
\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \leq C\left( \int_{\Omega} (1+u_{n})^{\frac{\sigma_{j}(p_{j}-1)(\lambda-q+p_{j} q)}{p_{j}-\sigma_{j}(p_{j}-1)}}\right)^{\frac{p_{j}-\sigma_{j}(p_{j}-1)}{p_{j}}}.
\end{align*}
We now introduce the parameter $\sigma_{j} = \theta\frac{p_{j}}{p_{j}-1}$, with $\theta \in (0,1)$. Under this choice, one readily verifies that
$$
\frac{p_{j}-\sigma_{j}(p_{j}-1)}{p_{j}} = 1 - \theta,\quad \forall\; j\in J,$$
and
$$ \frac{\sigma_{j}(p_{j}-1)(\lambda-q+p_{j} q)}{p_{j}-\sigma_{j}(p_{j}-1)} = \frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)],\quad \forall\; j\in J.
$$
Substituting into the previous estimate leads to
\begin{equation}
\label{E11}
\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \leq C\left( \int_{\Omega}(1+ u_{n})^{\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)]}\right)^{1-\theta},
\end{equation}
for every $j\in J$. Hence, we deduce the following inequality:
\begin{align*}
&\prod_{i\in J}\left(\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)}\;dx\right)^{\frac{1}{\theta p_{j}}}\\
&\quad \leq C\prod_{i\in J}\left( \int_{\Omega} (1+u_{n})^{\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)]}\;dx\right)^{\frac{1-\theta}{\theta p_{j}}}.
\end{align*}
By invoking the anisotropic inequality \eqref{(9)} with $\delta_{j} = \sigma_{j}(p_{j}-1) \leq p_{j}$ (due to $\theta < 1$), $t_{j} = q \geq 0$, and $\overline{\delta} = \theta \overline{p}$, we obtain
\begin{equation}\label{E12}
\left(\int_{\Omega}u_{n}^{r}\right)^{\frac{N}{\theta\overline{p}}-1} \leq C\prod_{i\in J}\left( \int_{\Omega} (1+u_{n})^{\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)]}\right)^{\frac{1-\theta}{\theta p_{j}}}.
\end{equation}
In order for the preceding inequality to hold uniformly for all $j \in J$, we must enforce that
$$
r = \frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)], \quad \forall\, j\in J.
$$
This leads to the following system of equations:
\begin{align}
& r=\frac{1+q}{b_{j}(N-1)-1+\frac{1}{\theta p_{j}}}, \quad \forall j\in J, \label{S1} \\
& r=\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)], \quad \forall j\in J, \label{S2} \\
&\sum_{i\in J}b_{j}=1, \quad b_{j} \geq 0, \quad \forall j\in J. \label{S3}
\end{align}
By combining equations \eqref{S1} and \eqref{S3}, one deduces
$$
r = \frac{N(1+q)\theta \overline{p}}{N-\theta\overline{p}}.
$$
In parallel, equation \eqref{S2} yields
$$
\theta = \frac{N(\overline{p}-\lambda+q)}{\overline{p}[N(1+q)-q\overline{p}-\lambda+q]}.
$$
Since $q \geq 0$ and $\lambda > 1$, it follows that $0 < \theta < 1$. Consequently, by combining these expressions, we arrive at
$$
r = \frac{N(\overline{p}-\lambda+q)}{N-\overline{p}}.
$$
Using this value of $r$ in inequality \eqref{E12}, we find
$$
\left(\int_{\Omega}u_{n}^{r}\right)^{\frac{N}{\theta\overline{p}}-1} \leq C\left(\int_{\Omega}u_{n}^{r}\right)^{\frac{(1-\theta)N}{\theta \overline{p}}}.
$$
Since the assumption $\overline{p} < N$ ensures that the left-hand exponent exceeds the right-hand one, we conclude that $\{u_{n}\}$ is bounded in $L^{r}(\Omega)$. This bound, together with estimate \eqref{E11}, implies that
$$
\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \leq C, \quad \forall\, j\in J.
$$
Given that $p_{j} > 2$ and $\sigma_{j} > 1$ for all $j\in\mathcal{E}$, we apply Hölder’s inequality to obtain
\begin{align*}
\int_{\Omega} u_{n}^{q \sigma_{j}} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)}
&= \int_{\left\{u_{n}<1\right\}} u_{n}^{q \sigma_{j}} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \\
&\qquad+ \int_{\left\{u_{n} \geq 1\right\}} u_{n}^{q \sigma_{j}} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)}\\
&\leq \int_{\Omega} \left\vert D_{j} \mathcal{T}_{1}(u_{n})\right\vert^{\sigma_{j}(p_{j}-1)}
+ \int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \\
&\leq C \left(\int_{\Omega} \left\vert D_{j} \mathcal{T}_{1}(u_{n})\right\vert^{p_{j}} \right)^{\frac{\sigma_{j}(p_{j}-1)}{p_{j}}} + C \\
&\leq C.
\end{align*}
Thus, it follows that the sequence $\left\lbrace u_{n}^{q}\left\vert D_{j} u_{n}\right\vert^{p_{j}-1}\right\rbrace$ is uniformly bounded in $L^{\sigma_{j}}(\Omega)$ for all $1 < \sigma_{j} < p'_{j}$ and for each $j \in J$.
\end{proof}
Let us specify some useful notation we will use from now on. For any \(n\in \mathbb{N}^{*}\) we define the following auxiliary functions:
\begin{equation*}
\mathcal{H}_{n}(l)=\int_{0}^{l}\frac{dt}{\left(t+\frac{1}{n}\right)^{\theta}},
\qquad
\mathcal{H}_{\infty}(l)=\int_{0}^{l}\frac{dt}{t^{\theta}},
\end{equation*}
and, let $\gamma=\frac{\nu}{\alpha}>0,$ we denote
\begin{equation*}
\Psi_{n}(l)=e^{-\gamma \mathcal{H}_{n}(l)},\qquad \Psi_{\infty}(l)=e^{-\gamma \mathcal{H}_{\infty}(l)}.
\end{equation*}
Note that, the function \(\mathcal{H}_{\infty}\) is well defined because \(\theta<1\). Moreover, it's clear that
$$\lim_{n\rightarrow \infty}\mathcal{H}_{n}(l)=\mathcal{H}_{\infty}(l)\text{ and }\lim_{n\rightarrow \infty}\Psi_{n}(l)=\Psi_{\infty}(l).$$
Observe that, for any \(\phi, u_{n}\in W_{0}^{1,(p_{j})}(\Omega)\cap L^{\infty}(\Omega)\) with \(\phi\ge0\) , we have \(\Psi_{n}(u_{n})\phi\in W_{0}^{1,(p_{j})}(\Omega)\cap L^{\infty}(\Omega)\).

We now give a brief description of the proof of our main theorem. The first step consists in removing the degeneracy of the principal operator and the singularity of the lower-order gradient term. For this purpose, we consider a sequence of approximate problems associated with \eqref{(1)}, where both difficulties are regularized. In the second step, we establish suitable a priori estimates for the approximate solutions. The main challenge here is to control the term
$$-\sum_{j \in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2}D_{j}u\right)$$
in an appropriate anisotropic Lebesgue space. Finally, by using Fatou’s lemma together with a standard compactness argument, we pass to the limit in the approximate problems and obtain a weak solution to the original equation.
\begin{remark}
We collect here several observations related to the assumptions of Theorem~\ref{T2}.

We first note that hypothesis \eqref{(4)} already ensures that
\[
1 < r_{j} \le p_{j} \qquad \text{for every } j \in J.
\]
Moreover, by combining assumptions \eqref{(2)} and \eqref{(4)}, we obtain the stronger condition
\[
r_{j} > p_{j} - 1, \qquad \forall\, j \in J.
\]

We also observe that the condition \(1 - \theta < q\) appearing in Theorem~\ref{T2}(ii) is equivalent to
\[
\frac{N(\overline{p}-\theta)}{\overline{p}(N-\theta)} p_{j}
\;<\;
\frac{N(\overline{p}-1+q)}{\overline{p}(N+q-1)} p_{j},
\qquad \forall\, j \in J.
\]
This relation will be useful when comparing the integrability exponents arising in the proof.

Furthermore, the hypothesis \(q \le 1\) in Theorem~\ref{T2}(ii) automatically implies that
\[
1 < \eta_{j} < p_{j},
\]
which plays an important role in establishing the required estimates.

It is also worth mentioning that, in the isotropic case \(p_{j} = 2\) for all \(j \in J\), Theorem~\ref{T2} reduces to known regularity results for classical elliptic equations. In particular, our conclusions coincide with those of Theorem~1.1 in \cite{R0}.

Finally, by applying the Sobolev embedding theorem, we can describe the integrability of the solutions.
For Theorem~\ref{T2}(i), the solution \(u\) also satisfies
\[
u \in L^{\overline{r}^{*}}(\Omega),
\qquad
\overline{r}^{*} = \frac{N(\overline{p}-\theta)}{N-\overline{p}}.
\]
For Theorem~\ref{T2}(ii), the solution enjoys the additional property
\[
u \in L^{s}(\Omega),
\qquad \text{for every }\; s < \frac{N(\overline{p}-1+q)}{\,N-\overline{p}\,}.
\]
\end{remark}
For ease of reading, Section 2 begins with a brief recall of the main analytical tools of anisotropic Sobolev spaces and the definition of the approximate problem. We then state the essential a priori estimates Lemmas as well as some convergences results, which form the main step on the proof, and finally we pass to the limit.

To prove Theorem~\ref{T2}, the authors give the following outline. First, they "remove the degeneracy of the principal operator and the singularity of the lower-order gradient term" by introducing "a sequence of approximate problems associated with \eqref{(1)}, where both difficulties are regularized." Second, they "establish suitable a priori estimates for the approximate solutions"; the main difficulty is "to control the term
$$-\sum_{j \in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2}D_{j}u\right)$$
in an appropriate anisotropic Lebesgue space." Finally, "by using Fatou’s lemma together with a standard compactness argument," they "pass to the limit in the approximate problems and obtain a weak solution to the original equation." They further indicate that Section~2 recalls tools on anisotropic Sobolev spaces and defines the approximate problem, then states the key a priori estimates and convergence results, and "finally we pass to the limit."

To prove the main theorem, the authors give the following outline. First, they “remove the degeneracy of the principal operator and the singularity of the lower-order gradient term” by introducing “a sequence of approximate problems associated with
\begin{equation}
\label{(1)}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+b(x)\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
, where both difficulties are regularized.” Second, they “establish suitable a priori estimates for the approximate solutions”; the main difficulty is “to control the term
$$-\sum_{j \in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2}D_{j}u\right)$$
in an appropriate anisotropic Lebesgue space.” Finally, “by using Fatou’s lemma together with a standard compactness argument,” they “pass to the limit in the approximate problems and obtain a weak solution to the original equation.” They further indicate that the next part recalls tools on anisotropic Sobolev spaces and defines the approximate problem, then states the key a priori estimates and convergence results, and “finally we pass to the limit.”

\label{T2}
Assume
\begin{align}
&\label{(2)}
q>0,\quad 0<\theta<1,\\
&\label{(3)}
f\geq0,\;\; f\not\equiv0,\quad f\in L^{1}(\Omega),\\
&\label{(4)}
2\leq p_{1}\leq p_{2}\leq ... \leq p_{N-1}\leq p_{N}\quad \mbox{and}\quad 2\leq \overline{p}:=N\left(\sum\limits_{j\in J}\frac{1}{p_{j}}\right)^{-1}<N.
\end{align}
--
\begin{align}
&\label{(5)}
 0<\alpha\leq a(x)\leq \beta,\\
&\label{(6)}
0<\mu\leq b(x)\leq \nu.
\end{align}
hold. Then,
\begin{itemize}
\item[(i)] If $0<q<1-\theta$, there exists a distributional solution $u\in W_{0}^{1,(r_{j})}(\Omega)$ to problem
\begin{equation}
\label{(1)}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+b(x)\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
such that
\begin{align}
\label{(17)}
r_{j}=\frac{N(\overline{p}-\theta)}{\overline{p}(N-\theta)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(ii)] If $1-\theta<q\leq1$, there exists a distributional solution $u\in W_{0}^{1,(\eta_{j})}(\Omega)$ to the above problem such that
\begin{align}
\label{(017)}
r_{j}<\frac{N(\overline{p}-1+q)}{\overline{p}(N+q-1)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(iii)] If $q>1$, there exists a distributional solution $u\in W_{0}^{1,(p_{j})}(\Omega)$ to the above problem.
\end{itemize}

Question:
Let \(J=\{1,\dots,N\}\). Consider the anisotropic Dirichlet problem
\[
\begin{cases}
-\displaystyle\sum_{j\in J} D_j\!
\left(\left[a(x)+u^{q}\right]\,|D_j u|^{p_j-2}D_j u\right)
+ b(x)\displaystyle\sum_{j\in J}\frac{|D_j u|^{p_j}}{u^{\theta}}=f & \text{in } \Omega,\\[2mm]
u>0 & \text{in } \Omega,\\
u=0 & \text{on } \partial\Omega.
\end{cases}
\]
Assume
\[
q>0,\qquad 0<\theta<1,
\]
\[
f\ge 0,\quad f\not\equiv 0,\quad f\in L^1(\Omega),
\]
\[
2\le p_1\le p_2\le \cdots \le p_N,
\qquad
2\le \overline p:=N\left(\sum_{j\in J}\frac1{p_j}\right)^{-1}<N,
\]
and the coefficients satisfy, a.e. in \(\Omega\),
\[
0<\alpha\le a(x)\le \beta,
\qquad
0<\mu\le b(x)\le \nu.
\]
Here \(W_0^{1,(s_j)}(\Omega)\) denotes the anisotropic Sobolev space of zero-trace functions such that \(D_j u\in L^{s_j}(\Omega)\) for each \(j\), and a distributional solution means a function with zero trace, positive a.e. in \(\Omega\), satisfying the equation in the sense of distributions. Which conclusion about existence and regularity is valid under these hypotheses?

The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j<\frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).

Choices:
B.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q\le 1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j\le \frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).
C.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) for some directional exponents \((\eta_j)_{j\in J}\);
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).
D.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j<\frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\), and moreover for every \(q>0\) one may choose the solution in the full energy space \(W_0^{1,(p_j)}(\Omega)\).
E.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta\le q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j<\frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).

critical threshold and strict upper bound in the intermediate regime

explicit quantitative upper bound on the exponents in case (ii)

loss of full anisotropic energy regularity for small/intermediate q caused by the delicate estimate on the nonlinear term

The stem does not explicitly reveal the correct option. It gives the full hypotheses of the PDE theorem, but no direct cue singles out choice A over the nearby variants.

This is essentially a theorem-conclusion identification item: the stem states the assumptions and asks for the corresponding existence statement. The distractors introduce threshold and exponent variations, so it is not a pure verbatim restatement, but it remains close to theorem recall.

The item requires some discrimination among subtle alternatives (strict vs non-strict inequalities, exact regime splits, equality vs upper-bound regularity), but it mainly tests recognition/recall of a specific result rather than genuine derivation or generative mathematical reasoning.

B, D, and E are plausible and mathematically targeted distractors. However, C is problematic because it reads like a weaker paraphrase that may also be taken as true unless the question explicitly asks for the most precise statement, creating potential ambiguity.

A technically sophisticated but mostly theorem-recall MCQ. It avoids answer leakage and has several strong near-miss distractors, but it does not strongly test generative reasoning, and the weaker paraphrase option introduces ambiguity.