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\title[A Sobolev-type inequality with Hardy potential]{On the critical behavior for a Sobolev-type inequality with Hardy potential}

\author{\firstname{Mohamed} \lastname{Jleli}}
\address{Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia}
\email{jleli@ksu.edu.sa}

\author{\firstname{Bessem} \lastname{Samet}}
\address[1]{Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia}
\email{bsamet@ksu.edu.sa}
\subjclass{35R45, 35A01, 35B33}

\thanks{The second author is supported by Researchers Supporting Project number (RSP2023R4), King Saud University, Riyadh, Saudi Arabia.}
\CDRGrant[King Saud University]{RSP2023R4}

\keywords{Sobolev-type inequality; Hardy potential; bounded domain; existence; nonexistence; critical exponent}

\begin{abstract}
We investigate the existence and nonexistence of weak solutions to the Sobolev-type inequality $ -\partial_t(\Delta u)-\Delta u+ \frac{\sigma}{|x|^2}u \geq |x|^{\mu}|u|^p$ in $(0,\infty)\times B$, under the inhomogeneous Dirichlet-type boundary condition $u(t,x)=f(x)$ on $(0,\infty)\times \partial B$, where $B$ is the unit open ball of $\mathbb{R}^N$, $N\geq 2$, $\sigma>-\bigl(\frac{N-2}{2}\bigr)^2$, $\mu\in \mathbb{R}$ and $p>1$. In particular, when $\sigma\neq 0$, we show that the dividing line with respect to existence and nonexistence is given by a critical exponent that depends on $N$, $\sigma$ and $\mu$.
\end{abstract}

\dateposted{2024-02-02}
\begin{document}
\maketitle

\section{Introduction}\label{sec1}

We are concerned with the study of existence and nonexistence of solutions to the Sobolev-type inequality
\begin{equation}\label{P}
-\partial_t(\Delta u)-\Delta u+ \frac{\sigma}{|x|^2}u \geq |x|^{\mu}|u|^p\quad \text{ in } (0,\infty)\times B,
\end{equation}
where $u=u(t,x)$, $B$ is the unit open ball of $\mathbb{R}^N$, $N\geq 2$, $\sigma>-\left(\frac{N-2}{2}\right)^2$, $\mu\in \mathbb{R}$ and $p>1$. Problem~\eqref{P} is considered under the Dirichlet-type boundary condition
\begin{equation}\label{BC}
u =f\quad \text{ on } (0,\infty)\times \partial B,
\end{equation}
where $f=f(x)\in L^1(\partial B)$. We mention below some motivations for investigating problems of type~\eqref{P}--\eqref{BC}.

The corresponding equation to~\eqref{P} belongs to the class of Sobolev-type equations of the form
\begin{equation}\label{eq1.3}
\partial_t Au+Bu=V(x)F(u),
\end{equation}
where $A$ and $B$ are linear elliptic operators and $F(u)$ is a nonlinear term with respect to $u$. In our case, we have $Au=-\Delta u$, $Bu=-\Delta u+\frac{\sigma}{|x|^2}u$, $V(x)=|x|^\mu$ and $F(u)=|u|^p$. Equations of type~\eqref{eq1.3} arise in many mathematical models. For instance, the Hoff equation~\cite{Hoff} ($Au=-\partial_{xx}u+u$, $Bu=0$, $V=1$, $F(u)=\alpha u+\beta u^3$), the Barenblatt--Zheltov--Kochina equation~\cite{BA}\pagebreak{} ($Au=-\Delta u+cu$, $Bu=-\Delta u$, $V=1$, $F(u)=0$) that describes nonstationary filtering processes in fissured-porous media, the semiconductor equation~\cite{KO-05} ($Au=-\Delta u+u$, $Bu=-\Delta u$, $V=1$, $Fu=\alpha u^3$) that describes nonstationary processes in crystalline semiconductors, the one-dimensional Boussinesq equation~\cite{DZ} ($Au=-\partial_{xx}u+u$, $Bu=0$, $V=1$, $F(u)=\alpha \partial_{xx}\left(|u|^{p-2}u\right)$), and many others.

Sobolev-type equations and inequalities have been studied in various contexts: numerical solutions~\cite{AI,BE,GU,ZAM}, asymptotic behaviour of solutions~\cite{AR,AR2,BR,DA}, inverse problems~\cite{FY,SV,UR,ZA2} and blow-up of solutions~\cite{AL,AR3,CA,JS,JS2,KO-05,KO-09,KO-16,MU}. In particular, the issue of nonexistence of (weak) solutions to various differential inequalities of Sobolev-type has been investigated in~\cite{KO-09}. For instance, the special case of~\eqref{P} with $\sigma=\mu=0$ has been studied in the whole space $\mathbb{R}^N$. Namely, it was shown that the Sobolev-type inequality
\begin{equation}\label{SP-case}
-\partial_t(\Delta u)-\Delta u \geq |u|^p\quad \text{ in } (0,\infty)\times \mathbb{R}^N
\end{equation}
subject to the initial condition
\[
u(0,x)=u_0(x)\quad \text{ in } \mathbb{R}^N,
\]
where $p>1$ and $u_0\in L^1(\mathbb{R}^N)$, admits no nontrivial solution, provided that $N\in\{1,2\}$; or
\[
N\geq 3,\quad p<\frac{N}{N-2}.
\]
An extension of the above result to a time-space-fractional version of~\eqref{SP-case} has been obtained in~\cite{AL}.

The issue of existence and nonexistence of solutions to evolution equations and inequalities with Hardy potential in unbounded domains has been considered in several papers. For instance, Hamidi and Laptev~\cite{HA} invetsigated the higher order evolution inequality
\begin{equation}\label{P-HL}
\partial_t^k u-\Delta u+\frac{\lambda}{|x|^2}u \geq |u|^p\quad \text{in }(0,\infty)\times \mathbb{R}^N
\end{equation}
subject to the initial condition
\begin{equation}\label{IC-HA}
\partial_t^{k-1}u(0,x)\geq 0\quad \text{in } \mathbb{R}^N.
\end{equation}
where $\partial_t^i u=\frac{\partial^i u}{\partial t^i}$, $N\geq 3$, $k\geq 1$, $p>1$ and $\lambda \geq -\left(\frac{N-2}{2}\right)^2$. Namely, it was proven that, if one of the following assumptions is satisfied:
\[
\lambda\geq 0,\quad 1<p\leq 1+\frac{2}{\frac{2}{k}+s^*};
\]
or
\[
-\left(\frac{N-2}{2}\right)^2\leq \lambda<0,\quad 1<p\leq 1+\frac{2}{\frac{2}{k}-s_*},
\]
where
\[
s^*=\frac{N-2}{2}+\sqrt{\lambda+\left(\frac{N-2}{2}\right)^2}, s_*=s^*+2-N,
\]
then~\eqref{P-HL}--\eqref{IC-HA} admits no nontrivial (weak) solution. In the parabolic case, among other problems, Abdellaoui et\,al.~\cite{ABD} (see also~\cite{ABD2}) considered problems of the form
\begin{equation}\label{P-ABD}
\partial_t(u^{p-1})-\Delta_p u=\lambda\frac{u^{p-1}}{|x|^p}+u^q\, (u>0)\quad \text{in }(0,\infty)\times \mathbb{R}^N,
\end{equation}
where $1<p<N$, $q>0$ and $0\leq \lambda<\left(\frac{N-p}{p}\right)^p$. Namely, it was shown that there exist two exponents $q^+(p,\lambda)$ and $F(p,\lambda)$ such that,
\begin{enumerate}\romanenumi
\item if $p-1<q< F(p,\lambda)<q^+(p,\lambda)$ and $u$ is a solution to~\eqref{P-ABD} satisfying a certain behavior, then $u$ blows-up in a finite time;
\item if $F(p,\lambda)<q< q^+(p,\lambda)$, then under suitable condition on $u(0,\cdot\,)$, \eqref{P-ABD} admits a global in time positive solution.
\end{enumerate}
We refer also to~\cite{JSV21}, where~\eqref{P-HL} with $k=2$ has been studied in an exterior domain of $\mathbb{R}^N$ under different types of inhomogeneous boundary conditions. Additional results related to parabolic and elliptic equations involving Hardy potential in bounded domains of $\mathbb{R}^N$ can be found in~\cite{ABD,ABD02,ABD2,ABD3,MERC}.

To the best of our knowledge, Sobolev-type equations/inequalities with Hardy potential have not previously been studied.

Before presenting our obtained results, let us give the meaning of solutions to the considered problem. Let
\[
\omega=(0,\infty)\times \bar{B}\backslash\{0\},\quad \Gamma=(0,\infty)\times \partial B.
\]
Observe that $\Gamma\subset \omega$. We introduce the set
\[
\Phi=\left\{\varphi\in C^3(\omega): \supp(\varphi)\subset\subset\omega,\, \varphi\geq 0,\, \varphi|_{\Gamma}=0\right\}.
\]
Here, by $\supp(\varphi)\subset\subset\omega$, we mean that $\supp(\varphi)$ is a compact subset of
\[
(0,\infty)\times \left\{x\in \mathbb{R}^N: 0<|x|\leq 1\right\}.
\]

Solutions to~\eqref{P} under the boundary condition~\eqref{BC} are defined as follows.

\begin{defi}\label{def1}
Let $N\geq 2$, $\sigma>-\left(\frac{N-2}{2}\right)^2$, $\mu\in \mathbb{R}$, $p>1$ and $f=f(x)\in L^1(\partial B)$. We say that $u\in L^p_{\loc}(\omega)$ is a weak solution to~\eqref{P} under the boundary condition~\eqref{BC}, if
\begin{equation}\label{ws-P1}
\int_{\omega} |x|^{\mu}|u|^p\varphi\,\dx\,\dt+\int_{\Gamma}\left(\partial_\nu \partial_t\varphi-\partial_\nu \varphi\right)\, f(x)\,\dS_x\,\dt
\leq \int_{\omega} u\left(\Delta \partial_t\varphi -\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right)\,\dx\,\dt
\end{equation}
for all $\varphi\in \Phi$, where $\nu$ is the outward unit normal to $\partial B$, relative to $B$, and $\partial_\nu$ is the normal derivative on $\partial B$.
\end{defi}

By integration by parts, it can be easily seen that any smooth solution to~\eqref{P}--\eqref{BC} is a weak solution in the sense of Definition~\ref{def1}.

For $\sigma>-\left(\frac{N-2}{2}\right)^2$, we introduce the parameter
\begin{equation}\label{sigmaN}
\sigma_N=-\frac{N-2}{2}+\sqrt{\sigma+\left(\frac{N-2}{2}\right)^2}.
\end{equation}
We introduce also the set
\[
L^{1,+}(\partial B)=\left\{w\in L^1(\partial B): \int_{\partial B}w(x)\,\dS_x>0\right\}.
\]

Our main result is sated in the following theorem.

\begin{theo}\label{T1}
Let $N\geq 2$, $\sigma>-\left(\frac{N-2}{2}\right)^2$, $\mu\in \mathbb{R}$ and $p>1$.
\begin{enumerate}\Romanenumi
\item \label{2I} Let $f\in L^{1,+}(\partial B)$. If
\begin{equation}\label{cd1-blowup}
\sigma_N p<\sigma_N-\mu-2,
\end{equation}
then problem~\eqref{P} under the boundary condition~\eqref{BC} admits no weak solution.
\item \label{2II} If
\begin{equation}\label{cd1-existence}
\sigma_N p>\sigma_N-\mu-2,
\end{equation}
then problem~\eqref{P} under the boundary condition~\eqref{BC} admits stationary solutions for some $f\in L^{1,+}(\partial B)$.
\end{enumerate}
\end{theo}

The proof of part~\eqref{2I} of Theorem~\ref{T1} is based on nonlinear capacity estimates specifically adapted to the operator $-\Delta\cdot+\frac{\sigma}{|x|^2}\cdot $, the domain $(0,\infty)\times B$, and the boundary condition~\eqref{BC}. Part~\eqref{2II} of Theorem~\ref{T1} is proved by the construction of explicit solutions.

Now, let us consider the case $\sigma=0$ and $N\geq 3$. In this case, by~\eqref{sigmaN}, one has $\sigma_N=0$. Hence, \eqref{cd1-blowup} reduces to $\mu<-2$, and~\eqref{cd1-existence} reduces to $\mu>-2$. Thus, from Theorem~\ref{T1}, we deduce the following result.

\begin{coro}\label{CR1}
Let $N\geq 3$, $\sigma=0$, $\mu\in \mathbb{R}$ and $p>1$.
\begin{enumerate}\Romanenumi
\item \label{3I} Let $f\in L^{1,+}(\partial B)$. If $\mu<-2$, then problem~\eqref{P} under the boundary condition~\eqref{BC} admits no weak solution.
\item \label{3II} If $\mu>-2$, then problem~\eqref{P} under the boundary condition~\eqref{BC} admits stationary solutions for some $f\in L^{1,+}(\partial B)$.
\end{enumerate}
\end{coro}

\begin{rema}
From Corollary~\ref{CR1}, we deduce that in the case $\sigma=0$ and $N\geq 3$, $\mu^*=-2$ is a critical parameter for problem~\eqref{P} under the boundary condition~\eqref{BC}.
\end{rema}

Next, let us consider the case $-\left(\frac{N-2}{2}\right)^2<\sigma<0$ and $N\geq 3$. In this case, by~\eqref{sigmaN}, one has $\sigma_N<0$. Hence, \eqref{cd1-blowup} reduces to $p>1-\frac{\mu+2}{\sigma_N}$, and~\eqref{cd1-existence} reduces to $p<1-\frac{\mu+2}{\sigma_N}$. Then, by Theorem~\ref{T1}, we obtain the following result.

\begin{coro}\label{CR2}
Let $N\geq 3$, $-\left(\frac{N-2}{2}\right)^2<\sigma<0$ and $\mu\in \mathbb{R}$.
\begin{enumerate}\Romanenumi
\item \label{5I} Let $f\in L^{1,+}(\partial B)$.
\begin{enumerate}
\item \label{5Ia}  If $\mu\leq -2$, then for all $p>1$, problem~\eqref{P} under the boundary condition~\eqref{BC} admits no weak solution.
\item \label{5Ib} If $\mu>-2$, then for all
\[
p>1-\frac{\mu+2}{\sigma_N},
\]
problem~\eqref{P} under the boundary condition~\eqref{BC} admits no weak solution.
\end{enumerate}
\item \label{5II} If $\mu>-2$, then for all
\[
1<p<1-\frac{\mu+2}{\sigma_N},
\]
problem~\eqref{P} under the boundary condition~\eqref{BC} admits stationary solutions for some $f\in L^{1,+}(\partial B)$.
\end{enumerate}
\end{coro}

\begin{rema}
From Corollary~\ref{CR2}, we deduce that in the case$-\left(\frac{N-2}{2}\right)^2<\sigma<0$ and $N\geq 3$, the dividing line with respect to existence and nonexistence is given by the critical exponent
\[
p^*=p^*(N,\sigma,\mu)=
\begin{cases}
1 &\text{if } \mu\leq -2,\\
1- \frac{\mu+2}{\sigma_N} &\text{if } \mu> -2.
\end{cases}
\]
Namely,
\begin{enumerate}\romanenumi
\item if $f\in L^{1,+}(\partial B)$ and $p>p^*$, then problem~\eqref{P} under the boundary condition~\eqref{BC} admits no weak solution;
\item if $1<p<p^*$, then problem~\eqref{P} under the boundary condition~\eqref{BC} admits solutions for some $f\in L^{1,+}(\partial B)$.
\end{enumerate}
\end{rema}

Finally, let us consider the case $\sigma>0$ and $N\geq 2$. In this case, by~\eqref{sigmaN}, one has $\sigma_N>0$. Hence, \eqref{cd1-blowup} reduces to $p<1-\frac{\mu+2}{\sigma_N}$, and~\eqref{cd1-existence} reduces to $p>1-\frac{\mu+2}{\sigma_N}$. Thus, we deduce from Theorem~\ref{T1} the following result.

\begin{coro}\label{CR3}
Let $N\geq 2$, $\sigma>0$ and $\mu\in \mathbb{R}$.
\begin{enumerate}\Romanenumi
\item Let $f\in L^{1,+}(\partial B)$. If $\mu<-2$, then for all
\[
1<p<1-\frac{\mu+2}{\sigma_N},
\]
problem~\eqref{P} under the boundary condition~\eqref{BC} admits no weak solution.
\item If $\mu\geq -2$, then for all $p>1$, problem~\eqref{P} under the boundary condition~\eqref{BC} admits stationary solutions for some $f\in L^{1,+}(\partial B)$.
\item If $\mu<-2$, then for all
\[
p>1-\frac{\mu+2}{\sigma_N},
\]
problem~\eqref{P} under the boundary condition~\eqref{BC} admits stationary solutions for some $f\in L^{1,+}(\partial B)$.
\end{enumerate}
\end{coro}

\begin{rema}
Let $N\geq 2$ and $\sigma>0$. From Corollary~\ref{CR3}, we deduce that, if $\mu< -2$, the dividing line with respect to existence and nonexistence is given by the critical exponent
\[
p_*=p_*(N,\sigma,\mu)=1- \frac{\mu+2}{\sigma_N}.
\]
Namely,
\begin{enumerate}\romanenumi
\item \label{8i} if $f\in L^{1,+}(\partial B)$ and $1<p<p_*$, then problem~\eqref{P} under the boundary condition~\eqref{BC} admits no weak solution;
\item \label{8ii} if $p>p^*$, then problem~\eqref{P} under the boundary condition~\eqref{BC} admits solutions for some $f\in L^{1,+}(\partial B)$.
\end{enumerate}
However, if $\mu\geq -2$, the problem admits no critical behavior, that is, for all $p>1$, stationary solutions exists for some $f\in L^{1,+}(\partial B)$.
\end{rema}

The rest of the paper is organized as follows. In Section~\ref{sec2}, we establish some useful preliminary estimates. In Section~\ref{sec3}, we prove Theorem~\ref{T1}.

Throughout the paper, the symbols $C$ or $C_i$ denote always generic positive constants, which are independent of the scaling parameters $T$ and $R$, and the solution $u$. Their values could be changed from one line to another. We will use frequently the notation $T,R,\ell\gg 1$, to indicate that the above parameters are sufficiently large.


\section{Preliminaries}\label{sec2}

Let $N\geq 2$, $\sigma>-\left(\frac{N-2}{2}\right)^2$, $\mu\in \mathbb{R}$ and $p>1$. We introduce the function
\[
K(x)=|x|^{2-N-\sigma_N}\left(1-|x|^{2\sigma_N+N-2}\right),\quad x\in \bar{B}\backslash\{0\},
\]
where $\sigma_N$ is given by~\eqref{sigmaN}. It can be easily seen that the function $K$ satisfies the following properties.

\begin{lemm}\label{L2.1}
The following properties hold:
\begin{enumerate}\romanenumi
\item \label{9i} $K\geq 0$;
\item \label{9ii} $-\Delta K+\frac{\sigma}{|x|^2}K=0$ in $B\backslash\{0\}$;
\item \label{9iii} $K|_{\partial B}=0$;
\item \label{9iv} $\partial_\nu K=2-N-2\sigma_N$.
\end{enumerate}
\end{lemm}

Next, we introduce two cut-off functions $\alpha$ and $\beta$ satisfying:
\begin{equation}\label{pptsalpha}
\alpha\in C^\infty([0,\infty)),\quad \alpha\geq 0,\quad \supp(\alpha)\subset\subset (0,1)
\end{equation}
and
\begin{equation}\label{pptsbeta}
\beta\in C^\infty([0,\infty)),\quad 0\leq \beta\leq 1,\quad \beta(s)=0\ \text{ if } 0\leq s\leq \frac{1}{2},\quad \beta(s)=1\ \text{ if } s\geq 1.
\end{equation}
For $T,R,\ell \gg 1$, let
\begin{equation}\label{alphaT-f}
\alpha_T(t)=\alpha^\ell\left(\frac{t}{T}\right),\quad t\geq 0
\end{equation}
and
\begin{equation}\label{betaR-f}
\beta_R(x)=K(x)\beta^\ell(R|x|),\quad x\in B\backslash\{0\}.
\end{equation}
We consider the family of functions $\{\varphi_{T,R,\ell}\}_{T,R,\ell \gg 1}$, where
\begin{equation}\label{testf}
\varphi_{T,R,\ell}(t,x)=\varphi(t,x)=\alpha_T(t)\beta_R(x),\quad (t,x)\in \omega.
\end{equation}

\begin{lemm}\label{L2.2}
For $T,R,\ell \gg 1$, the function $\varphi$ defined by~\eqref{testf} belongs to $\Phi$. Moreover, we have
\begin{equation}\label{nd1}
\partial_\nu\varphi(t,x)=(2-N-2\sigma_N)\alpha_T(t),\quad (t,x)\in \Gamma
\end{equation}
and
\begin{equation}\label{nd1-t}
\partial_\nu\partial_t\varphi(t,x)=(2-N-2\sigma_N)\alpha_T'(t),\quad (t,x)\in \Gamma.
\end{equation}
\end{lemm}

\begin{proof}
By the definition of $\varphi$, it can be easily seen that for $T,R,\ell \gg 1$, we have
\[
\varphi\in C^3(\omega),\quad \supp(\varphi)\subset\subset\omega.
\]
Furthermore, by~\eqref{pptsalpha}, \eqref{pptsbeta}, \eqref{alphaT-f}, \eqref{betaR-f}, \eqref{testf} and Lemma~\ref{L2.1}$\MK$\eqref{9i}, \eqref{9iii}, we have
\[
\varphi\geq 0,\quad \varphi|_{\Gamma}=0.
\]
Consequently, we have $\varphi\in \Phi$. On the other hand, by~\eqref{pptsbeta} and~\eqref{betaR-f}, we have
\begin{equation}\label{S1-L2.2}
\beta_R(x)=K(x),\quad \frac{1}{R}\leq |x|<1.
\end{equation}
Then~\eqref{nd1} and~\eqref{nd1-t} follow from~\eqref{testf}, \eqref{S1-L2.2} and Lemma~\ref{L2.1}$\MK$\eqref{9iv}.
\end{proof}

For $T,R,\ell\gg 1$, let $\varphi$ be the function defined by~\eqref{testf}.

\begin{lemm}\label{L2.3}
The following estimate holds:
\begin{equation}\label{est-L2.3}
\int_{\supp\left(\Delta \partial_t\varphi\right)} |x|^{\frac{-\mu}{p-1}} \varphi^{\frac{-1}{p-1}}\left|\Delta \partial_t\varphi\right|^{\frac{p}{p-1}}\,\dx\,\dt\leq C T^{1-\frac{p}{p-1}}\left(\ln R+R^{\sigma_N-2+\frac{\mu+2p}{p-1}}\right).
\end{equation}
\end{lemm}

\begin{proof}
By the definition of $\varphi$, we obtain
\begin{multline}\label{S1-L2.3}
\int_{\supp\left(\Delta \partial_t\varphi\right)} |x|^{\frac{-\mu}{p-1}} \varphi^{\frac{-1}{p-1}}\left|\Delta \partial_t\varphi\right|^{\frac{p}{p-1}}\,\dx\,\dt\\
=\left(\int_0^T\alpha_T^{\frac{-1}{p-1}}(t)|\alpha_T'(t)|^{\frac{p}{p-1}}\,\dt\right)\left(\int_{\supp\left(\Delta \beta_R\right)} |x|^{\frac{-\mu}{p-1}}\beta_R^{\frac{-1}{p-1}}(x)|\Delta \beta_R(x)|^{\frac{p}{p-1}}\,\dx\right).
\end{multline}
On the other hand, by~\eqref{alphaT-f}, we have
\[
\alpha_T'(t)=\ell T^{-1} \alpha^{\ell-1}\left(\frac{t}{T}\right)\alpha'\left(\frac{t}{T}\right),
\]
which implies by~\eqref{pptsalpha} and~\eqref{alphaT-f} that
\[
\alpha_T^{\frac{-1}{p-1}}(t)|\alpha_T'(t)|^{\frac{p}{p-1}}\leq C T^{-\frac{p}{p-1}} \alpha^{\ell-\frac{p}{p-1}}\left(\frac{t}{T}\right),\quad 0<t<T.
\]
Integrating, we get
\[
\int_0^T\alpha_T^{\frac{-1}{p-1}}(t)|\alpha_T'(t)|^{\frac{p}{p-1}}\,\dt\leq CT^{-\frac{p}{p-1}}\int_0^T \alpha^{\ell-\frac{p}{p-1}}\left(\frac{t}{T}\right)\,\dt=CT^{1-\frac{p}{p-1}}\int_0^1 \alpha^{\ell-\frac{p}{p-1}}(s)\,\ds,
\]
that is,
\begin{equation}\label{S2-L2.3}
\int_0^T\alpha_T^{\frac{-1}{p-1}}(t)|\alpha_T'(t)|^{\frac{p}{p-1}}\,\dt\leq C T^{1-\frac{p}{p-1}}.
\end{equation}
Furthermore, by~\eqref{betaR-f}, we have
\begin{align*}
\Delta(\beta_R(x))&=\Delta\left(K(x)\beta^\ell(R|x|)\right)\\
&=\beta^\ell(R|x|)\Delta K(x)+K(x)\Delta\, \beta^\ell(R|x|)+2 \nabla K(x)\cdot \nabla\, \beta^\ell(R|x|),
\end{align*}
where $\cdot$ denotes the inner product in $\mathbb{R}^N$, which implies by Lemma~\ref{L2.1}$\MK$\eqref{9ii} that
\[
\Delta(\beta_R(x))=\sigma|x|^{-2}K(x)\beta^\ell(R|x|) +K(x)\Delta\, \beta^\ell(R|x|)+2 \nabla K(x)\cdot \nabla\,\beta^\ell(R|x|).
\]
Hence, from~\eqref{pptsbeta}, we deduce that
\begin{equation}\label{S3-L2.3}
\int_{\supp\left(\Delta \beta_R\right)} |x|^{\frac{-\mu}{p-1}}\beta_R^{\frac{-1}{p-1}}(x)|\Delta \beta_R(x)|^{\frac{p}{p-1}}\,\dx\leq C(I_1+I_2+I_3),
\end{equation}
where
\begin{align*}
I_1&= \int_{\frac{1}{2R}<|x|<1}|x|^{-\frac{\mu+2p}{p-1}}K(x) \beta^\ell(R|x|)\,\dx,\\
I_2&= \int_{\frac{1}{2R}<|x|<\frac{1}{R}} |x|^{\frac{-\mu}{p-1}} K(x)\left|\Delta\, \beta^\ell(R|x|)\right|^{\frac{p}{p-1}} \beta^{\frac{-\ell}{p-1}}(R|x|)\,\dx,\\
I_3 &= \int_{\frac{1}{2R}<|x|<\frac{1}{R}} |x|^{\frac{-\mu}{p-1}} K^{\frac{-1}{p-1}}(x) |\nabla K(x)|^{\frac{p}{p-1}}\beta^{\frac{-\ell}{p-1}}(R|x|)\left|\nabla\,\beta^\ell(R|x|)\right|^{\frac{p}{p-1}}\,\dx.
\end{align*}
Let us estimate the terms $I_i$, $i=1,2,3$. Since $0\leq \beta\leq 1$, by the definition of $K$, we obtain
\begin{align*}
I_1 &\leq \int_{\frac{1}{2R}<|x|<1}|x|^{-\frac{\mu+2p}{p-1}}K(x) \,\dx\\
&\leq \int_{\frac{1}{2R}<|x|<1}|x|^{-\frac{\mu+2p}{p-1}+2-N-\sigma_N} \,\dx\\
&= C \int_{r=\frac{1}{2R}}^1 r^{-\frac{\mu+2p}{p-1}+1-\sigma_N}\,\dr\\
&= C \begin{cases}
1 &\text{if } -\frac{\mu+2p}{p-1}+2-\sigma_N>0,\\
R^{\sigma_N-2+\frac{\mu+2p}{p-1}} &\text{if } -\frac{\mu+2p}{p-1}+2-\sigma_N<0,\\
\ln R &\text{if } -\frac{\mu+2p}{p-1}+2-\sigma_N=0,
\end{cases}
\end{align*}
which yields
\begin{equation}\label{S4-L2.3}
I_1\leq C \left(\ln R+R^{\sigma_N-2+\frac{\mu+2p}{p-1}}\right).
\end{equation}
On the other hand, by~\eqref{sigmaN}, \eqref{betaR-f} and the definition of $K$, for $\frac{1}{2R}<|x|<\frac{1}{R}$, we have
\begin{equation}\label{S5-L2.3}
\left|\Delta\, \beta^\ell(R|x|)\right|\leq C R^2 \beta^{\ell-2}(R|x|), \quad \left|\nabla\, \beta^\ell(R|x|)\right|\leq C R \beta^{\ell-2}(R|x|)
\end{equation}
and
\begin{equation}\label{S6-L2.3}
C_1 R^{\sigma_N+N-2}\leq K(x)\leq C_2 R^{\sigma_N+N-2},\quad |\nabla K(x)|\leq C R^{\sigma_N+N-1}.
\end{equation}
Thus, due to $0\leq \beta\leq 1$, and using~\eqref{S5-L2.3} and~\eqref{S6-L2.3}, we obtain
\begin{equation}\label{S7-L2.3}
I_2 \leq C R^{\sigma_N-2+\frac{\mu+2p}{p-1}}
\end{equation}
and
\begin{equation}\label{S8-L2.3}
I_3 \leq C R^{\sigma_N-2+\frac{\mu+2p}{p-1}}.
\end{equation}
Finally, in view of~\eqref{S1-L2.3}, \eqref{S2-L2.3}, \eqref{S3-L2.3}, \eqref{S4-L2.3}, \eqref{S7-L2.3} and~\eqref{S8-L2.3}, we obtain~\eqref{est-L2.3}.
\end{proof}

\begin{lemm}\label{L2.4}
The following estimate holds:
\begin{equation}
\int_{\supp(\varphi)} |x|^{\frac{\mu}{p-1}} \varphi^{\frac{-1}{p-1}}\left|-\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right|^{\frac{p}{p-1}}\,\dx\,\dt\leq C T R^{\sigma_N-2+\frac{\mu+2p}{p-1}}.
\end{equation}
\end{lemm}

\begin{proof}
By the definition of the function $\varphi$, we obtain
\begin{multline}\label{S1-L2.4}
\int_{\supp(\varphi)} |x|^{\frac{\mu}{p-1}} \varphi^{\frac{-1}{p-1}}\left|-\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right|^{\frac{p}{p-1}}\,\dx\,\dt\\
=\left(\int_0^T\alpha_T(t)\,\dt\right)\left(\int_{\supp(\beta_R)}|x|^{\frac{\mu}{p-1}} \beta_R^{\frac{-1}{p-1}}(x)\left|-\Delta \beta_R(x)+\frac{\sigma}{|x|^2}\beta_R(x)\right|^{\frac{p}{p-1}}\,\dx\right).
\end{multline}
On the other hand, by~\eqref{alphaT-f}, we have
\begin{align*}
\int_0^T\alpha_T(t)\,\dt&= \int_0^T \alpha^\ell\left(\frac{t}{T}\right)\,\dt\\
&= T\int_0^1 \alpha^\ell(s)\,\ds,
\end{align*}
that is,
\begin{equation}\label{S2-L2.4}
\int_0^T\alpha_T(t)\,\dt=CT.
\end{equation}
moreover, using similar calculations as that done in the proof of Lemma~\ref{L2.3}, we obtain
\begin{equation}\label{S3-L2.4}
\int_{\supp(\beta_R)}|x|^{\frac{\mu}{p-1}} \beta_R^{\frac{-1}{p-1}}(x)\left|-\Delta \beta_R(x)+\frac{\sigma}{|x|^2}\beta_R(x)\right|^{\frac{p}{p-1}}\,\dx\leq C R^{\sigma_N-2+\frac{\mu+2p}{p-1}}.
\end{equation}
Hence, \eqref{est-L2.3} follows from~\eqref{S1-L2.4}, \eqref{S2-L2.4} and~\eqref{S3-L2.4}.
\end{proof}


\section{Proof of the main result}\label{sec3}

\begin{proof}[Proof of Theorem~\ref{T1}]\ 
\begin{proof}[(\ref{2I})]
Suppose that $u\in L^p_{\loc}(\omega)$ is a weak solution to~\eqref{P} under the boundary condition~\eqref{BC}. Then, by~\eqref{ws-P1}, for all $\varphi\in \Phi$, there holds
\begin{multline}\label{S1-T1-I}
\int_{\omega} |x|^{\mu}|u|^p\varphi\,\dx\,\dt+\int_{\Gamma}\left(\partial_\nu \partial_t\varphi-\partial_\nu \varphi\right)\, f(x)\,\dS_x\,\dt\\
\leq \int_{\omega} |u| \left|\Delta \partial_t\varphi \right|\,\dx\,\dt+\int_\omega |u|\left|-\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right|\,\dx\,\dt.
\end{multline}
On the other hand, by means of Young's inequality, we have
\begin{align}\label{S2-T1-I}
\nonumber \int_{\omega} |u| \left|\Delta \partial_t\varphi \right|\,\dx\,\dt&= \int_{\omega} |x|^{\frac{\mu}{p}}|u|\varphi^{\frac{1}{p}}\, |x|^{\frac{-\mu}{p}}\varphi^{\frac{-1}{p}}\left|\Delta \partial_t\varphi \right|\,\dx\,\dt\\
&\leq \frac{1}{2}\int_{\omega} |x|^{\mu}|u|^p\varphi\,\dx\,\dt+C \int_{\supp\left(\Delta \partial_t\varphi\right)} |x|^{\frac{-\mu}{p-1}}\varphi^{\frac{-1}{p-1}}\left|\Delta \partial_t\varphi \right|^{\frac{p}{p-1}}\,\dx\,\dt
\end{align}
and
\begin{multline}\label{S3-T1-I}
\int_\omega |u|\left|-\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right|\,\dx\,\dt\\
\leq \frac{1}{2}\int_{\omega} |x|^{\mu}|u|^p\varphi\,\dx\,\dt+C \int_{\supp(\varphi)} |x|^{\frac{\mu}{p-1}} \varphi^{\frac{-1}{p-1}}\left|-\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right|^{\frac{p}{p-1}}\,\dx\,\dt,
\end{multline}
provided that
\begin{equation}\label{finiteIn}
\begin{aligned}
\int_{\supp\left(\Delta \partial_t\varphi\right)} |x|^{\frac{-\mu}{p-1}}\varphi^{\frac{-1}{p-1}}\left|\Delta \partial_t\varphi \right|^{\frac{p}{p-1}}\,\dx\,\dt&<\infty,\\
\int_{\supp(\varphi)} |x|^{\frac{\mu}{p-1}} \varphi^{\frac{-1}{p-1}}\left|-\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right|^{\frac{p}{p-1}}\,\dx\,\dt&<\infty.
\end{aligned}
\end{equation}
In view of~\eqref{S1-T1-I}, \eqref{S2-T1-I} and~\eqref{S3-T1-I}, we obtain
\begin{equation}\label{S4-T1-I}
\begin{aligned}[b]
&\int_{\Gamma}\left(\partial_\nu \partial_t\varphi-\partial_\nu \varphi\right)\, f(x)\,\dS_x\,\dt\\
&\leq C\left(\int_{\supp\left(\Delta \partial_t\varphi\right)} |x|^{\frac{-\mu}{p-1}}\varphi^{\frac{-1}{p-1}}\left|\Delta \partial_t\varphi \right|^{\frac{p}{p-1}}\,\dx\,\dt+\int_{\supp(\varphi)} |x|^{\frac{\mu}{p-1}} \varphi^{\frac{-1}{p-1}}\left|-\Delta \varphi+\frac{\sigma}{|x|^2}\varphi\right|^{\frac{p}{p-1}}\,\dx\,\dt\right).
\end{aligned}
\end{equation}
Next, for $T,R,\ell\gg 1$, we consider the function $\varphi$ defined by~\eqref{testf}. By Lemma~\ref{L2.2}, we know that $\varphi\in \Phi$. Moreover, by Lemmas~\ref{L2.3} and~\ref{L2.4}, \eqref{finiteIn} holds. Consequently, for $T,R,\ell\gg 1$, \eqref{S4-T1-I} holds for the function $\varphi$ defined by~\eqref{testf}. On the other hand, thanks to~\eqref{nd1} and~\eqref{nd1-t}, we have
\[
\int_{\Gamma}\left(\partial_\nu \partial_t\varphi-\partial_\nu \varphi\right)\, f(x)\,\dS_x\,\dt=(2-N-2\sigma_N) \int_0^\infty \int_{\partial B} \left(\alpha_T'(t)-\alpha_T(t)\right)f(x)\,\dS_x\,\dt.
\]
Notice that by~\eqref{sigmaN}, we have $2-N-2\sigma_N<0$. Hence, by~\eqref{pptsalpha} and~\eqref{alphaT-f}, we get
\begin{equation}\label{S5-T1-I}
\begin{aligned}
\int_{\Gamma}\left(\partial_\nu \partial_t\varphi-\partial_\nu \varphi\right)\, f(x)\,\dS_x\,\dt
&=C\left(\int_0^T \left(\alpha^\ell\left(\frac{t}{T}\right)-\ell T^{-1}\alpha^{\ell-1}\left(\frac{t}{T}\right) \alpha'\left(\frac{t}{T}\right)\right)\,\dt\right)\left(\int_{\partial B}f(x)\,\dS_x\right)\\
&=C T\left(\int_0^1 \left(\alpha^\ell(s)-\ell T^{-1}\alpha^{\ell-1}(s)\alpha'(s)\right)\,\ds\right)\left(\int_{\partial B}f(x)\,\dS_x\right).
\end{aligned}
\end{equation}
Furthermore, by the dominated convergence theorem, we have
\[
\lim_{T\to \infty} \int_0^1 \left(\alpha^\ell(s)-\ell T^{-1}\alpha^{\ell-1}(s)\alpha'(s)\right)\,\ds=\int_0^1\alpha^\ell(s)\,\ds>0.
\]
Thus, for $T\gg 1$, one has
\[
\int_0^1 \left(\alpha^\ell(s)-\ell T^{-1}\alpha^{\ell-1}(s)\alpha'(s)\right)\,\ds\geq C.
\]
Since $f\in L^{1,+}(\partial B)$, we deduce from~\eqref{S5-T1-I} that
\begin{equation}\label{S6-T1-I}
\int_{\Gamma}\left(\partial_\nu \partial_t\varphi-\partial_\nu \varphi\right)\, f(x)\,\dS_x\,\dt\geq C T\int_{\partial B}f(x)\,\dS_x.
\end{equation}
Then, using~\eqref{S4-T1-I}, \eqref{S6-T1-I}, Lemmas~\ref{L2.3} and~\ref{L2.4}, we obtain
\[
T\int_{\partial B}f(x)\,\dS_x\leq C\left[T^{1-\frac{p}{p-1}}\left(\ln R+R^{\sigma_N-2+\frac{\mu+2p}{p-1}}\right)+T R^{\sigma_N-2+\frac{\mu+2p}{p-1}}\right],
\]
that is,
\begin{equation}\label{S7-T1-I}
\int_{\partial B}f(x)\,\dS_x\leq C \left(T^{\frac{-p}{p-1}}\ln R+T^{\frac{-p}{p-1}} R^\iota +R^\iota\right),
\end{equation}
where
\[
\iota= \sigma_N-2+\frac{\mu+2p}{p-1}.
\]
Notice that due to~\eqref{cd1-blowup}, we have $\iota<0$. Hence, taking $T=R$ in~\eqref{S7-T1-I}, and passing to the limit as $R\to \infty$, we obtain $\int_{\partial B}f(x)\,\dS_x\leq 0$, which is a contradiction with $f\in L^{1,+}(\partial B)$. This completes the proof of part~\eqref{2I} of Theorem~\ref{T1}.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{2II})] Let
\begin{equation}\label{delta}
\max\left\{2-N-\sigma_N,\frac{-(\mu+2)}{p-1}\right\}<\delta<\sigma_N
\end{equation}
and
\begin{equation}\label{eps}
0<\varepsilon< \left(-\delta^2+(2-N)\delta+\sigma\right)^{\frac{1}{p-1}}.
\end{equation}
Notice that by~\eqref{sigmaN}, we have $2-N-\sigma_N<\sigma_N$. Moreover, due to~\eqref{cd1-existence}, there holds $\frac{-(\mu+2)}{p-1}<\sigma_N$. Hence, the set of $\delta$ satisfying~\eqref{delta} is nonempty. Notice also that $2-N-\sigma_N$ and $\sigma_N$ are the roots of the polynomial function
\[
F(\delta)=-\delta^2+(2-N)\delta+\sigma.
\]
Hence, for all $\delta$ satisfying~\eqref{delta}, one has $F(\delta)>0$. Thus, the set of $\varepsilon$ satisfying~\eqref{eps} is nonempty. We consider functions of the form
\[
u_{\delta,\varepsilon}(x)=\varepsilon |x|^\delta,\quad x\in B\backslash\{0\}.
\]
Elementary calculations show that
\begin{equation}\label{OK}
-\Delta u_{\delta,\varepsilon}+ \frac{\sigma}{|x|^2}u_{\delta,\varepsilon}=\varepsilon F(\delta)|x|^{\delta-2}.
\end{equation}
Then, thanks to~\eqref{delta}, \eqref{eps} and~\eqref{OK}, we obtain
\[
-\Delta u_{\delta,\varepsilon}+ \frac{\sigma}{|x|^2}u_{\delta,\varepsilon}\geq \varepsilon^p |x|^{\delta p+\mu}=|x|^\mu u_{\delta,\varepsilon}^p.
\]
Thus, $u_{\delta,\varepsilon}$ is a stationary solution to~\eqref{P}--\eqref{BC} with $f\equiv \varepsilon$. This completes the proof of part~\eqref{2II} of Theorem~\ref{T1}.
\end{proof}
\let\qed\relax
\end{proof}

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