First, Lemma <ref>(b), Lemma <ref> and dominated convergence theorem give[we use $ \Delta f (w+W_{Y_1} + \cdots W_{Y_n}) $ to replace $  f(w+\Delta w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w+\Delta w)} -  f(w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w)}$ for simplicity. $a_1,\ldots,a_k$ are arbitrary finite numbers.]

\begin{align}
&\lim_{\Delta w \rightarrow 0}\frac{1}{\Delta w} \mathbb{E} \left[   \Delta f (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w),c_{n-1}(w+\Delta w)} \right] =   \mathbb{E} \left[    f' (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w)} \right], \label{eq-der2forgmain} \\
& \lim_{\Delta w \rightarrow 0}\frac{1}{\Delta w} \mathbb{E} \left[   \Delta f (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c^{k1}(w),c^{k2}(w+\Delta w)} \right] = 0, \ k1,k2 \in \{1,2,\cdots,n-1\}. \label{eq-der2forgmain2}
\end{align}

Similarly, for any $k\in \{1,2,\ldots,n-1\}$,

\begin{align}
\nonumber  & \lim_{\Delta w \rightarrow 0} \frac{1}{\Delta w} \mathbb{E} \left[   \Delta f (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w),c^k(w+\Delta w)} \right] \\
\nonumber  = &  \lim_{\Delta w \rightarrow 0}  \frac{1}{\Delta w} \mathbb{E} \left[ - f (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w),c_{k-1}(w+\Delta w), |w+\Delta w +W_{Y_1}+\cdots+W_{Y_k}| <a_k}  \right] \\
\nonumber &  \Delta w \rightarrow 0+: \\ \nonumber = &  \lim_{\Delta w \rightarrow 0}  \frac{1}{\Delta w}\mathbb{E} \Big[  - f (-a_k+W_{Y_{k+1}} + \cdots W_{Y_n}) {\text{\large $\mathds{1}$}}_{ |-a_k+W_{Y_{k+1}}| \ge a_{k+1}, \cdots,|-a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} } \\ \nonumber & \times  {\text{\large $\mathds{1}$}}_{c_{k-1}(w), c_{k-1}(w+\Delta w), -a_k - \Delta w < w +W_{Y_1}+\cdots+W_{Y_k} < -a_k  } \Big]  \\
\nonumber = &  \mathbb{E} \left[  - f (-a_k+W_{Y_{k+1}} + \cdots W_{Y_{n}}) {\text{\large $\mathds{1}$}}_{ |-a_k+W_{Y_{k+1}}| \ge a_{k+1}, \cdots,|-a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} } \right] \\  & \times  \lim_{\Delta w \rightarrow 0} \frac{1}{\Delta w} \mathbb{E} \left[  {\text{\large $\mathds{1}$}}_{c_{k-1}(w), -a_k - \Delta w < w +W_{Y_1}+\cdots+W_{Y_k} < -a_k  } \right]; \\
\nonumber &   \Delta w \rightarrow 0-: \\ = & \mathbb{E} \left[   - f (a_k+W_{Y_{k+1}} + \cdots W_{Y_{n}}) {\text{\large $\mathds{1}$}}_{ \cdots,|a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} }  \right]   \lim_{\Delta w \rightarrow 0}  \frac{1}{\Delta w}  \mathbb{E} \left[  {\text{\large $\mathds{1}$}}_{ c_{k-1}(w), a_k  < w +W_{Y_1}+\cdots+W_{Y_k} < a_k - \Delta w } \right].   \end{align}

In other case,

\begin{align}
\nonumber  & \lim_{\Delta w \rightarrow 0} \frac{1}{\Delta w} \mathbb{E} \left[   \Delta f (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w+\Delta w),c^k(w)} \right] \\
\nonumber  = & \lim_{\Delta w \rightarrow 0}  \frac{1}{\Delta w} \mathbb{E} \left[  f (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w+\Delta w), c^{k-1}(w), |w +W_{Y_1}+\cdots+W_{Y_k}| <a_k}  \right] \\
\nonumber & \Delta w \rightarrow 0+: \\ = &  \mathbb{E} \left[  f (a_k+W_{Y_{k+1}} + \cdots W_{Y_n}) {\text{\large $\mathds{1}$}}_{ \cdots,|a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} } \right]  \lim_{\Delta w \rightarrow 0}  \frac{1}{\Delta w} \mathbb{E} \left[  {\text{\large $\mathds{1}$}}_{ c_{k-1}(w), a_k - \Delta w < w +W_{Y_1}+\cdots+W_{Y_k} < a_k  } \right]; \\
\nonumber &  \Delta w \rightarrow 0-: \\ \nonumber = & \mathbb{E} \left[  f (-a_k+W_{Y_{k+1}} + \cdots W_{Y_n}) {\text{\large $\mathds{1}$}}_{  |-a_k+W_{Y_{k+1}}| \ge a_{k+1}, \cdots,|-a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} } \right] \\ & \times  \lim_{\Delta w \rightarrow 0}  \frac{1}{\Delta w} \mathbb{E} \left[  {\text{\large $\mathds{1}$}}_{c_{k-1}(w), -a_k  < w +W_{Y_1}+\cdots+W_{Y_k} < -a_k - \Delta w }  \right].
\end{align}

\begin{align}
\nonumber & \lim_{\Delta w \rightarrow 0}  \frac{1}{\Delta w} \mathbb{E} \left[   \Delta f (w+W_{Y_1} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w+\Delta w),c^k(w)} +  {\text{\large $\mathds{1}$}}_{ c_{n-1}(w),c^k (w+\Delta w)} \right] \\
\nonumber = &  f_{w+W_{Y_1} + \cdots W_{Y_k}}(a_k )  \mathbb{E} \left[  f (a_k+W_{Y_{k+1}} + \cdots W_{Y_n}) {\text{\large $\mathds{1}$}}_{|a_k+W_{Y_{k+1}}  |\ge a_{k+1}, \cdots,|a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} }  \right] \\
\nonumber & -  f_{w+W_{Y_1} + \cdots W_{Y_k}}(-a_k ) \mathbb{E} \left[ f (-a_k+W_{Y_{k+1}} + \cdots W_{Y_n})  {\text{\large $\mathds{1}$}}_{ |-a_k+W_{Y_{k+1}}  |\ge a_{k+1}, \cdots,|-a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} } \right] \\
\nonumber = & (  f_{w+W_{Y_1} + \cdots W_{Y_k}}(a_k ) + f_{w+W_{Y_1} + \cdots W_{Y_k}}(-a_k ) )  \\ & \times \mathbb{E} \left[    f (a_k+W_{Y_{k+1}} + \cdots W_{Y_n}) {\text{\large $\mathds{1}$}}_{ |a_k+W_{Y_{k+1}}  |\ge a_{k+1}, \cdots,|a_k+W_{Y_{k+1}} + \cdots W_{Y_{n-1}} |\ge a_{n-1} } \right], \label{eq-der2forg}
\end{align}

which is a constant term multiplied by $ f_{w+W_{Y_1} + \cdots W_{Y_k}}(a_k ) + f_{w+W_{Y_1} + \cdots W_{Y_k}}(-a_k )$, where $ f_{w+W_{Y_1} + \cdots W_{Y_k}}(a_k)$ is defined in Definition <ref>. The last equation in (<ref>) holds because $W_Y$ is symmetric and $f(\cdot)$ is odd[Note that if $f(\cdot)$ is even, then the first term of the last equation in (<ref>) becomes $ f_{w+W_{Y_1} + \cdots W_{Y_k}}(a_k ) - f_{w+W_{Y_1} + \cdots W_{Y_k}}(-a_k )$.].
Recall that $\partial_x g(w,a)$ contains two odd polynomial terms $2 \mathbb{E} \left[ Y \right] w  $ and $2 \text{mse}_{\text{opt}} w - 2/3 w^3 $ related to $w$. Therefore,
the derivative of $ \mathbb{E} \left[ \partial_x g(w+W_{Y_1+\cdots +Y_k},v_{n+1-k}) {\text{\large $\mathds{1}$}}_{ c_{n,k}(w)  } \right]$ that appears in (<ref>) is expressed as the sum of forms (<ref>), (<ref>), and (<ref>). The value of (<ref>) is $0$. According to Lemma <ref>(c), the value of (<ref>) is a constant multiplied by $f_{w+W_{Y_1} + \cdots W_{Y_k}}(x )$, a continuously differentiable function in $w$ for some parameter $x$.
For (<ref>), note that the term of (<ref>) is continuous in $w$. We can take the derivative and apply the previous calculations (<ref>)—(<ref>) again[Despite that $f'(w)$ becomes an even polynomial function with $O(w^2)$, except the minor sign change of the last equality of (<ref>) as described in the previous footnote, the calculations (<ref>)—(<ref>) remain the same.]. Then, the term in (<ref>) is still continuously differentiable. This shows that $ \mathbb{E} \left[ \partial_x g(w+W_{Y_1+\cdots +Y_k},v_{n+1-k}) {\text{\large $\mathds{1}$}}_{ c_{n,k}(w)  } \right]$ is continuously differentiable. Thus, $ \frac{d^2}{dw^2} \mathbb{E} \left[   J_n(w+W_Y) \right],  \frac{d^3}{dw^3} \mathbb{E} \left[   J_n(w+W_Y) \right]$ are both continuous. This ends the proof of Lemma <ref>(a).

§.§ Proof of Lemma <ref>(b),(c)

We then use induction to prove Lemma <ref>(b),(c).
Let us denote $\beta=  \text{mse}_{\text{opt}} -  \mathbb{E} \left[ Y \right]$, Note that the free boundary method implies that $v_1=\sqrt{3\beta}$, and $g(w,v_1)$ is continuously differentiable. In addition,

\begin{align}
\partial_{xx}g(w,v_1) & = \left\{ \begin{array}{lll}
2 \mathbb{E} \left[ Y \right] & \ w > v_1, \vspace{1mm} \\
-2 w^2 + 2 \text{mse}_{\text{opt}}  & \ 0 \le w<v_1.
\end{array}
\right. \\
\partial_{xxx}g(w,v_1) & = \left\{ \begin{array}{lll}
0 & \ w > v_1, \vspace{1mm} \\
-4 w & \ 0 \le w<v_1,
\end{array}
\right.
\end{align}

Then, we have $G''_1(w) = 2 \mathbb{E} \left[ Y \right] $ and for all $w\ge v_1$,

\begin{align}
G''_1(w) - (-2 w^2 + 2 \text{mse}_{\text{opt}}) = -2 \beta + 2w^2 \ge -2\beta + 6\beta \ge 0.
\end{align}

For all $w \ge 0$,

\begin{align}
G'''_1(w) - (-4w) = 0+ 4w  \ge 0.
\end{align}

This satisfies the initial condition of Lemma <ref>(b),(c).
By Lemma <ref>,

\begin{align}
\nonumber  G''_{n+1}(w) & = 2 \mathbb{E} \left[ Y \right]+  \frac{d^2}{dw^2} \mathbb{E} \left[ J_n (w+W_{Y}) \right] \\
\nonumber & =   \frac{d}{dw} \mathbb{E} \left[  J'_n (w+W_{Y}) \right] \\
& =  \lim_{\Delta w \rightarrow 0} \frac{1}{\Delta w} \mathbb{E} \left[   J'_n (w+\Delta w+W_{Y}) -  J'_n (w+W_{Y}) \right].
\end{align}

Since $J'_n(\cdot)$ is continuous, there exists $0\le \epsilon\le \Delta w$ or $\Delta w \le \epsilon\le 0$, such that

\begin{align}
\nonumber & \frac{1}{\Delta w} \left(  J'_n (w+\Delta w+W_{Y}) -   J'_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}|>v_n} \\  \nonumber = &   J''_n (w+\epsilon+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+\Delta w  +W_{Y}|>v_n,|w+W_{Y}|>v_n}, \\
\nonumber  & \frac{1}{\Delta w}  \left(  J'_n (w+\Delta w+W_{Y}) -  J'_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}|<v_n} \\ = &  J''_n(w+\epsilon+W_{Y})  {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}|<v_n}
\end{align}

We have shown that $  \frac{d^2}{dw^2} \mathbb{E} \left[ J_n(w+W_{Y})  \right] $ is continuous, so $ J''_{n+1}(w) $ is continuous for $|w|\ne v_{n+1}$. Applying the same analysis for $ \frac{d^2}{dw^2} \mathbb{E}_{W_{Y_2}} \left[ J_n(w+W_{Y}+ W_{Y_2})  \right] $ (to replace $\frac{d^2}{dw^2} \mathbb{E} \left[ J_n(w+W_{Y})  \right] $) into the proof of Lemma <ref>(a) described in Appendix <ref>, both $ G''_n(w+W_Y+\Delta w)$ and $J''_n(w+W_Y+\Delta w)$ are bounded by a finite random variable. Therefore, using dominated convergence theorem,

\begin{align}
\nonumber  &  \lim_{\Delta w \rightarrow 0}\mathbb{E} \Big[  \frac{1}{\Delta w} \left(  J'_n (w+\Delta w+W_{Y}) -  J'_n (w+W_{Y}) \right) \times \\ \nonumber & \left( {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}|>v_n} + {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}|<v_n} \right) \Big] \\
= & \mathbb{E} \left[  J''_n (w+W_{Y}) \times \left( {\text{\large $\mathds{1}$}}_{|w+\Delta w  +W_{Y}|>v_n,|w+W_{Y}|>v_n} + {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}|<v_n} \right) \right]. \label{eq-der2a}
\end{align}

By Lemma <ref>, the two remaining events vanishes as $\Delta w \rightarrow 0$.
Thus, for small $\Delta w$, we have

\begin{align}
\nonumber & \Big| \ \mathbb{E} \Big{[} \frac{1}{\Delta w} \left(   J'_n (w+\Delta w+W_{Y}) -  J'_n (w+W_{Y}) \right) \\  \nonumber &  \left( {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}| < v_n} +  {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|< v_n,|w+W_{Y}| > v_n} \right)   \Big{]}  \ \Big| \\
\le & \max_{|v_n-x|\le |\Delta w| , x\ne v_n} |J''(x)| \ \mathbb{E}
\left[ {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}| < v_n} +  {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|< v_n,|w+W_{Y}| > v_n} \right]
\rightarrow 0. \label{eq-der2b}
\end{align}

By (<ref>),(<ref>), we have an interesting result:

\begin{align}
\frac{d^2}{dw^2}  \mathbb{E} \left[ J_n (w+W_{Y}) \right] & = \mathbb{E} \left[  J''_n (w+W_{Y}) \right], \\
G''_{n+1}(w) & = 2 \mathbb{E} \left[ Y \right]+  \mathbb{E} \left[  J''_n (w+W_{Y}) \right].
\end{align}

Then, we consider the third derivative:

\begin{align}
\nonumber G_{n+1}'''(w) & = 0+  \frac{d^3}{dw^3} \mathbb{E} \left[ J_n (w+W_{Y}) \right]
=   \frac{d}{dw} \mathbb{E} \left[ J''_n (w+W_{Y}) \right] \\
& =  \lim_{\Delta w \rightarrow 0} \frac{1}{\Delta w} \mathbb{E} \left[  J''_n (w+\Delta w+W_{Y}) -   J''_n (w+W_{Y}) \right].
\end{align}

For this derivation, there exists $0\le \epsilon\le \Delta w$ or $\Delta w \le \epsilon\le 0$, such that

\begin{align}
\nonumber & \frac{1}{\Delta w} \left(   J''_n (w+\Delta w+W_{Y}) -   J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}|>v_n} \\ \nonumber = &   G'''_n (w+\epsilon+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+\Delta w  +W_{Y}|>v_n,|w+W_{Y}|>v_n}, \\
\nonumber & \frac{1}{\Delta w}  \left(  J''_n (w+\Delta w+W_{Y}) -   J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}|<v_n} \\ = & -4(w+\epsilon+W_Y)  {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}|<v_n}.
\end{align}

Recall that $G'''_n(\cdot)$ is continuous.
Applying the same analysis for $ \frac{d^3}{dw^3} \mathbb{E}_{W_{Y_2}} \left[ J_n(w+W_{Y}+ W_{Y_2})  \right] $ (to replace $\frac{d^3}{dw^3} \mathbb{E} \left[ J_n(w+W_{Y})  \right] $) into the proof of Lemma <ref>(a) described in Appendix <ref>, both $ G'''_n(w+W_Y+\Delta w)$ and $J'''_n(w+W_Y+\Delta w)$ are bounded by a finite random variable. Therefore, using dominated convergence theorem (similar to (<ref>)),

\begin{align}
\nonumber &  \lim_{\Delta w \rightarrow 0}\mathbb{E} \left[  \frac{1}{\Delta w} \left(   J''_n (w+\Delta w+W_{Y}) -  J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}|>v_n} \right] \\
\nonumber = & \mathbb{E} \left[  \lim_{\Delta w \rightarrow 0} G'''_n (w+\epsilon+W_{Y}) \times {\text{\large $\mathds{1}$}}_{|w+\Delta w  +W_{Y}|>v_n,|w+W_{Y}|>v_n}  \right] \\
\nonumber = &  \mathbb{E} \left[  G'''_n (w+W_{Y}) \times {\text{\large $\mathds{1}$}}_{|w+W_{Y}|>v_n}  \right],
\\ \nonumber
&  \lim_{\Delta w \rightarrow 0}\mathbb{E} \left[  \frac{1}{\Delta w} \left(  J''_n (w+\Delta w+W_{Y}) -  J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}|<v_n} \right] \\ = & \mathbb{E} \left[  -4(w+W_Y) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|<v_n} \right]. \label{eq-g3a}
\end{align}

We then discuss the two remaining events.
If $\Delta w > 0$,

\begin{align}
\nonumber &  \frac{1}{\Delta w} \left(  J''_n (w+\Delta w+W_{Y}) -   J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}| < v_n} \\ = & \left(   G''_n (v_n) + 2v^2_{n} - 2 \text{mse}_{\text{opt}} + o(\Delta w) \right)  \frac{1}{\Delta w}  {\text{\large $\mathds{1}$}}_{ v_n - w - \Delta w < W_Y < v_n - w }, \\
\nonumber &  \frac{1}{\Delta w} \left(   J''_n (w+\Delta w+W_{Y}) -   J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}| > v_n} \\ = & - \left(  G''_n (-v_n) + 2v^2_{n} - 2 \text{mse}_{\text{opt}} + o(\Delta w) \right)  \frac{1}{\Delta w}  {\text{\large $\mathds{1}$}}_{ -v_n - w- \Delta w < W_Y < -v_n - w },
\end{align}

If $\Delta w < 0$,

\begin{align}
\nonumber &  \frac{1}{\Delta w} \left( J''_n (w+\Delta w+W_{Y}) -  J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}| < v_n} \\ = & \left(   G''_n (- v_n) + 2v^2_{n} - 2 \text{mse}_{\text{opt}} + o(\Delta w) \right)  \frac{1}{\Delta w}  {\text{\large $\mathds{1}$}}_{ -v_n - w  < W_Y < v_n - w - \Delta w }, \\
\nonumber &  \frac{1}{\Delta w} \left(  J''_n (w+\Delta w+W_{Y}) -  J''_n (w+W_{Y}) \right) {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|<v_n,|w+W_{Y}| > v_n} \\ = & - \left(   G''_n (v_n) + 2v^2_{n} - 2 \text{mse}_{\text{opt}} + o(\Delta w) \right)  \frac{1}{\Delta w}  {\text{\large $\mathds{1}$}}_{ v_n - w < W_Y < v_n - w - \Delta w }.
\end{align}

\begin{align}
\nonumber &  \lim_{\Delta w \rightarrow 0} \mathbb{E} \Big[ \frac{1}{\Delta w} \left(   J''_n (w+\Delta w+W_{Y}) -  J''_n (w+W_{Y}) \right) \\ \nonumber & \times \left( {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|>v_n,|w+W_{Y}| < v_n} +  {\text{\large $\mathds{1}$}}_{|w+\Delta w + W_{Y}|< v_n,|w+W_{Y}| > v_n} \right)   \Big]  \\
& = - \left( G''_n (v_n) + 2v^2_{n} - 2 \text{mse}_{\text{opt}} \right) (f_{W_Y}(-v_n - w)  - f_{W_Y}(v_n - w)) \ge 0. \label{eq-remain}
\end{align}
The last inequality of (<ref>) holds due to the induction hypothesis of $G''_n(\cdot)$ and Lemma <ref>.

Combining (<ref>) and (<ref>),
for all $w \ge 0$,

\begin{align}
\nonumber  G'''_{n+1}(w) = & 0+\alpha \mathbb{E} \left[ J'''_n(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|>v_n} +   J'''_n(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|<v_n} \right] \\ % \nonumber & + \left( \frac{d^2}{dw^2} J_n(v_{n}-) -  \frac{d^2}{dw^2} J_n(v_{n}+) \right) (f_{-v_n - w} - f_{v_n - w}) \\
\nonumber & - \left( G''_n (v_n) + 2v^2_{n} - 2 \text{mse}_{\text{opt}} \right) (f_{W_Y}(-v_n - w)  - f_{W_Y}(v_n - w))\\
\nonumber \ge & \alpha \mathbb{E} \left[ J'''_n(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|>v_n} +   J'''_n(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|<v_n} \right] \\
\nonumber = & \alpha \mathbb{E} \left[   G'''_n(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|>v_n} -4(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|<v_n}  \right] \\
= & \alpha \mathbb{E} \left[  \left( G'''_n(w+W_{Y})+4(w+W_Y) \right) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|>v_n} -4(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|\ne v_n}  \right]. \label{eq-g3ninequality}
\end{align}

Note that $G'''_n(w)+4w$ is an odd function, and by hypothesis, $G'''_n(w)+4w\ge0$ for all $w \ge 0$. By Lemma <ref>, $f_{w+W_Y}(x) \ge f_{w+W_Y}(-x)$ for all $w\ge 0$ and $x\ge 0$. Therefore,

\begin{align}
\nonumber &  \mathbb{E} \left[  \left( G'''_n(w+W_{Y})+4(w+W_Y) \right) {\text{\large $\mathds{1}$}}_{|w+W_{Y}|>v_n} \right] \\ \nonumber  = & ( \int_{x>v_n} + \int_{x<- v_n} ) (G'''_n(x)+4x) f_{w+W_Y}(x) dx \\ \nonumber = & \int_{x>v_n} (G'''_n(x)+4x) f_{w+W_Y}(x) dx + \int_{x>v_n} (G'''_n(-x)-4x) f_{w+W_Y}(-x) dx \\ = &  \int_{x>v_n} (G'''_n(x)+4x) (f_{w+W_Y}(x) - f_{w+W_Y}(-x)) dx \ge 0. \label{eq-g3ninequality2}
\end{align}

Finally, (<ref>) and (<ref>) give

\begin{align}
\nonumber G'''_n(w) \ge & \alpha  \mathbb{E} \left[ -4(w+W_{Y}) {\text{\large $\mathds{1}$}}_{|w+W_{Y}| \ne v_n} \right] \\
\nonumber  = & \alpha  \mathbb{E} \left[ -4(w+W_{Y})  \right] \\
\nonumber = & -4\alpha w\\
\ge & -4w.
\end{align}
The last inequality is strict if $w>0$. This ends the proof of Lemma <ref>(c).

Let us define $ f_{n+1}(w) = G'_{n+1}(w) + 2/3w^3-\text{mse}_{\text{opt}} w$ for simplicity. Then, $f_{n+1}''(w) = G'''_{n+1}(w)+4w\ge 0$ for $w\in[0,\infty)$. This implies that $f_{n+1}(w)$ is convex in $w\in[0,\infty)$ and strictly convex in $w>0$. By Lemma <ref>(a), $G'_{n+1}(w)$ is continuous and odd. Thus, $f_{n+1}(0)=0$.
By the definition of free boundary method (<ref>), $J'_{n+1}(v_{n+1}-)=J'_{n+1}(v_{n+1}+)=G'_{n+1}(v_{n+1})$. Thus, $f_{n+1}(v_{n+1})=0$. Therefore, we have $f'(v_{n+1}) = G''_{n+1}(v_{n+1}) + 2v_{n+1}^2-\text{mse}_{\text{opt}}  \ge 0$ and $f'(w) > 0$ for all $w > v_{n+1}$, and $f'(w) < 0$ for $w\in (0,v_{n+1})$. This ends the proof of Lemma <ref>(b).

§.§ Proof of Lemma <ref>(d)

Now we show Lemma <ref>(d).

We now use induction to show that $v_n \le \sqrt{3 \text{mse}_{\text{opt}} } $ for all $n=1,2,\cdots$.
Note that $v_1 = \sqrt{3\beta} = \sqrt{3(\text{mse}_{\text{opt}} - \mathbb{E} \left[ Y \right])}\le \sqrt{3\text{mse}_{\text{opt}}}$.
The second threshold $v_2$ is the root of

\begin{align}
0 = & \frac{2}{3} w^3 - 2\beta w + \alpha  \mathbb{E} \left[  \partial_x g( w+W_{Y_1},v_1  )    \right]. \label{eq-v2}
\end{align}

Note that $2 \text{mse}_{\text{opt}} w - \frac{2}{3} w^3$ is positive at $0\le w \le \sqrt{3 \text{mse}_{\text{opt}} }$. Therefore, if $v \le \sqrt{3 \text{mse}_{\text{opt}} }$, $\partial_x g(w,v)$ is always positive at $w \ge 0$.
Since $v_1 \le \sqrt{3\text{mse}_{\text{opt}}}$, $\partial_x g(w,v_1) \ge 0$ for all $w \ge 0$. Recall that $\partial_x g(w,v)$ is an odd function on $w$ for any $v \ge 0$.
Therefore, utilizing the same analysis as (<ref>), for all $w \ge 0$,

\begin{align}
\mathbb{E} \left[  \partial_x g( w+W_{Y_1},v_1  )    \right] \ge 0.
\end{align}

The first term $ \frac{2}{3} w^3 - 2\beta w >0$ for all $w \ge \sqrt{3\text{mse}_{\text{opt}}}$. To keep the equation (<ref>) holds, we have $v_2 \le \sqrt{3\text{mse}_{\text{opt}}} $.

Suppose that $v_2,\cdots, v_n \le \sqrt{3\text{mse}_{\text{opt}}}$.
Now, we will show that $v_{n+1} \le \sqrt{3\text{mse}_{\text{opt}}}$. Note that $v_{n+1}$ is the root of

\begin{align}
\nonumber
0 = & \frac{2}{3} w^3 - 2\beta w + \alpha  \mathbb{E} \left[  \partial_x g( w+W_{Y_1},v_n  )    \right] \\ & +
\sum_{k=2}^{n} \alpha^k \mathbb{E} \left[  \partial_x g( w+W_{Y_1}+\cdots W_{Y_{k}}, v_{n+1-k} ) {\text{\large $\mathds{1}$}}_{ \left\{ |w+ Y_1| \ge v_n, \cdots, |w+W_{Y_1}+\cdots+W_{Y_{k-1}} | \ge v_{n+2-k}\right\} }    \right].
\end{align}

Since the hypothesis tells that $v_n \le \sqrt{3\text{mse}_{\text{opt}}}$, $ \mathbb{E} \left[  \partial_x g( w+W_{Y_1},v_n  )    \right] \ge 0$.
To show that $v_{n+1} \le \sqrt{3\text{mse}_{\text{opt}}} $, it is sufficient to show that for $k=2,\cdots,n$,

\begin{align}
& \mathbb{E} \left[  \partial_x g( w+W_{Y_1}+\cdots W_{Y_{k}}, v_{n+1-k} ) {\text{\large $\mathds{1}$}}_{ \left\{ |w+ Y_1| \ge v_n, \cdots, |w+W_{Y_1}+\cdots+W_{Y_{k-1}} | \ge v_{n+2-k}\right\} }    \right] \ge 0. \label{eq-der1single>0}
\end{align}

Since $v_{n+1-k} \le   \sqrt{3\text{mse}_{\text{opt}}}$, $\partial_x g(w,v_{n+1-k})\ge 0$ for all $w \ge 0$. Therefore, the inequality (<ref>) is shown by Lemma <ref>(c) and that $\partial_x g(w,v) $ is an odd function for any $v\ge0$.

Now, we will jointly show that $G_{n+1}'(w) \ge G_{n}'(w)$, and $v_{n+1} \le v_n$. First, $v_1 = \sqrt{3\beta}, G'_1(w) = 2  \mathbb{E} \left[ Y  \right] w, G'_2 (w) = G'_1(w) + \alpha  \mathbb{E} \left[ \partial_x g(w+W_Y, v_1)   \right]$. Since $v_1 \le \sqrt{3\text{mse}_{\text{opt}}}$, $ \mathbb{E} \left[ \partial_x g(w+W_Y, v_1)   \right] \ge 0$, and we directly have $G'_2(w) \ge G'_1(w)$. For simplicity, let us define $f_n(w)$ as

\begin{align}
f_{n}(w)=G'_{n}(w) -( -\frac{2}{3}w^3+2 \text{mse}_{\text{opt}} w).
\end{align}

If $v_{2} > v_1$, then we have $f_1(v_{2}) \le f_{2}(v_{2})=0$, which contradicts to $f_1(w) >0$ for $w > v_1$. Therefore, $v_{2} \le v_1$.

Then, suppose that $G'_n(w) \ge G'_{n-1}(w)$ for $w \ge 0$, and $v_n \le v_{n-1}$. We have

\begin{align}
\nonumber G'_{n+1}(w) - G'_n(w) & = \alpha  \mathbb{E} \left[ J'_n(w+W_Y) - J_{n-1}'(w+W_Y)  \right]\\
\nonumber & =  \alpha  \mathbb{E} \big{[} G'_n(w+W_Y) {\text{\large $\mathds{1}$}}_{|w+W_Y |\ge v_n} - G'_{n-1}(w+W_Y) {\text{\large $\mathds{1}$}}_{|w+W_Y |\ge v_{n-1}} \\ & + J'_n(w+W_Y) {\text{\large $\mathds{1}$}}_{|w+W_Y | < v_n} - J'_{n-1}(w+W_Y) {\text{\large $\mathds{1}$}}_{|w+W_Y | < v_{n-1}} \big]. \label{eq-179}
\end{align}
Note that $G'_n (w)$ is odd.
If $|w+W_Y| \ge v_{n-1}$, utilizing the same analysis as (<ref>), we have

\begin{align}
\nonumber  & \mathbb{E} \left[ (J'_n(w+W_Y) - J_{n-1}'(w+W_Y)) {\text{\large $\mathds{1}$}}_{|w+W_Y | \ge v_{n-1}}   \right]  \\ = &
\mathbb{E} \left[  (G'_n(w+W_Y) - G'_{n-1}(w+W_Y)) {\text{\large $\mathds{1}$}}_{|w+W_Y | \ge v_{n-1}} \right] \ge 0. \label{eq-180-1}
\end{align}

If $|w+W_Y| < v_{n}$, we have $J'_n(w+W_Y) = 2\text{mse}_{\text{opt}} (w+W_Y) - 2/3 (w+W_Y)^3 $, irrelevant to $n$. Thus

\begin{align}
\nonumber & \mathbb{E} \left[ (J'_n(w+W_Y) - J_{n-1}'(w+W_Y)) {\text{\large $\mathds{1}$}}_{|w+W_Y | < v_{n}}   \right] \\ = & \mathbb{E} \left[
J'_n(w+W_Y) {\text{\large $\mathds{1}$}}_{|w+W_Y | < v_n} - J'_{n-1}(w+W_Y) {\text{\large $\mathds{1}$}}_{|w+W_Y | < v_{n}} \right]= 0. \label{eq-180-2}
\end{align}

If $v_n \le |w+W_Y| < v_{n-1}$,

\begin{align}
\nonumber & \mathbb{E} \left[ (J'_n(w+W_Y) - J_{n-1}'(w+W_Y)) {\text{\large $\mathds{1}$}}_{v_{n} \le |w+W_Y | < v_{n-1}}   \right] \\ \nonumber = &  \mathbb{E} \left[ (G'_n(w+W_Y) - J'_{n-1}(w+W_Y)) {\text{\large $\mathds{1}$}}_{v_{n} \le |w+W_Y | < v_{n-1}}   \right] \\ = &  \mathbb{E} \left[ (G'_n(w+W_Y) -  2\text{mse}_{\text{opt}} (w+W_Y) + \frac{2}{3} (w+W_Y)^3 ) {\text{\large $\mathds{1}$}}_{v_{n} \le |w+W_Y | < v_{n-1}}   \right] \ge 0. \label{eq-180-3}
\end{align}

The last inequality holds because $f_n(x)$ is an odd function and non-negative for $x \ge v_n$. Inserting (<ref>),(<ref>),(<ref>) into (<ref>), we finally have $G'_{n+1}(w) \ge G'_n(w)$ for $w\ge 0$.

Recall that the free boundary method (<ref>) implies that $v_{n+1}$ is the root of $f_{n+1}'(w)=G'_{n+1}(w) -( -\frac{2}{3}w^3+2 \text{mse}_{\text{opt}} w)=0$. Note that $f_{n}'(w) = G'_{n}(w) -( -\frac{2}{3}w^3+2 \text{mse}_{\text{opt}} w) \le f_{n+1}'(w)  $. If $v_{n+1} > v_n$, then we have $f_n'(v_{n+1}) \le f_{n+1}'(v_{n+1})=0$, which contradicts to $f_n'(w) >0$ for $w > v_n$. Therefore, $v_{n+1} \le v_n$, and we have that $\{v_n\}_n$ is decreasing.

§ PROOF OF LEMMA <REF>

By (<ref>), $ \tilde{J}_n(w,q) = \tilde{G}_n(w,q)$ if $|w|> v_n$. It remains to show that $ \tilde{J}_n(w,q) \le \tilde{G}_n(w,q)$ for $|w| \le v_n$ (by symmetry, we will assume $w\ge0$).

Define $f(w)\triangleq \tilde{G}_n(w,q) - \tilde{J}_n(w,q) = G_n(w) - J_n(w)$. It is easy to see that $f(v_n)=0$, and $f(w)$ is not a function of $y$.

By Lemma <ref>, we have shown that $J'_n(w) =  -2/3w^3+2 \text{mse}_{\text{opt}} w, J''_n(w) =  -2w^2+2 \text{mse}_{\text{opt}}, J'''_n(w) =  -4w $ if $|w|\le v$.
Therefore, by Lemma <ref>(c), $f'''(w)\ge 0$ for $w\in[0,v]$. This implies that $f'(w)$ is convex in $w\in[0,v_n]$. Since $f'(0)=f'(v_n)=0$, we have $f'(w)\le 0$ for $w\in[0,v_n]$. Note that $f(v_n)=0$. So $f(w)$ is non-increasing in $w\in[0,v_n]$, and thus $f(w) \ge 0$ for $w\in[0,v_n]$. This implies that $\tilde{G}(w,q) - \tilde{J}(w,q) \ge 0$, which completes our proof.

§ PROOF OF LEMMA <REF>

$\tilde{J}_n(w,q)$ is continuously differentiable, and twice condinuously differentiable except at $(\pm v_n,q)$. However, since the Lebesgue measure of reaching $(\pm v_n,q)$ is zero, the values $ \frac{\partial^2}{\partial w^2}  \tilde{J}_n(\pm w,q)$ can be chosen in the sequel arbitrary <cit.>.

In Lemma <ref>, it is easy to see that $ \partial_x \tilde{J}_n(w,q) = J' (w)$, not a function of $q$, and $ \partial_x  \tilde{J}_n(w,q) = O(w)$. Therefore, for any given time $t$,

\begin{align}
\mathbb{E} \left[  \int_0^t \left[ \partial_x  \tilde{J}_n(w+W_r, q+Q_r) \right]^2  dt \right] <\infty.
\end{align}

The integral $\int_0^t W_r^2 dr$ is increasing in $t$. Using Itô's formula <cit.>, almost surely,

\begin{align}
\nonumber  & \tilde{J}_n(w+W_t,q+Q_t) -  \tilde{J}_n(w,q) \\ = &  \int_0^t  (w+W_r)^2 - \text{mse}_{\text{opt}}+ \frac{1}{2} \partial_{xx} \tilde{J}_{n}(w+W_r, q+Q_r) dr +  \int_0^t  \partial_x  \tilde{J}_n(w+W_r,q+Q_r) dW_r.
\end{align}

By <cit.>, the process $ \int_0^t \partial_x  \tilde{J}_n(w+W_r,q+Q_r)  dW_r$ is a martingale and thus

\begin{align}
\mathbb{E} \left[  \int_0^t \partial_x \tilde{J}_n(w+W_r,q+Q_r)  dW_r  \right] = 0.
\end{align}

\begin{align}
\mathbb{E} \left[  \tilde{J}_n(w+W_t,q+Q_t)  \right]  -  \tilde{J}_n(w,q)  &  =  \mathbb{E} \left[  \int_0^t  (w+W_r)^2 - \text{mse}_{\text{opt}}+\frac{1}{2}\partial_{xx}  \tilde{J}_{n}(w+W_r,q+Q_r) dr \right]. \label{eq-ito}
\end{align}

If $|w+W_r|<v_n$, according to Lemma <ref>(a) we have $\partial_x  \tilde{J}_{n}(w+W_r, q+Q_r) = -2/3 (w+W_r)^3 + 2\text{mse}_{\text{opt}} (w+W_r)$, and $\partial_{xx}  \tilde{J}_{n}(w+W_r, q+Q_r) = -2 (w+W_r)^2 + 2\text{mse}_{\text{opt}}$ (correspond to the first equation of free boundary method (<ref>)). Therefore,

\begin{align}
(w+W_r)^2 - \text{mse}_{\text{opt}}+\partial_{xx} \tilde{J}_{n}(w+W_r, q+Q_r) =0.
\end{align}

If $|w+W_r| \ge v_n$, according to Lemma <ref>(a),(b), we get

\begin{align}
\nonumber & 2((w+W_r)^2 - \text{mse}_{\text{opt}})+\partial_{xx}  \tilde{J}_{n}(w+W_r, q+Q_r) \\
= &  2((w+W_r)^2 - \text{mse}_{\text{opt}})+ G_n''(w+W_r) \ge 0.
\end{align}

Applying to (<ref>), we get $\mathbb{E}^{(w,y)}[\tilde{J}_n(w+W_t,q+Q_t)]\ge \tilde{J}_n(w,q)$. This ends our proof.

§ PROOF OF LEMMA <REF>

Note that for simplicity, we have set $X_\tau = w+ W_\tau $ as a Wiener process that starts from $X_0=w$. When $w^2 < \bar{b}$,

\begin{align}
\nonumber \frac{v(X_\tau+W_Y)}{v(w)} = & \frac{1}{\bar{b}} \max\{(X_\tau+W_Y)^2,\bar{b}\} = \max\{ \frac{X_\tau^2}{\bar{b}} + \frac{2X_\tau W_Y}{\bar{b}} + \frac{W_Y^2}{\bar{b}},1 \} \\ \nonumber \le & \max\{ 1+\frac{2\sqrt{\bar{b}}}{\bar{b}} |W_Y| + \frac{W_Y^2}{\bar{b}},1\} \\ = & 1+\frac{2 |W_Y|}{\sqrt{\bar{b}}}  + \frac{W_Y^2}{\bar{b}}.
\end{align}

When $w^2 < \bar{b}$, $X_\tau=w$, and we have

\begin{align}
\nonumber  \frac{v(X_\tau+W_Y)}{v(w)} = & \frac{1}{w^2} \max\{(w+W_Y)^2,\bar{b}\} = \max\{ 1 + \frac{2 W_Y}{w} + \frac{W_Y^2}{w^2},\frac{\bar{b}}{w^2} \} \\ \nonumber \le & \max\{ 1+\frac{2}{\sqrt{\bar{b}}} |W_Y| + \frac{W_Y^2}{\bar{b}},1\} \\ = & 1+\frac{2 |W_Y|}{\sqrt{\bar{b}}}  + \frac{W_Y^2}{\bar{b}}.
\end{align}

This ends the proof of Lemma <ref>(a).

Note that $g(w; \tau)$ is bounded in $w\in [-\bar{b},\bar{b}]$, and $g(w; \tau) = 2\mathbb{E} \left[ Y \right] w^2 + const$ for $|w|^2 \ge \bar{b}$. Therefore, there exists $k>0$ such that $\|g(w;\tau)\| \le k$. Recall that we denote $\tilde{W}_{n+1}$ as the state value at $n+1$th stage with $\tilde{W}_1 = w$, and $\tilde{W}_{n+1} = \tilde{W}_{n}+W_{\tau'} + W_Y$ for a stopping time $\tau'$. Then,

\begin{align}
\nonumber \mathbb{E} \left[ g(W_{n+1};\tau) \right] = & \mathbb{E} \left[ g(\tilde{W}_{n}+W_{\tau'} + W_Y;\tau) \right]
\le  k \mathbb{E} \left[ v(\tilde{W}_{n}+W_{\tau'} + W_Y) \right] \\
\nonumber \le & k \frac{\rho}{\alpha} \mathbb{E} \left[ v(\tilde{W}_n) \right]
\le  k \frac{\rho^2}{\alpha^2}  \mathbb{E} \left[ v(\tilde{W}_{n-1}) \right] \cdots \\
\le & k \frac{\rho^n}{\alpha^n} v(w).
\end{align}

We have shown that each optimal stopping times for solving $T(T^n 0(w))$ are some hitting times with bounded and convergent thresholds, so each stopping time belongs to the assumption of Lemma <ref>(a). We have

\begin{align}
\nonumber T^n 0(w) = & \sum_{k=1}^{n} \alpha^{k-1}  \mathbb{E} \left[ g(\tilde{W}_k;\tau_k) \right]
\le   \sum_{j=1}^{n} \rho^k v(w) \le \frac{1}{1-\rho} v(w),\\
T^\infty 0(w) \le &  \sum_{k=1}^{\infty} \rho^k v(w) \le \frac{1}{1-\rho} v(w).
\end{align}

Thus, both $\|T^n 0\|$ and $\| T^\infty 0\|$ are finite. For any stopping time $\tau$ within the assumption of Lemma <ref>(a),

\begin{align}
\nonumber T_\tau T^n 0(w) - T_\tau T^{n-1} 0(w) = & \alpha \mathbb{E} \left[ T^n 0(X_\tau + W_Y) - T^{n-1} (X_\tau + W_Y) \right]  \\
\nonumber = &  \mathbb{E} \left[ \frac{ T^n 0(X_\tau + W_Y) - T^{n-1} (X_\tau + W_Y)}{v(X_\tau + W_Y)} v(X_\tau + W_Y) \right] \\
\nonumber \le & \alpha \| T^n0 - T^{n-1} 0 \| \times \mathbb{E} \left[  v(X_\tau + W_Y)  \right] \\
\le & \rho v(w) \| T^n0 - T^{n-1} 0 \|.
\end{align}

This gives that

\begin{align}
\frac{T_\tau T^n 0(w)}{v(w)} \le \rho \| T^n 0 - T^{n-1} 0 \| + \frac{T_\tau T^{n-1} 0(w)}{v(w)}.\label{eq-contraction1a}
\end{align}

Take the minimum for left and right side of (<ref>) over all the stopping times $\tau$ with bounded threshold $\bar{b}$, then

\begin{align}
\frac{T^{n+1} 0(w)}{v(w)} \le \rho \| T^n 0 - T^{n-1} 0 \| + \frac{T^{n} 0(w)}{v(w)}.\label{eq-contraction1}
\end{align}
By symmetry,

\begin{align}
\frac{T^{n} 0(w)}{v(w)} \le \rho \| T^n 0 - T^{n-1} 0 \| + \frac{T^{n+1} 0(w)}{v(w)}.\label{eq-contraction1-5}
\end{align}

\begin{align}
\|T^{n+1} 0 - T^n 0\| \le \rho \|T^n 0 - T^{n-1} 0\|\cdots \le \rho^n \|T 0\|.\label{eq-contraction2}
\end{align}

This completes the proof of Lemma <ref>(b).
Due to (<ref>), the sequence $\{ T^n 0(w)\}$ is a Cauchy sequence, and thus $T^n 0(w)$ converges pointwise to $T^\infty 0(w)$, which is also measurable, and we have shown that $\| T^\infty 0(w) \| <\infty$. Therefore, using <cit.>, $\| T^n 0 - T^\infty 0\| \rightarrow 0$. We replace $T^n 0(w)$ by $T^\infty 0(w)$ in (<ref>) and use symmetry, we then find that[Here, we do not require $T T^\infty 0(w)$ to be measurable.]

\begin{align}
\| T T^\infty 0 - T^n 0 \| \le \rho \| T^\infty 0 - T^{n-1} 0\| \rightarrow 0.
\end{align}

Therefore, $J^* = T^\infty 0$ is the solution to the Bellman equation $T J^* = J^*$, and the $\rho-$convergence rate is immediately given. The solution is unique: If there exists any other measurable function $\tilde{J}(w)$ that satisfies the Bellman equation (with $\| \tilde{J}(w) \|<\infty$), we replace $T^n 0(w)$ by $T^\infty 0(w)$ and replace $T^{n-1}0$ by $\tilde{J}(w)$ in (<ref>), and we have

\begin{align}
\| T^\infty 0 - \tilde{J} \| = \| T T^\infty 0 - T \tilde{J} \| \le \rho \| T^\infty 0 - \tilde{J} \|,
\end{align}
which implies that $T^\infty 0 = \tilde{J}$. These completes the proof of Lemma <ref>(c).

§ PROOF OF THEOREM <REF>

We denote $\Pi_{j,\text{signal-agnostic}} \subset \Pi_{\text{signal-agnostic}}$ as a collection of sampling times $S_{j,1},S_{j,2},\cdots$ at $j$th epoch such that the inter-sampling times $S_{j,1} - S_{j-1,M_{j-1}},S_{j,2}  - S_{j-1,M_{j-1}},\ldots$ are independent of the history information before $S_{j-1,M_{j-1}}$. Note that the subscripts $(j,1),(j,2),\ldots$ are illustrated in Section <ref>.

Similar to Proposition <ref>, we have the following result:

There exists an optimal policy in $\Pi_{j,\text{signal-agnostic}}$ such that $\{ S_{j,M_{j}} - S_{j-1,M_{j-1}} \}_j$ are i.i.d. Moreover, problem (<ref>) when $\Pi = \Pi_{\text{signal-agnostic}}$ is equivalent to the following problem:

\begin{align}
\text{mse}_{\text{opt}} = & \inf_{\pi \in \Pi_{j,\text{signal-agnostic}}} \frac{ \mathbb{E} \left[  \int_{D_{j-1,M_{j-1}}}^{D_{j,M_j}} \Delta_t dt \right]}{ \mathbb{E} \left[   D_{j,M_j} - D_{j-1,M_{j-1}} \right] }. \label{eq-singlepoch-age}
\end{align}

The proof of Proposition <ref> is a special case of (thus included in) the proof of Proposision (<ref>) and is omitted. Problem (<ref>) has a much simpler form to Problem (<ref>) because (i) the sampling times are independent of the Wiener process, and (ii) we replace the square estimation error $(W_t - \hat{W}_t)^2$ by the linear age $\Delta_t$, the time period between $t$ and the sampling time $S_{j-1,M_{j-1}}$. By (<ref>) and <cit.>, we complete the proof of Theorem <ref>.

§ OPTIMAL SIGNAL-AWARE SAMPLING POLICY WITH SAMPLING RATE CONSTRAINT

In this section, we extend our problem (<ref>) by considering a sampling rate constraint:

\begin{align}
\text{mse}_{\text{opt}} =  \inf_{\pi \in \Pi} & \limsup_{T\rightarrow \infty}  \frac{1}{T} \mathbb{E} \left[  \int_{0}^{T} (W_t - \hat{W}_t)^2 dt \right],  \label{avg-aware2}  \\ \nonumber
\text{s.t. } & \liminf_{n\rightarrow \infty} \frac{1}{n} \mathbb{E}[S_n] \ge \frac{1}{f_{\text{max}}}.
\end{align}

Here, $f_{\text{max}}$ is the maximum allowed sampling rate. We impose the constraint because in practice, there is a need to restrict the energy consumption for the transmission.
It is easy to see that when $f_{\text{max}}=\infty$, problem (<ref>) is an uncontrained problem and is equivalent to problem (<ref>).
When we take into account the sampling rate constraint (<ref>) for general values of $f_{\text{max}}$, we provide the following result that extends Theorem <ref>:

Suppose that $ S_{i+1} - S_i$ is a regenerative process, then the sampling policy stated in Theorem <ref> is an optimal solution to (<ref>). Moreover, $\beta$ is the root of (<ref>), if the following condition holds:

\begin{align}
\mathbb{E} \left[\lim_{n\rightarrow \infty} l_n (W_Y,\beta) \right] \ge \frac{1}{f_{\text{max}}(1-\alpha)}, \label{eq-constr}
\end{align}

where $l_0(w,\beta)=0$ and

\begin{align}
l_{n}(w,\beta) =  \mathbb{E} \left[ Y \right] +
\max(v_n(\beta)^2 - w^2,0)   + \alpha \mathbb{E} \left[ l_{n-1} (\max\{ |w|,v_{n}(\beta) \}+W_Y,\beta) \right], \ n=1,2,\ldots
\end{align}

Otherwise,  $\beta$ is the root of

\begin{align}
\mathbb{E} \left[\lim_{n\rightarrow \infty} l_n(W_Y,\beta) \right]= \frac{1}{f_{\text{max}}(1-\alpha)}.  \label{eq-constr-root}
\end{align}

See Appendix <ref>.

Theorem <ref> states that the optimal solution to the constrained problem is similar to that of the unconstrained one: it is optimal to wait until the instantaneous estimation error $|W_t - \hat{W_t}|$ exceeds a threshold. The threshold $v(\beta)$ remains constant and is independent of whether the last transmission failed or not. Moreover, the threshold $v(\beta)$ is the same for a given $\beta$.

The difference lies in the choice of $\beta$. Intuitively, the value $\mathbb{E} \left[\lim_{n\rightarrow \infty} l_n (W_Y,\beta) \right]$ in (<ref>) is the expected time interval between the two successful delivery times. This expected time is approached via the sequence of functions $l_n$. $f_{\text{max}}(1-\alpha)$ is the maximum rate allowed between the two successful delivery times. Thus, the condition (<ref>) judges whether the sampling rate constraint is satisfied.
If satisfied, then the choice of $\beta$ is the same as that of Theorem <ref>. Otherwise, the sampling policy provided in Theorem <ref> is infeasible for the contratined problem (<ref>). Then, $\beta$ is chosen such that the average sampling rate is reduced to approach the constraint.
Note that we can still efficiently compute $\beta$.
Since $\mathbb{E} \left[ Y \right] +
\max(v_n(\beta)^2 - w^2,0) $ is positive and bounded, it is easy to see that $l_n(w,\beta)$ is positive and bounded. Theorem <ref> tells that $v_n(\beta)$ is convergent. Therefore, $l_n$ converges uniformly, and we can efficiently compute $\lim_{n\rightarrow \infty} l_n$.

In the case of reliable channel with $\alpha=0$, Theorem <ref> reduces to <cit.>. For the special case that the sampling time is independent of the Wiener process, we will discuss in Section <ref> below.

§ PROOF OF THEOREM <REF>

Note that in Section <ref>, we have illustrated how to substitute the subscript $i$ by the subscripts $(j,1),(j,2),\ldots$.
By the assumption of the regenerative process, using the same idea as in <cit.>, we have:

Our problem (<ref>) with $c = \text{mse}_{\text{opt}}$ is equivalent to

\begin{align}
\inf_{\pi\in \Pi_{\text{signal-aware}}} & \lim_{r \rightarrow \infty}  \frac{1}{r}\sum_{j=1}^{r} \mathbb{E} \left[  \int^{D_{j,M_j}}_{D_{j-1,M_{j-1}}}  (W_t - W_{S_{j-1,M_{j-1}}})^2 dt  -   c( D_{j,M_j} - D_{j-1,M_{j-1}} ) \right], \label{fraction-multi}\\
\nonumber & \text{s.t. } \lim_{r \rightarrow \infty} \frac{1}{r} \sum_{j=1}^{r}    \mathbb{E} \left[ D_{j,M_j} - D_{j-1,M_{j-1}}  \right] \ge \frac{1}{f_{\text{max}}(1-\alpha)}.
\end{align}

Note that the subscripts $(j,1),(j,2),\ldots$ are illustrated in Section <ref>.
We define the Lagrangian

\begin{align}
\nonumber L(\pi; \gamma) \triangleq & \lim_{r\rightarrow \infty}  \frac{1}{r}\sum_{j=1}^{r} \mathbb{E} \left[  \int^{D_{j,M_j}}_{D_{j-1,M_{j-1}}}  (W_t - W_{S_{j-1,M_{j-1}}})^2 dt  -   (\text{mse}_{\text{opt}}+\gamma)( D_{j,M_j} - D_{j-1,M_{j-1}} ) \right] \\ & +\frac{\gamma}{f_{\text{max}}(1-\alpha)}, \label{eq-lagrang}
\end{align}

where $\gamma\ge 0$ is the dual variable.

According to <cit.>, the policy $\pi^*$ along with the dual variable $\gamma^*$ is the optimal solution to (<ref>) if the following conditions hold:

\begin{align}
&  \lim_{r \rightarrow \infty} \frac{1}{r}  \sum_{j=1}^{r}    \mathbb{E} \left[ D_{j,M_j} - D_{j-1,M_{j-1}}  \right] - \frac{1}{f_{\text{max}}(1-\alpha)} \ge 0, \label{duality1} \\
& \pi^* \in \Pi_{\text{signal-aware}}, \gamma^* \ge 0, \label{duality2} \\
& L(\pi^*; \gamma^*) = \inf_{\pi\in\Pi_{\text{signal-aware}}} L(\pi; \gamma^*), \label{duality3} \\
& \gamma^* \left\{ \lim_{r \rightarrow \infty} \frac{1}{r}  \sum_{j=1}^{r}    \mathbb{E} \left[ D_{j,M_j} - D_{j-1,M_{j-1}}  \right] - \frac{1}{f_{\text{max}}(1-\alpha)}  \right\} = 0.\label{duality4}
\end{align}

Using <cit.>, the primal problem (<ref>) for any given $\gamma$ is equivalent to problem (<ref>). Thus, problem (<ref>)  has been exactly solved by Theorem <ref> in Section <ref>, by replacing $\text{mse}_{\text{opt}}$ with $\text{mse}_{\text{opt}}+\gamma$. Each sampling (waiting) time is a hitting time. Therefore, (<ref>)—(<ref>) is equivalent to:

\begin{align}
&  \mathbb{E} \left[ D_{j,M_j} - D_{j-1,M_{j-1}}  \right] - \frac{1}{f_{\text{max}}(1-\alpha)} \ge 0, \gamma^* \ge 0,\label{duality1-1} \\
& \gamma^* \left\{  \mathbb{E} [ D_{j,M_j} - D_{j-1,M_{j-1}}  ] - \frac{1}{f_{\text{max}}(1-\alpha)}  \right\} = 0.\label{duality4-1}
\end{align}

Suppose that the estimation error at $D_{j-1,M_{j-1}}$ (which has the same distribution as $W_Y$) is $w$, and $\beta \triangleq \text{mse}_{\text{opt}}+\gamma^*$. Then,

\begin{align}
\nonumber l_0 (w,\beta) & \triangleq 0,\\
\nonumber l_n(w,\beta) & \triangleq \mathbb{E} [D_{j,n} - D_{j-1,M_{j-1}}] = \mathbb{E} [\tau_n+Y] +  \alpha \mathbb{E} [ l_{n-1}(w+W_{\tau_n+Y},\beta) ] \\ & = \max(v_n(\beta)^2 - w^2,0)+\mathbb{E}[Y]  +\alpha \mathbb{E} [ l_{n-1}(\max(|w|,v_n(\beta))+W_{Y},\beta) ], \end{align}
where $\tau_n$ is the optimal waiting time of the problem (<ref>).
This gives that

\begin{align}
\mathbb{E} [ D_{j,M_j} - D_{j-1,M_{j-1}} ] & =  \mathbb{E} [ \lim_{n\rightarrow \infty} l_n(W_Y,\beta) ]. \label{duality5}
\end{align}

As a result, (<ref>), (<ref>), and (<ref>) imply the following solution:
(i) We first set $\gamma^*=0$ and use Algorithm <ref> to obtain $\text{mse}_{\text{opt}}$. If the first constraint in (<ref>) is satisfied, then the solution provided in Theorem <ref> ($\beta=\text{mse}_{\text{opt}}$) is the solution to (<ref>); (ii) If this constraint is not satisfied, then $\gamma^* \ge 0$, and $\beta \triangleq \text{mse}_{\text{opt}}+\gamma^*$ is the root of $ \mathbb{E} [ \lim_{n\rightarrow \infty} l_n(W_Y,\beta)  ] - \frac{1}{f_{\text{max}}(1-\alpha)}= 0$.

Finally, we will show that $\gamma^* \ge 0$ under the condition (ii).
In Appendix <ref>, we use induction to show that $v_n(\beta)$ is a decreasing sequence. Using the same proof idea again, we then find that $v_n(\beta)$ is increasing in $\beta$, and thus the optimal threshold $v(\beta)$ is increasing in $\beta$. The intuitive idea is that $\beta$ is the coefficient of the negative term in (<ref>), and if $\beta$ increases, it will be beneficial to wait for a longer time.
Under condition (ii), since $v(\beta)$ should be increased to reach the constraint, $\beta$ also increases. Combined with Lemma <ref>(a), $\beta \ge \text{mse}_{\text{opt}}$. Thus, it always holds that $\gamma^* \ge 0$.

Therefore, the proposed solution satisfies all of the conditions in (<ref>)—(<ref>) and thus optimal for the problem (<ref>) with the sampling rate constraint.
This ends the proof of Theorem <ref>.


# Higher order evolution inequalities with Hardy potential in the exterior of
a half-ball

Lotfi Jlali  (L. Jlali) Department of Mathematics and Statistics, College of
Science Imam Mohammad Ibn Saud Islamic University (IMSIU) Riyadh 11566, Saudi
Arabia<EMAIL_ADDRESS>and  Bessem Samet  (B. Samet) Department of
Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh,
11451, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

We consider semilinear higher order (in time) evolution inequalities posed in
an exterior domain of the half-space $\mathbb{R}_{+}^{N}$, $N\geq 2$, and
involving differential operators of the form
$\mathcal{L}_{\lambda}=-\Delta+\lambda/|x|^{2}$, where $\lambda\geq-N^{2}/4$.
A potential function of the form $|x|^{\tau}$, $\tau\in\mathbb{R}$, is allowed
in front of the power nonlinearity. Under inhomogeneous Dirichlet-type
boundary conditions, we show that the dividing line with respect to existence
or nonexistence is given by a Fujita-type critical exponent that depends on
$\lambda,N$ and $\tau$, but independent of the order of the time derivative.

###### Key words and phrases:

Higher order evolution inequalities; Hardy potential; exterior domain; half-
space; critical exponent

###### 2010 Mathematics Subject Classification:

35R45; 35A01; 35B44; 35B33

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 A priori estimate
    2. 2.2 Test functions
    3. 2.3 Estimates of $J_{i}(\varphi)$
    4. 2.4 Estimates of $J_{i}(\psi)$ in the critical case
  3. 3 Proofs of the main results
    1. 3.1 Proofs of the nonexistence results
    2. 3.2 Proof of the existence result

##  1\. Introduction

Let
$\mathbb{R}_{+}^{N}=\\{x=(x_{1},x_{2},\cdots,x_{N})\in\mathbb{R}^{N}:x_{N}>0\\}$
with $N\geq 2$. We consider the exterior domain

$\Omega=\\{x\in\mathbb{R}_{+}^{N}:|x|>1\\}.$

The boundary of $\Omega$ is denoted by
$\partial\Omega=\Gamma_{0}\cup\Gamma_{1}$, where

$\Gamma_{0}=\\{x\in\mathbb{R}^{N}:x_{N}=0,|x|>1\\},\,\,\Gamma_{1}=\\{x\in\mathbb{R}_{+}^{N}:|x|=1\\}.$

For $\lambda\geq-\frac{N^{2}}{4}$, we consider differential operators
$\mathcal{L}_{\lambda}$ of the form

$\mathcal{L}_{\lambda}=-\Delta+\frac{\lambda}{|x|^{2}}.$

Notice that the value $-\frac{N^{2}}{4}$ appears in the Hardy-type inequality
(see e.g. [7])

$\int_{\mathbb{R}_{+}^{N}}|\nabla\varphi|^{2}\,dx-\frac{N^{2}}{4}\int_{\mathbb{R}_{+}^{N}}\frac{\varphi^{2}}{|x|^{2}}\,dx\geq
0$

for all $\varphi\in C_{0}^{\infty}(\mathbb{R}_{+}^{N})$.

In this paper, we are concerned with the study of existence and nonexistence
of weak solutions to higher order evolution inequalities of the form

(1.1) $\frac{\partial^{k}u}{\partial
t^{k}}+\mathcal{L}_{\lambda}u\geq|x|^{\tau}|u|^{p}\quad\mbox{in
}(0,\infty)\times\Omega,\\\ $

where $u=u(t,x)$, $k\geq 1$ is a natural number, $\tau\in\mathbb{R}$ and
$p>1$. Problem (1.1) is considered under the Dirichlet-type boundary
conditions

(1.2) $u\geq 0\mbox{ on }(0,\infty)\times\Gamma_{0},\,\,u\geq w\mbox{ on
}(0,\infty)\times\Gamma_{1},$

where $w=w(x)\in L^{1}(\Gamma_{1})$. We provide below some motivations for
investigating problems of type (1.1)-(1.2).

The issue of existence and nonexistence of higher order (in time) evolution
equations and inequalities in the whole space $\mathbb{R}^{N}$ has been
considered in many papers. For instance, Caristi [6] considered evolution
inequalities of the form

(1.3) $\frac{\partial^{k}u}{\partial
t^{k}}-|x|^{\sigma}\Delta^{m}u\geq|u|^{q}\quad\mbox{in
}(0,\infty)\times\mathbb{R}^{N},$

where $k\geq 2$ and $m\geq 1$ are natural numbers, $\sigma\leq 2m$ and $q>1$.
When $\sigma=2m$ (critical degeneracy case), it was shown that under the
condition

$\int_{\mathbb{R}^{N}}\frac{\partial^{k-1}u}{\partial
t^{k-1}}(0,x)|x|^{-N}\,dx>0,$

if one of the following assumptions is satisfied:

  * (i)

$N\neq 2(j+1)$, $j\in\\{0,1,\cdots,m-1\\}$, $q\leq k+1$;

  * (ii)

$N=2(j+1)$ for some $j\in\\{0,1,\cdots,m-1\\}$,

then (1.3) has no nontrivial weak solution. In the case $\sigma<2m$
(subcritical degeneracy case), it was shown that under the condition

$\int_{\mathbb{R}^{N}}\frac{\partial^{k-1}u}{\partial
t^{k-1}}(0,x)|x|^{-\sigma}\,dx>0,$

if

$\left(k(N-2m)+2m-\sigma\right)q\leq Nk+2m-(k+1)\sigma,$

then (1.3) has no nontrivial weak solution. In [8], Filippucci and Ghergu
considered higher order evolution inequalities of the form

(1.4) $\frac{\partial^{k}u}{\partial
t^{k}}+(-\Delta)^{m}u\geq(K*|u|^{p})|u|^{q}\quad\quad\mbox{in
}(0,\infty)\times\mathbb{R}^{N}$

subject to the initial conditions

(1.5) $\frac{\partial^{i}u}{\partial t^{i}}u(0,x)=u_{i}(x)\quad\mbox{in
}\mathbb{R}^{N},\,\,i=0,1,,\cdots,k-1,$

where $k,m$ are positive integers and $p,q>0$. Under suitable conditions on
the positive weight $K$, the authors proved the following results:

  * (i)

If $k$ is even and $q\geq 1$, then (1.4)–(1.5) admits some positive solutions
$u\in C^{\infty}((0,\infty)\times\mathbb{R}^{N})$ that satisfy $u_{k-1}<0$;

  * (ii)

If $p+q>2$ and

$\limsup_{R\to\infty}R^{\frac{2N+\frac{2m}{k}}{p+q}-N+2m\left(1-\frac{1}{k}\right)}K(R)>0,$

then (1.4)–(1.5) has no nontrivial solution such that

$u_{k-1}\geq 0;\,\,\mbox{ or }\,\,u_{k-1}\in
L^{1}(\mathbb{R}^{N}),\,\int_{\mathbb{R}^{N}}u_{k-1}(x)\,dx>0.$

Hamidi and Laptev [9] investigated problem (1.1) in the case
$\Omega=\mathbb{R}^{N}$, $N\geq 3$ and
$\lambda\geq-\left(\frac{N-2}{2}\right)^{2}$, namely

(1.6) $\frac{\partial^{k}u}{\partial
t^{k}}+\mathcal{L}_{\lambda}u\geq|u|^{p}\quad\mbox{in
}(0,\infty)\times\mathbb{R}^{N}$

subject to the initial condition

(1.7) $\frac{\partial^{k-1}u}{\partial t^{k-1}}(0,x)\geq 0\quad\mbox{in
}\mathbb{R}^{N}.$

It was proven that, if one of the following assumptions is satisfied:

$\lambda\geq 0,\,\,1<p\leq 1+\frac{2}{\frac{2}{k}+s^{*}};$

or

$-\left(\frac{N-2}{2}\right)^{2}\leq\lambda<0,\,\,1<p\leq
1+\frac{2}{\frac{2}{k}-s_{*}},$

where

$s^{*}=\frac{N-2}{2}+\sqrt{\lambda+\left(\frac{N-2}{2}\right)^{2}},s_{*}=s^{*}+2-N,$

then (1.6)-(1.7) admits no nontrivial weak solution. In the parabolic case,
among other problems, Abdellaoui et al. [1] (see also [3]) considered problems
of the form

(1.8) $\frac{\partial(u^{p-1})}{\partial
t}-\Delta_{p}u=\lambda\frac{u^{p-1}}{|x|^{p}}+u^{q}\,(u>0)\quad\mbox{in
}(0,\infty)\times\mathbb{R}^{N},$

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,

  * (i)

if $p-1<q<F(p,\lambda)<q^{+}(p,\lambda)$ and $u$ is a solution to (1.8)
satisfying a certain behavior, then $u$ blows-up in a finite time;

  * (ii)

if $F(p,\lambda)<q<q^{+}(p,\lambda)$, then then under suitable condition on
$u(0,\cdot)$, (1.8) admits a global in time positive solution.

Evolution equations and inequalities have been also studied in other infinite
domains. For instance, in exterior domains of $\mathbb{R}^{N}$ (see e.g. [10,
11, 12, 13, 15, 19]) and cones (see e.g. [5, 14, 16, 17]). In particular, in
[11], the authors investigated hyperbolic inequalities of the form

(1.9) $\frac{\partial^{2}u}{\partial
t^{2}}+\mathcal{L}_{\lambda}u\geq|u|^{p}\quad\mbox{in
}(0,\infty)\times\mathbb{R}^{N}\backslash B_{1}$

under the boundary condition

(1.10) $\alpha\frac{\partial u}{\partial\nu}+\beta u\geq w\quad\mbox{on
}(0,\infty)\times\partial B_{1},$

where $B_{1}=\\{x\in\mathbb{R}^{N}:|x|<1\\}$, $N\geq 2$,
$\lambda\geq-\left(\frac{N-2}{2}\right)^{2}$, $\alpha,\beta\geq 0$ are
constants, $(\alpha,\beta)\neq(0,0)$, $\nu$ is the outward unit normal vector
to $\partial B_{1}$, relative to $\Omega=\mathbb{R}^{N}\backslash B_{1}$, and
$w\in L^{1}(\partial B_{1})$. It was shown that (1.9)-(1.10) admits a Fujita-
type critical exponent

$p_{c}(\lambda,N)=\left\\{\begin{array}[]{llll}\infty&\mbox{if}&N-2+2\lambda_{N}=0,\\\
1+\frac{4}{N-2+2\lambda_{N}}&\mbox{if}&N-2+2\lambda_{N}>0,\end{array}\right.$

where

$\lambda_{N}=\sqrt{\lambda+\left(\frac{N-2}{2}\right)^{2}}.$

More precisely, it was proven that,

  * (i)

if $1<p<p_{c}(\lambda,N)$ and $\displaystyle\int_{\partial
B_{1}}w(x)\,d\sigma>0$, then (1.9)-(1.10) admits no global weak solution;

  * (ii)

if $p>p_{c}(\lambda,N)$, then (1.9)-(1.10) admits global solutions for some
$w>0$.

Some results related to parabolic and elliptic equations involving the
operator $\mathcal{L}_{\lambda}$ in bounded domains of $\mathbb{R}^{N}$ can be
found in [1, 2, 3, 4, 18].

To the best of our knowledge, problems of type (1.1)-(1.2) in an exterior
domain of the half-space have not been previously considered.

Before stating our obtained results, we need to define weak solutions to
(1.1)-(1.2). Let

$Q=(0,\infty)\times\overline{\Omega}\quad\mbox{and}\quad\Gamma_{Q}^{i}=(0,\infty)\times\Gamma_{i},\,i=0,1.$

Notice that $\Gamma_{Q}^{i}\subset Q$ for all $i=0,1$. Let us introduce the
set $\Phi$ defined as follows: A function $\varphi=\varphi(t,x)$ belongs to
$\Phi$, if

  * (A1)

$\varphi\geq 0$, $\varphi\in C_{t,x}^{k,2}(Q)$;

  * (A2)

$\operatorname{supp}(\varphi)\subset\subset Q$;

  * (A3)

${\varphi}_{|{\Gamma_{Q}^{i}}}=0$, $i=0,1$;

  * (A4)

${\displaystyle\frac{\partial\varphi}{\partial\nu_{i}}}|_{\,{\Gamma_{Q}^{i}}}\leq
0$, $i=0,1$, where $\nu_{i}$ denotes the outward unit normal vector to
$\Gamma_{i}$, relative to $\Omega$.

We define weak solutions to (1.1)-(1.2) as follows.

######  Definition 1.1.

We say that $u\in L^{p}_{\operatorname{loc}}(Q)$ is a weak solution to
(1.1)-(1.2), if

(1.11)
$\int_{Q}|x|^{\tau}|u|^{p}\varphi\,dx\,dt-\int_{\Gamma_{Q}^{1}}\frac{\partial\varphi}{\partial\nu_{1}}w(x)\,dS_{x}\,dt\leq(-1)^{k}\int_{Q}u\frac{\partial^{k}\varphi}{\partial
t^{k}}\,dx\,dt+\int_{Q}u\mathcal{L}_{\lambda}\varphi\,dx\,dt$

for all $\varphi\in\Phi$.

Using standard integrations by parts, it can be easily seen that any smooth
solution to (1.1)-(1.2) is a weak solution, in the sense of Definition 1.1.

For $\lambda\geq-\frac{N^{2}}{4}$, let us introduce the parameter

(1.12) $\mu=-\frac{N}{2}+\sqrt{\lambda+\frac{N^{2}}{4}}.$

For $w\in L^{1}(\Gamma_{1})$, let

$I_{w}=\int_{\Gamma_{1}}w(x)x_{N}\,dS_{x}.$

We introduce the set

$L^{1,+}(\Gamma_{1})=\left\\{w\in L^{1}(\Gamma_{1}):I_{w}>0\right\\}.$

Our main results are stated in the following theorems.

######  Theorem 1.2.

Let $k\geq 1$, $N\geq 2$, $\tau\in\mathbb{R}$, $\lambda\geq-\frac{N^{2}}{4}$
and $p>1$.

  * (I)

Let $w\in L^{1,+}(\Gamma_{1})$. If

(1.13) $(N+\mu-1)p<N+\mu+1+\tau,$

then (1.1)-(1.2) admits no weak solution.

  * (II)

If

(1.14) $(N+\mu-1)p>N+\mu+1+\tau,$

then (1.1)-(1.2) admits nonnegative (stationary) solutions for some $w\in
L^{1,+}(\Gamma_{1})$.

When $\lambda>-\frac{N^{2}}{4}$ and $(N+\mu-1)p=N+\mu+1+\tau$, we have the
following nonexistence result.

######  Theorem 1.3.

Let $k\geq 1$, $N\geq 2$, $\tau\in\mathbb{R}$, $\lambda>-\frac{N^{2}}{4}$ and
$p>1$. If $w\in L^{1,+}(\Gamma_{1})$ and

(1.15) $(N+\mu-1)p=N+\mu+1+\tau,$

then (1.1)-(1.2) admits no weak solution.

The proof of the nonexistence results given by Theorem 1.2 (I) and Theorem
1.3, relies on nonlinear capacity estimates specifically adapted to the
operator $\mathcal{L}_{\lambda}$, the domain $\Omega$ and the considered
boundary conditions. The existence result provided by Theorem 1.2 (II) is
established by the construction of explicit solutions.

######  Remark 1.4.

Theorems 1.2 and 1.3 leave open the issue of existence/nonexistence in the
critical case:

$\lambda=-\frac{N^{2}}{4},\,\,(N+\mu-1)p=N+\mu+1+\tau.$

######  Remark 1.5.

  * (i)

If $\lambda>-\frac{N^{2}}{4}$, then

$N+\mu-1=\frac{N-2}{2}+\sqrt{\lambda+\frac{N^{2}}{4}}>0.$

Hence, (1.13) and (1.15) reduce to

$\tau>-2,\,\,1<p\leq 1+\frac{\tau+2}{N+\mu-1},$

and (1.14) reduces to

$\tau\leq-2,\,\,p>1;\quad\mbox{or}\quad\tau>-2,\,\,p>1+\frac{\tau+2}{N+\mu-1}.$

Consequently, when $\tau>-2$, (1.1)-(1.2) admits a Fujita-type critical
exponent given by

$p^{*}(\lambda,N,\tau)=1+\frac{\tau+2}{N+\mu-1}.$

It is interesting to observe that $p^{*}(\lambda,N,\tau)$ is independent of
the value of $k$.

  * (ii)

When $\lambda=-\frac{N^{2}}{4}$, we have

$N+\mu-1=\frac{N-2}{2}\geq 0.$

    * (a)

If $N=2$, we deduce from Theorem 1.2 that (1.1)-(1.2) admits a critical value
$\tau^{*}=-2$ in the following sense: if $w\in L^{1,+}(\Gamma_{1})$ and
$\tau>\tau^{*}$, then (1.1)-(1.2) admits no weak solution; if $\tau<\tau^{*}$,
then (1.1)-(1.2) admits solutions for some $w\in L^{1,+}(\Gamma_{1})$.

    * (b)

If $N\geq 3$, then $N+\mu-1>0$. Hence, when $\tau>-2$, (1.1)-(1.2) admits as
Fujita-type critical exponent the real number

$p^{*}(\lambda,N,\tau)=p^{*}\left(-\frac{N^{2}}{4},N,\tau\right)=1+\frac{2(\tau+2)}{N-2}.$

Clearly, Theorems 1.2 and 1.3 yield existence and nonexistence results for the
corresponding elliptic inequality

(1.16) $\mathcal{L}_{\lambda}u\geq|x|^{\tau}|u|^{p}\quad\mbox{in }\Omega$

under the boundary conditions

(1.17) $u\geq 0\mbox{ on }\Gamma_{0},\,\,u\geq w\mbox{ on }\Gamma_{1}.$

######  Corollary 1.6.

Let $N\geq 2$, $\tau\in\mathbb{R}$, $\lambda\geq-\frac{N^{2}}{4}$ and $p>1$.

  * (I)

Let $w\in L^{1,+}(\Gamma_{1})$. If (1.13) holds, then (1.16)-(1.17) admits no
weak solution.

  * (II)

If (1.14) holds, then (1.16)-(1.17) admits nonnegative solutions for some
$w\in L^{1,+}(\Gamma_{1})$.

######  Corollary 1.7.

Let $N\geq 2$, $\tau\in\mathbb{R}$, $\lambda>-\frac{N^{2}}{4}$ and $p>1$. If
$w\in L^{1,+}(\Gamma_{1})$ and (1.15) holds, then (1.16)-(1.17) admits no weak
solution.

The rest of the paper is organized as follows. In Section 2, some preliminary
results useful for the proof of Theorem 1.2 (I) and Theorem 1.3 are
established. Namely, we first establish an a priori estimate for problem
(1.1)-(1.2) (see Lemma 2.1). Next, we introduce two families of test functions
belonging to $\Phi$ and depending on three sufficiently large parameters
$T,R$, $\ell$, and a nonnegative function $H$, solution to

$\mathcal{L}_{\lambda}H=0\mbox{ in }\Omega,\,\,H=0\mbox{ on
}\Gamma_{0}\cup\Gamma_{1}.$

The first family of test functions will be used for proving Theorem 1.2 (I).
The second one will be used for proving Theorem 1.3. For both families of the
introduced test functions, some useful estimates are provided. Finally,
Section 3 is devoted to the proofs of Theorems 1.2 and 1.3.

Throughout this paper, the symbols $C$, $C_{i}$ denote always generic positive
constants, which are independent on the scaling parameters $T,R$ and the
solution $u$. Their values could be changed from one line to another. The
notation $R\gg 1$ means that $R$ is sufficiently large.

##  2\. Preliminaries

Let $k\geq 1$, $N\geq 2$, $\tau\in\mathbb{R}$, $p>1$ and
$\lambda\geq-\frac{N^{2}}{4}$. For $\varphi\in\Phi$, let

(2.1) $\displaystyle J_{1}(\varphi)$ $\displaystyle=$
$\displaystyle\int_{\operatorname{supp}\left(\frac{\partial^{k}\varphi}{\partial
t^{k}}\right)}|x|^{\frac{-\tau}{p-1}}\varphi^{\frac{-1}{p-1}}\left|\frac{\partial^{k}\varphi}{\partial
t^{k}}\right|^{\frac{p}{p-1}}\,dx\,dt,$ (2.2) $\displaystyle J_{2}(\varphi)$
$\displaystyle=$
$\displaystyle\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\varphi\right)}|x|^{\frac{-\tau}{p-1}}\varphi^{\frac{-1}{p-1}}\left|\mathcal{L}_{\lambda}\varphi\right|^{\frac{p}{p-1}}\,dx\,dt.$

###  2.1. A priori estimate

We have the following a priori estimate.

######  Lemma 2.1.

Let $u\in L^{p}_{\operatorname{loc}}(Q)$ be a weak solution to (1.1)-(1.2).
Then,

(2.3)
$-\int_{\Gamma_{Q}^{1}}\frac{\partial\varphi}{\partial\nu_{1}}w(x)\,dS_{x}\,dt\leq
C\sum_{i=1}^{2}J_{i}(\varphi)$

for all $\varphi\in\Phi$, provided that $J_{i}(\varphi)<\infty$, $i=1,2$.

###### Proof.

Let $u\in L^{p}_{\operatorname{loc}}(Q)$ be a weak solution to (1.1)-(1.2). By
(1.11), for all $\varphi\in\Phi$, there holds

(2.4)
$\int_{Q}|x|^{\tau}|u|^{p}\varphi\,dx\,dt-\int_{\Gamma_{Q}^{1}}\frac{\partial\varphi}{\partial\nu_{1}}w(x)\,dS_{x}\,dt\leq\int_{Q}|u|\left|\frac{\partial^{k}\varphi}{\partial
t^{k}}\right|\,dx\,dt+\int_{Q}|u|\left|\mathcal{L}_{\lambda}\varphi\right|\,dx\,dt.$

By means of Young’s inequality, we obtain

(2.5) $\displaystyle\int_{Q}|u|\left|\frac{\partial^{k}\varphi}{\partial
t^{k}}\right|\,dx\,dt$ $\displaystyle=$
$\displaystyle\int_{Q}\left(|x|^{\frac{\tau}{p}}|u|\varphi^{\frac{1}{p}}\right)\left(|x|^{\frac{-\tau}{p}}\varphi^{\frac{-1}{p}}\left|\frac{\partial^{k}\varphi}{\partial
t^{k}}\right|\right)\,dx\,dt$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{Q}|x|^{\tau}|u|^{p}\varphi\,dx\,dt+CJ_{1}(\varphi).$

Similarly, we havve

(2.6)
$\displaystyle\int_{Q}|u|\left|\mathcal{L}_{\lambda}\varphi\right|\,dx\,dt\leq\frac{1}{2}\int_{Q}|x|^{\tau}|u|^{p}\varphi\,dx\,dt+CJ_{2}(\varphi).$

Thus, (2.3) follows from (2.4), (2.5) and (2.6). ∎

###  2.2. Test functions

We first introduce the function $H$ defined in $\overline{\Omega}$ by

$H(x)=x_{N}h(|x|)=x_{N}\left\\{\begin{array}[]{llll}|x|^{\mu}\left(1-|x|^{-N-2\mu}\right)&\mbox{if}&\lambda>-\frac{N^{2}}{4},\\\
\\\
|x|^{\frac{-N}{2}}\ln|x|&\mbox{if}&\lambda=-\frac{N^{2}}{4},\end{array}\right.$

where the parameter $\mu$ is defined by (1.12). We collect below some useful
properties satisfied by $H$.

######  Lemma 2.2.

The following properties hold:

  * (i)

$H\geq 0$, $\mathcal{L}_{\lambda}H=0$ in $\Omega$,
$H_{|\Gamma_{0}}=H_{|\Gamma_{1}}=0$;

  * (ii)

$R\gg 1,\,x\in\Omega,\,|x|<R\implies H(x)\leq x_{N}|x|^{\mu}\ln R$;

  * (iii)

$R\gg 1,\,x\in\Omega,\,|x|>R\implies H(x)\geq Cx_{N}|x|^{\mu}$;

  * (iv)

$\displaystyle\frac{\partial H}{\partial\nu_{0}}(x)=-h(|x|)$;

  * (v)

$\displaystyle\frac{\partial H}{\partial\nu_{1}}(x)=-Cx_{N}$.

###### Proof.

Property (i) follows from elementary calculations. Properties (ii) and (iii)
follow immediately from the definition of $H$. On the other hand, by the
definition of $H$, for all $x\in\Omega$, we obtain

(2.7) $\nabla
H(x)=h(|x|)e_{N}+x_{N}\left\\{\begin{array}[]{llll}|x|^{\mu-2}\left(\mu+(N+\mu)|x|^{-N-2\mu}\right)x&\mbox{if}&\lambda>-\frac{N^{2}}{4},\\\
\\\
\left(\mu|x|^{\mu-2}\ln|x|+|x|^{\mu-2}\right)x&\mbox{if}&\lambda=-\frac{N^{2}}{4},\end{array}\right.$

where $e_{N}=(0,\cdots,0,1)\in\mathbb{R}^{N}$. Using (2.7), we obtain

$\frac{\partial H}{\partial\nu_{0}}(x)=-\nabla H(x)\cdot
e_{N}|_{x_{N}=0}=-h(|x|),$

where $\cdot$ denotes the inner product in $\mathbb{R}^{N}$, which proves
property (iv). Again, using (2.7), we get

$\displaystyle\frac{\partial H}{\partial\nu_{1}}(x)$ $\displaystyle=$
$\displaystyle-\nabla H(x)\cdot x|_{|x|=1}$ $\displaystyle=$ $\displaystyle-
x_{N}h(|x|)|_{|x|=1}-x_{N}\left\\{\begin{array}[]{llll}2\mu+N&\mbox{if}&\lambda>-\frac{N^{2}}{4},\\\
\\\ 1&\mbox{if}&\lambda=-\frac{N^{2}}{4}\end{array}\right.$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{llll}-(2\mu+N)x_{N}&\mbox{if}&\lambda>-\frac{N^{2}}{4},\\\
\\\ -x_{N}&\mbox{if}&\lambda=-\frac{N^{2}}{4}.\end{array}\right.$

Hence, from $2\mu+N>0$, property (v) follows. ∎

Let $\xi,\vartheta,\iota\in C^{\infty}(\mathbb{R})$ be three cut-off functions
satisfying

(2.10) $\displaystyle 0\leq\xi\leq 1,\quad\xi(s)=1\mbox{ if }|s|\leq
1,\quad\xi(s)=0\mbox{ if }|s|\geq 2;$ (2.11) $\displaystyle 0\leq\vartheta\leq
1,\quad\vartheta(s)=1\mbox{ if }s\leq 0,\quad\vartheta(s)=0\mbox{ if }s\geq
1;$ (2.12) $\displaystyle\iota\geq
0,\quad\operatorname{supp}(\iota)\subset\subset(0,1).$

For $T,R,\ell\gg 1$, let

(2.13) $\displaystyle\alpha_{T}(t)$ $\displaystyle=$
$\displaystyle\iota^{\ell}\left(\frac{t}{T}\right),\quad t>0,$ (2.14)
$\displaystyle\beta_{R}(x)$ $\displaystyle=$ $\displaystyle
H(x)\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right),\quad x\in\overline{\Omega},$
(2.15) $\displaystyle\gamma_{R}(x)$ $\displaystyle=$ $\displaystyle
H(x)\vartheta^{\ell}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right),\quad
x\in\overline{\Omega}.$

We consider test functions of the forms

(2.16) $\varphi(t,x)=\alpha_{T}(t)\beta_{R}(x),\quad(t,x)\in Q$

and

(2.17) $\psi(t,x)=\alpha_{T}(t)\gamma_{R}(x),\quad(t,x)\in Q.$

Test functions of the form (2.16) will be used in the proof of part (I) of
Theorem 1.2. In the proof of Theorem 1.3, we will make use of test functions
of the form (2.17).

######  Lemma 2.3.

For $T,R,\ell\gg 1$, the function $\varphi$ defined by (2.16), belongs to
$\Phi$.

###### Proof.

By the definition of the function $\varphi$, it can be easily seen that (A1)
and (A2) are satisfied. On the other hand, by Lemma 2.2 (i), we obtain
${\varphi}_{|{\Gamma_{Q}^{i}}}=0$, $i=0,1$, which shows that (A3) is also
satisfied. Furthermore, in view of (2.10) and (2.14), we have

$\beta_{R}(x)=H(x),\quad x_{N}>0,\,1<|x|\leq R,$

which implies by Lemma 2.2 (v) and (2.16) that

$\frac{\partial\beta_{R}}{\partial\nu_{1}}(x)=\frac{\partial
H}{\partial\nu_{1}}(x)=-Cx_{N}$

and

(2.18) $\frac{\partial\varphi}{\partial\nu_{1}}(t,x)=-Cx_{N}\alpha_{T}(t)\leq
0,\quad(t,x)\in\Gamma_{Q}^{1}.$

This shows that $\varphi$ satisfies (A4) for $i=1$. Finally, we have to show
that (A4) is satisfied for $i=0$. By (2.14), for all $x\in\Omega$, we have

$\nabla\beta_{R}(x)=\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\nabla
H(x)+H(x)\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right),$

which implies by Lemma 2.2 (i), (iv) that

$\frac{\partial\beta_{R}}{\partial\nu_{0}}(x)=\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)|_{x\in\Gamma_{0}}\frac{\partial
H}{\partial\nu_{0}}(x)=-\left(h(|x|)\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\right)|_{x\in\Gamma_{0}}\leq
0.$

Hence, by (2.16), we deduce that

$\frac{\partial\varphi}{\partial\nu_{0}}(t,x)=\alpha_{T}(t)\frac{\partial\beta_{R}}{\partial\nu_{0}}(x)\leq
0,\quad(t,x)\in\Gamma_{Q}^{0},$

which shows that $\varphi$ satisfies (A4) for $i=0$. The proof of Lemma 2.3 is
then completed. ∎

######  Lemma 2.4.

For $T,R,\ell\gg 1$, the function $\psi$ defined by (2.17) belongs to $\Phi$.

###### Proof.

By the definition of the function $\psi$, and making use of Lemma 2.2 (i), it
can be easily seen that (A1), (A2) and (A3) are satisfied. On the other hand,
by (2.11) and (2.15), we have

$\gamma_{R}(x)=H(x),\quad x_{N}>0,\,1<|x|\leq\sqrt{R},$

which implies by Lemma 2.2 (v) and (2.17) that

$\frac{\partial\gamma_{R}}{\partial\nu_{1}}(x)=\frac{\partial
H}{\partial\nu_{1}}(x)=-Cx_{N}$

and

(2.19) $\frac{\partial\psi}{\partial\nu_{1}}(t,x)=-Cx_{N}\alpha_{T}(t)\leq
0,\quad(t,x)\in\Gamma_{Q}^{1}.$

This shows that $\psi$ satisfies (A4) for $i=1$. Proceeding as in the proof of
Lemma 2.3, we can show that (A4) is also satisfied for $i=0$. ∎

######  Lemma 2.5.

For $R,\ell\gg 1$ and $R<|x|<\sqrt{2}R$, $x_{N}>0$, the following estimates
hold:

(2.20)
$\displaystyle\left|H(x)\Delta\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\right|$
$\displaystyle\leq$ $\displaystyle CR^{-2}\ln
R\,x_{N}|x|^{\mu}\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right),$ (2.21)
$\displaystyle\left|\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\right|$
$\displaystyle\leq$ $\displaystyle CR^{-2}\ln
R\,x_{N}|x|^{\mu}\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right).$

###### Proof.

By (2.10), and Lemma 2.2 (ii), for $R<|x|<\sqrt{2}R$, $x_{N}>0$, we have

$\left|\Delta\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\right|\leq
CR^{-2}\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right),\,\,H(x)\leq
Cx_{N}|x|^{\mu}\ln R,$

which yields (2.20). On the other hand, by (2.7), for $R<|x|<\sqrt{2}R$,
$x_{N}>0$, and $\lambda>-\frac{N^{2}}{4}$, we have

$\displaystyle\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)$
$\displaystyle=C\xi^{\ell-1}\left(\frac{|x|^{2}}{R^{2}}\right)\xi^{\prime}\left(\frac{|x|^{2}}{R^{2}}\right)R^{-2}\left[h(|x|)e_{N}+x_{N}|x|^{\mu-2}\left(\mu+(N+\mu)|x|^{-N-2\mu}\right)x\right]\cdot
x$
$\displaystyle=C\xi^{\ell-1}\left(\frac{|x|^{2}}{R^{2}}\right)\xi^{\prime}\left(\frac{|x|^{2}}{R^{2}}\right)R^{-2}\left[h(|x|)x_{N}+x_{N}|x|^{\mu}\left(\mu+(N+\mu)|x|^{-N-2\mu}\right)\right].$

Then, by (2.10) and using that $N+2\mu>0$, we get

$\displaystyle\left|\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\right|$
$\displaystyle\leq
CR^{-2}\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right)x_{N}|x|^{\mu}\left[1+\left(|\mu|+(N+\mu)|x|^{-N-2\mu}\right)\right]$
$\displaystyle\leq
CR^{-2}x_{N}|x|^{\mu}\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right)$
$\displaystyle\leq CR^{-2}x_{N}|x|^{\mu}\ln
R\,\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right),$

which proves (2.21). Similarly, by (2.7), for $R<|x|<\sqrt{2}R$, $x_{N}>0$,
and $\lambda=-\frac{N^{2}}{4}$, we have

$\displaystyle\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)$
$\displaystyle=C\xi^{\ell-1}\left(\frac{|x|^{2}}{R^{2}}\right)\xi^{\prime}\left(\frac{|x|^{2}}{R^{2}}\right)R^{-2}\left[h(|x|)e_{N}+x_{N}|x|^{\mu-2}\left(\mu\ln|x|+1\right)x\right]\cdot
x$
$\displaystyle=C\xi^{\ell-1}\left(\frac{|x|^{2}}{R^{2}}\right)\xi^{\prime}\left(\frac{|x|^{2}}{R^{2}}\right)R^{-2}\left[h(|x|)x_{N}+x_{N}|x|^{\mu}\left(\mu\ln|x|+1\right)\right],$

which implies by (2.10) that

$\displaystyle\left|\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\right|$
$\displaystyle\leq
CR^{-2}\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right)x_{N}|x|^{\mu}\left(2+|\mu|\ln|x|\right)$
$\displaystyle\leq CR^{-2}x_{N}|x|^{\mu}\ln
R\,\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right),$

which proves (2.21). ∎

Using (2.7), (2.11) and similar calculations as above, we obtain the following
estimates.

######  Lemma 2.6.

For $R,\ell\gg 1$ and $\sqrt{R}<|x|<R$, $x_{N}>0$, the following estimates
hold:

$\displaystyle\left|H(x)\Delta\,\vartheta^{\ell}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right)\right|$
$\displaystyle\leq$ $\displaystyle Cx_{N}(\ln
R)^{-1}|x|^{\mu-2}\vartheta^{\ell-2}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right),$
$\displaystyle\left|\nabla
H(x)\cdot\nabla\,\vartheta^{\ell}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right)\right|$
$\displaystyle\leq$ $\displaystyle Cx_{N}(\ln
R)^{-1}|x|^{\mu-2}\vartheta^{\ell-2}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right).$

###  2.3. Estimates of $J_{i}(\varphi)$

For $T,R,\ell\gg 1$, we shall estimate the terms $J_{i}(\varphi)$, $i=1,2$,
where $\varphi$ is the function defined by (2.16).

######  Lemma 2.7.

The following estimate holds:

(2.22)
$\int_{\operatorname{supp}\left(\frac{d^{k}\alpha_{T}}{dt^{k}}\right)}\alpha_{T}^{\frac{-1}{p-1}}(t)\left|\frac{d^{k}\alpha_{T}}{dt^{k}}(t)\right|^{\frac{p}{p-1}}\,dt\leq
CT^{1-\frac{kp}{p-1}}.$

###### Proof.

By (2.12) and (2.13), we obtain

$\displaystyle\int_{\operatorname{supp}\left(\frac{d^{k}\alpha_{T}}{dt^{k}}\right)}\alpha_{T}^{\frac{-1}{p-1}}(t)\left|\frac{d^{k}\alpha_{T}}{dt^{k}}(t)\right|^{\frac{p}{p-1}}\,dt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\iota^{\frac{-\ell}{p-1}}\left(\frac{t}{T}\right)\left|\frac{d^{k}}{dt^{k}}\left[\iota^{\ell}\left(\frac{t}{T}\right)\right]\right|^{\frac{p}{p-1}}\,dt$
$\displaystyle\leq$ $\displaystyle
CT^{\frac{-kp}{p-1}}\int_{0}^{T}\iota^{\frac{-\ell}{p-1}}\left(\frac{t}{T}\right)\iota^{\frac{(\ell-k)p}{p-1}}\left(\frac{t}{T}\right)\,dt$
$\displaystyle=$ $\displaystyle
CT^{1-\frac{kp}{p-1}}\int_{0}^{1}\iota^{\ell-\frac{kp}{p-1}}(s)\,ds,$

which yields (2.22). ∎

######  Lemma 2.8.

The following estimate holds:

(2.23)
$\int_{\operatorname{supp}(\beta_{R})}|x|^{\frac{-\tau}{p-1}}\beta_{R}(x)\,dx\leq
C\ln R\left(\ln R+R^{\frac{(\mu+N+1)p-\mu-N-1-\tau}{p-1}}\right).$

###### Proof.

By (2.10) and (2.14), we obtain

$\displaystyle\int_{\operatorname{supp}(\beta_{R})}|x|^{\frac{-\tau}{p-1}}\beta_{R}(x)\,dx$
$\displaystyle=$
$\displaystyle\int_{1<|x|<\sqrt{2}R,\,x_{N}>0}|x|^{\frac{-\tau}{p-1}}H(x)\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\,dx$
$\displaystyle\leq$
$\displaystyle\int_{1<|x|<\sqrt{2}R,\,x_{N}>0}|x|^{\frac{-\tau}{p-1}}H(x)\,dx.$

Then, making use of Lemma 2.2 (ii), we get

$\displaystyle\int_{\operatorname{supp}(\beta_{R})}|x|^{\frac{-\tau}{p-1}}\beta_{R}(x)\,dx$
$\displaystyle\leq$ $\displaystyle\ln
R\int_{1<|x|<\sqrt{2}R,\,x_{N}>0}|x|^{\mu-\frac{\tau}{p-1}}x_{N}\,dx$
$\displaystyle\leq$ $\displaystyle\ln
R\int_{1<|x|<\sqrt{2}R}|x|^{\mu+1-\frac{\tau}{p-1}}\,dx$ $\displaystyle=$
$\displaystyle C\ln R\int_{r=1}^{\sqrt{2}R}r^{\mu+N-\frac{\tau}{p-1}}\,dr$
$\displaystyle=$ $\displaystyle C\ln R\left\\{\begin{array}[]{llll}\ln
R&\mbox{if}&\tau=(\mu+N+1)(p-1),\\\ 1&\mbox{if}&\tau>(\mu+N+1)(p-1),\\\
R^{\frac{(\mu+N+1)p-\mu-N-1-\tau}{p-1}}&\mbox{if}&\tau<(\mu+N+1)(p-1),\end{array}\right.$

which proves (2.23). ∎

From (2.1), (2.16), Lemmas 2.7 and 2.8, we deduce the following estimate.

######  Lemma 2.9.

The following estimate holds:

$J_{1}(\varphi)\leq CT^{1-\frac{kp}{p-1}}\ln R\left(\ln
R+R^{\frac{(\mu+N+1)p-\mu-N-1-\tau}{p-1}}\right).$

######  Lemma 2.10.

The following estimate holds:

(2.25)
$\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\beta_{R}\right)}|x|^{\frac{-\tau}{p-1}}\beta_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\beta_{R}(x)\right|^{\frac{p}{p-1}}\,dx\leq
CR^{\frac{(N+\mu-1)p-N-1-\mu-\tau}{p-1}}(\ln R)^{\frac{p}{p-1}}.$

###### Proof.

By (2.14), for all $x\in\operatorname{supp}(\beta_{R})$, we have

$\displaystyle\mathcal{L}_{\lambda}\beta_{R}(x)$ $\displaystyle=$
$\displaystyle-\Delta\left(H(x)\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\right)+\frac{\lambda}{|x|^{2}}H(x)\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)$
$\displaystyle=$
$\displaystyle-\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\Delta
H(x)-H(x)\Delta\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)-2\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)$
$\displaystyle+\frac{\lambda}{|x|^{2}}H(x)\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)$
$\displaystyle=$
$\displaystyle\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)\mathcal{L}_{\lambda}H(x)-H(x)\Delta\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)-2\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right),$

which implies by Lemma 2.2 (i) and (2.10) that

(2.26)
$\mathcal{L}_{\lambda}\beta_{R}(x)=-H(x)\Delta\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)-2\nabla
H(x)\cdot\nabla\,\xi^{\ell}\left(\frac{|x|^{2}}{R^{2}}\right)$

and

(2.27)
$\displaystyle\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\beta_{R}\right)}|x|^{\frac{-\tau}{p-1}}\beta_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\beta_{R}(x)\right|^{\frac{p}{p-1}}\,dx$
$\displaystyle=\int_{R<|x|<\sqrt{2}R,\,x_{N}>0}|x|^{\frac{-\tau}{p-1}}\beta_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\beta_{R}(x)\right|^{\frac{p}{p-1}}\,dx.$

On the other hand, by Lemma 2.5, for $R<|x|<\sqrt{2}R$, $x_{N}>0$, we have

$\left|\mathcal{L}_{\lambda}\beta_{R}(x)\right|\leq CR^{-2}\ln
R\,x_{N}|x|^{\mu}\xi^{\ell-2}\left(\frac{|x|^{2}}{R^{2}}\right),$

which yields

(2.28) $\left|\mathcal{L}_{\lambda}\beta_{R}(x)\right|^{\frac{p}{p-1}}\leq
CR^{-\frac{2p}{p-1}}(\ln R)^{\frac{p}{p-1}}x_{N}^{\frac{p}{p-1}}|x|^{\frac{\mu
p}{p-1}}\xi^{\frac{(\ell-2)p}{p-1}}\left(\frac{|x|^{2}}{R^{2}}\right).$

Furthermore, by (2.14) and Lemma 2.2 (iii), for $R<|x|<\sqrt{2}R$, we have

(2.29) $\displaystyle\beta_{R}^{\frac{-1}{p-1}}(x)$ $\displaystyle=$
$\displaystyle
H^{\frac{-1}{p-1}}(x)\xi^{\frac{-\ell}{p-1}}\left(\frac{|x|^{2}}{R^{2}}\right)$
$\displaystyle\leq$ $\displaystyle
Cx_{N}^{\frac{-1}{p-1}}|x|^{\frac{-\mu}{p-1}}\xi^{\frac{-\ell}{p-1}}\left(\frac{|x|^{2}}{R^{2}}\right).$

Thus, in view of (2.27), (2.28) and (2.29), we obtain

$\displaystyle\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\beta_{R}\right)}|x|^{\frac{-\tau}{p-1}}\beta_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\beta_{R}(x)\right|^{\frac{p}{p-1}}\,dx$
$\displaystyle\leq CR^{-\frac{2p}{p-1}}(\ln
R)^{\frac{p}{p-1}}\int_{R<|x|<\sqrt{2}R,\,x_{N}>0}|x|^{\frac{-(\tau+\mu)+\mu
p}{p-1}}x_{N}\xi^{\ell-\frac{2p}{p-1}}\left(\frac{|x|^{2}}{R^{2}}\right)\,dx$
$\displaystyle\leq CR^{-\frac{2p}{p-1}}(\ln
R)^{\frac{p}{p-1}}\int_{R<|x|<\sqrt{2}R,\,x_{N}>0}|x|^{\frac{p-1-(\tau+\mu)+\mu
p}{p-1}}\,dx$ $\displaystyle\leq CR^{-\frac{2p}{p-1}}(\ln
R)^{\frac{p}{p-1}}R^{\frac{p-1-(\tau+\mu)+\mu p}{p-1}}R^{N}$
$\displaystyle=CR^{\frac{(N+\mu-1)p-N-1-\mu-\tau}{p-1}}(\ln
R)^{\frac{p}{p-1}},$

which proves (2.25). ∎

######  Lemma 2.11.

The following estimate holds:

(2.30) $J_{2}(\varphi)\leq CTR^{\frac{(N+\mu-1)p-N-1-\mu-\tau}{p-1}}(\ln
R)^{\frac{p}{p-1}}.$

###### Proof.

By (2.2) and (2.16), we have

(2.31)
$J_{2}(\varphi)=\left(\int_{\operatorname{supp}(\alpha_{T})}\alpha_{T}(t)\,dt\right)\left(\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\beta_{R}\right)}|x|^{\frac{-\tau}{p-1}}\beta_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\beta_{R}(x)\right|^{\frac{p}{p-1}}\,dx\right).$

On the other hand, by (2.12) and (2.13), there holds

(2.32) $\displaystyle\int_{\operatorname{supp}(\alpha_{T})}\alpha_{T}(t)\,dt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\iota\left(\frac{t}{T}\right)^{\ell}\,dt$
$\displaystyle=$ $\displaystyle T\int_{0}^{1}\iota(s)^{\ell}\,ds.$

Hence, by Lemma 2.10, (2.31) and (2.32), we obtain (2.30). ∎

###  2.4. Estimates of $J_{i}(\psi)$ in the critical case

In this subsection, for $\lambda>\frac{-N^{2}}{4}$ and $T,R,\ell\gg 1$, we
shall estimate the terms $J_{i}(\psi)$, $i=1,2$, in the critical case
$(N+\mu-1)p=N+\mu+1+\tau$ (see Theorem 1.3), where $\psi$ is the function
defined by (2.17).

The proof of the following lemma is similar to that of Lemma 2.8. We omit the
details.

######  Lemma 2.12.

Let $\lambda>-\frac{N^{2}}{4}$ and $(N+\mu-1)p=N+\mu+1+\tau$. The following
estimate holds:

$\int_{\operatorname{supp}(\gamma_{R})}|x|^{\frac{-\tau}{p-1}}\gamma_{R}(x)\,dx\leq
C\left(\ln R+R^{\frac{2p}{p-1}}\right).$

Using (2.1), (2.17), Lemmas 2.7 and 2.12, we deduce the following estimate.

######  Lemma 2.13.

Let $\lambda>-\frac{N^{2}}{4}$ and $(N+\mu-1)p=N+\mu+1+\tau$. The following
estimate holds:

$J_{1}(\psi)\leq CT^{1-\frac{kp}{p-1}}\left(\ln R+R^{\frac{2p}{p-1}}\right).$

######  Lemma 2.14.

Let $\lambda>-\frac{N^{2}}{4}$ and $(N+\mu-1)p=N+\mu+1+\tau$. The following
estimate holds:

(2.33)
$\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\gamma_{R}\right)}|x|^{\frac{-\tau}{p-1}}\gamma_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\gamma_{R}(x)\right|^{\frac{p}{p-1}}\,dx\leq
C(\ln R)^{\frac{-1}{p-1}}.$

###### Proof.

By (2.15), Lemma 2.2 (i), and following the proof of Lemma 2.10, for all
$x\in\operatorname{supp}(\gamma_{R})$, we obtain

(2.34)
$\mathcal{L}_{\lambda}\gamma_{R}(x)=-H(x)\Delta\,\vartheta^{\ell}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right)-2\nabla
H(x)\cdot\nabla\,\vartheta^{\ell}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right),$

which implies by (2.11) that

(2.35)
$\displaystyle\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\gamma_{R}\right)}|x|^{\frac{-\tau}{p-1}}\gamma_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\gamma_{R}(x)\right|^{\frac{p}{p-1}}\,dx$
$\displaystyle=\int_{\sqrt{R}<|x|<R,\,x_{N}>0}|x|^{\frac{-\tau}{p-1}}\gamma_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\gamma_{R}(x)\right|^{\frac{p}{p-1}}\,dx.$

On the other hand, by (2.34) and Lemma 2.6, for $\sqrt{R}<|x|<R,\,x_{N}>0$, we
have

$\left|\mathcal{L}_{\lambda}\gamma_{R}(x)\right|\leq Cx_{N}(\ln
R)^{-1}|x|^{\mu-2}\vartheta^{\ell-2}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right),$

which yields

(2.36) $\left|\mathcal{L}_{\lambda}\gamma_{R}(x)\right|^{\frac{p}{p-1}}\leq
Cx_{N}^{\frac{p}{p-1}}(\ln
R)^{-\frac{p}{p-1}}|x|^{\frac{(\mu-2)p}{p-1}}\vartheta^{\frac{(\ell-2)p}{p-1}}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right).$

Furthermore, by (2.15) and Lemma 2.2 (iii), for $\sqrt{R}<|x|<R,\,x_{N}>0$, we
have

(2.37) $\gamma_{R}^{\frac{-1}{p-1}}(x)\leq
Cx_{N}^{-\frac{1}{p-1}}|x|^{-\frac{\mu}{p-1}}\vartheta^{-\frac{\ell}{p-1}}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right).$

Hence, in view of (2.11), (2.35), (2.36) and (2.37), we get

$\displaystyle\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\gamma_{R}\right)}|x|^{\frac{-\tau}{p-1}}\gamma_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\gamma_{R}(x)\right|^{\frac{p}{p-1}}\,dx$
$\displaystyle\leq C(\ln
R)^{\frac{-p}{p-1}}\int_{\sqrt{R}<|x|<R,\,x_{N}>0}|x|^{\frac{(\mu-2)p-\tau-\mu}{p-1}}x_{N}\vartheta^{\ell-\frac{2p}{p-1}}\left(\frac{\ln\left(\frac{|x|}{\sqrt{R}}\right)}{\ln(\sqrt{R})}\right)\,dx$
$\displaystyle\leq C(\ln
R)^{\frac{-p}{p-1}}\int_{\sqrt{R}<|x|<R}|x|^{\frac{(\mu-1)p-\tau-\mu-1}{p-1}}\,dx$
$\displaystyle=C(\ln
R)^{\frac{-p}{p-1}}\int_{r=\sqrt{R}}^{R}r^{\frac{(N+\mu-1)p-N-\mu-1-\tau}{p-1}}r^{-1}\,dr.$

Since $(N+\mu-1)p=N+\mu+1+\tau$, the above estimate yields

$\displaystyle\int_{\operatorname{supp}\left(\mathcal{L}_{\lambda}\gamma_{R}\right)}|x|^{\frac{-\tau}{p-1}}\gamma_{R}^{\frac{-1}{p-1}}(x)\left|\mathcal{L}_{\lambda}\gamma_{R}(x)\right|^{\frac{p}{p-1}}\,dx$
$\displaystyle\leq$ $\displaystyle C(\ln
R)^{\frac{-p}{p-1}}\int_{r=\sqrt{R}}^{R}r^{-1}\,dr$ $\displaystyle=$
$\displaystyle C(\ln R)^{\frac{-1}{p-1}},$

which proves (2.33). ∎

Using (2.2), (2.17), (2.32) and Lemma 2.14, we obtain the following estimate.

######  Lemma 2.15.

Let $\lambda>-\frac{N^{2}}{4}$ and $(N+\mu-1)p=N+\mu+1+\tau$. The following
estimate holds:

$J_{2}(\psi)\leq CT(\ln R)^{\frac{-1}{p-1}}.$

##  3\. Proofs of the main results

###  3.1. Proofs of the nonexistence results

In this subsection, we prove part (I) of Theorem 1.2, and Theorem 1.3.

###### Proof of part (I) of Theorem 1.2.

We use the contradiction argument. Namely, we suppose that $u\in
L^{p}_{\operatorname{loc}}(Q)$ is a weak solution to (1.1)-(1.2). Then, by
Lemma 2.1, (2.3) holds for all $\varphi\in\Phi$ (with $J_{i}(\varphi)<\infty$,
$i=1,2$). Hence, from Lemma 2.3, we deduce that for $T,R,\ell\gg 1$,

(3.1)
$-\int_{\Gamma_{Q}^{1}}\frac{\partial\varphi}{\partial\nu_{1}}w(x)\,dS_{x}\,dt\leq
C\sum_{i=1}^{2}J_{i}(\varphi),$

where $\varphi$ is the function given by (2.16). On the other hand, by (2.18)
and (2.32), we have

(3.2)
$\displaystyle-\int_{\Gamma_{Q}^{1}}\frac{\partial\varphi}{\partial\nu_{1}}w(x)\,dS_{x}\,dt$
$\displaystyle=$ $\displaystyle
C\int_{0}^{\infty}\int_{\Gamma_{1}}w(x)x_{N}\alpha_{T}(t)\,dS_{x}\,dt$
$\displaystyle=$ $\displaystyle
C\left(\int_{0}^{\infty}\alpha_{T}(t)\,dt\right)\left(\int_{\Gamma_{1}}w(x)x_{N}\,dS_{x}\right)$
$\displaystyle=$ $\displaystyle CT\int_{\Gamma_{1}}w(x)x_{N}\,dS_{x}$
$\displaystyle=$ $\displaystyle CTI_{w}.$

Hence, by (3.1), (3.2), Lemmas 2.9 and 2.11, we obtain

$TI_{w}\leq C\left[T^{1-\frac{kp}{p-1}}\ln R\left(\ln
R+R^{\frac{(\mu+N+1)p-\mu-N-1-\tau}{p-1}}\right)+TR^{\frac{(N+\mu-1)p-N-1-\mu-\tau}{p-1}}(\ln
R)^{\frac{p}{p-1}}\right],$

that is,

(3.3) $I_{w}\leq C\left(T^{-\frac{kp}{p-1}}(\ln
R)^{2}+T^{-\frac{kp}{p-1}}R^{a}\ln R+R^{b}(\ln R)^{\frac{p}{p-1}}\right),$

where

$a=\frac{(\mu+N+1)p-(\mu+N+1+\tau)}{p-1}$

and

$b=\frac{(N+\mu-1)p-(N+\mu+1+\tau)}{p-1}.$

Taking $T=R^{\theta}$, where

(3.4) $\theta>\max\left\\{\frac{a(p-1)}{kp},0\right\\},$

(3.3) reduces to

(3.5) $I_{w}\leq C\left(R^{\frac{-\theta kp}{p-1}}(\ln
R)^{2}+R^{a-\frac{\theta kp}{p-1}}\ln R+R^{b}(\ln R)^{\frac{p}{p-1}}\right).$

Notice that from the choice (3.4) of the parameter $\theta$, one has
$a-\frac{\theta kp}{p-1}<0$. Moreover, due to (1.13), one has $b<0$. Hence,
passing to the limit as $R\to\infty$ in (3.5), we obtain $I_{w}\leq 0$, which
contradicts the condition $w\in L^{1,+}(\Gamma_{1})$. Consequently,
(1.1)-(1.2) admits no weak solution. This completes the proof of part (I) of
Theorem 1.2. ∎

###### Proof of Theorem 1.3.

We use also the contradiction argument by supposing that $u\in
L^{p}_{\operatorname{loc}}(Q)$ is a weak solution to (1.1)-(1.2). Then, from
Lemmas 2.1 and 2.4, we deduce that for $T,R,\ell\gg 1$,

(3.6)
$-\int_{\Gamma_{Q}^{1}}\frac{\partial\psi}{\partial\nu_{1}}w(x)\,dS_{x}\,dt\leq
C\sum_{i=1}^{2}J_{i}(\psi),$

where $\psi$ is the function given by (2.17). On the other hand, by (2.19) and
(2.32), we obtain

(3.7)
$-\int_{\Gamma_{Q}^{1}}\frac{\partial\psi}{\partial\nu_{1}}w(x)\,dS_{x}\,dt=CTI_{w}.$

Hence, making use of (3.6), (3.7), Lemmas 2.13 and 2.15, we obtain

$TI_{w}\leq C\left[T^{1-\frac{kp}{p-1}}\left(\ln
R+R^{\frac{2p}{p-1}}\right)+T(\ln R)^{\frac{-1}{p-1}}\right],$

that is,

(3.8) $I_{w}\leq C\left(T^{-\frac{kp}{p-1}}\ln
R+T^{-\frac{kp}{p-1}}R^{\frac{2p}{p-1}}+(\ln R)^{\frac{-1}{p-1}}\right).$

Thus, taking $T=R^{\theta}$, where $\theta>\frac{2}{k}$, and passing to the
limit as $R\to\infty$ in (3.8), we obtain a contradiction with $w\in
L^{1,+}(\Gamma_{1})$. This completes the proof of Theorem 1.3. ∎

###  3.2. Proof of the existence result

In this subsection, we give the

###### Proof of part (II) of Theorem 1.2.

We first consider
(i) The case $\lambda>-\frac{N^{2}}{4}$.
For $\delta$ and $\epsilon$ satisfying respectively

(3.9) $\max\left\\{-\mu,\frac{\tau+p+1}{p-1}\right\\}<\delta<\mu+N$

and

(3.10) $0<\epsilon<\left(-\delta^{2}+N\delta+\lambda\right)^{\frac{1}{p-1}},$

let

(3.11) $u_{\delta,\epsilon}(x)=\epsilon x_{N}|x|^{-\delta},\quad x\in\Omega.$

Notice that by (1.12), since $\lambda>-\frac{N^{2}}{4}$, one has $-\mu<\mu+N$.
Moreover, due to (1.14), one has

$\frac{\tau+p+1}{p-1}<\mu+N.$

Hence, the set of $\delta$ satisfying (3.9) is nonempty. On the other hand,
observe that $-\mu$ and $\mu+N$ are the roots of the polynomial function

$P(\delta)=-\delta^{2}+N\delta+\lambda,$

which implies that $P(\delta)>0$ for any $\delta$ satisfying (3.9), so
$P(\delta)^{\frac{1}{p-1}}$ is well-defined, and the set of $\epsilon$
satisfying (3.10) is nonempty. Elementary calculations show that

(3.12) $\mathcal{L}_{\lambda}u_{\delta,\epsilon}(x)=\epsilon
P(\delta)x_{N}|x|^{-\delta-2},\quad x\in\Omega.$

Then, in view of (3.9), (3.10), (3.11) and (3.12), for all $x\in\Omega$, we
obtain

$\displaystyle\mathcal{L}_{\lambda}u_{\delta,\epsilon}(x)$ $\displaystyle\geq$
$\displaystyle\epsilon\epsilon^{p-1}x_{N}|x|^{-\delta-2}$ $\displaystyle=$
$\displaystyle\left(|x|^{\tau}\epsilon^{p}x_{N}^{p}|x|^{-\delta
p}\right)\left(x_{N}^{1-p}|x|^{-\delta-2+\delta p-\tau}\right)$
$\displaystyle\geq$
$\displaystyle|x|^{\tau}u_{\delta,\epsilon}^{p}(x)|x|^{\delta(p-1)-(\tau+p+1)}$
$\displaystyle\geq$ $\displaystyle|x|^{\tau}u_{\delta,\epsilon}^{p}(x),$

which shows that for any $\delta$ and $\epsilon$ satisfying respectively (3.9)
and (3.10), functions of the form (3.11) are stationary solutions to
(1.1)-(1.2) with

$w(x)=\epsilon x_{N},\quad x\in\Gamma_{1}.$

Next, we consider
(ii) The case $\lambda=-\frac{N^{2}}{4}$.
For

(3.13) $0<\kappa<1,\,\,\rho>1,\,\,\varepsilon>0,$

we consider functions of the form

(3.14) $u_{\kappa,\rho,\varepsilon}(x)=\varepsilon
x_{N}|x|^{\mu}[\ln(\rho|x|)]^{\kappa},\quad x\in\Omega.$

Taking into consideration that $\lambda=-\frac{N^{2}}{4}$ (so
$\mu=-\frac{N}{2}$), elementary calculations show that

(3.15)
$\mathcal{L}_{\lambda}u_{\kappa,\rho,\varepsilon}(x)=\varepsilon\kappa(1-\kappa)x_{N}|x|^{\mu-2}[\ln(\rho|x|)]^{\kappa-2},\quad
x\in\Omega.$

In view of (3.13), (3.14) and (3.15), for all $x\in\Omega$, we obtain

(3.16) $\displaystyle\mathcal{L}_{\lambda}u_{\kappa,\rho,\varepsilon}(x)$
$\displaystyle=$
$\displaystyle|x|^{\tau}u_{\kappa,\rho,\varepsilon}^{p}(x)\left(\varepsilon^{1-p}\kappa(1-\kappa)x_{N}^{1-p}|x|^{\mu-2-\mu
p-\tau}[\ln(\rho|x|)]^{\kappa-2-\kappa p}\right)$ $\displaystyle\geq$
$\displaystyle|x|^{\tau}u_{\kappa,\rho,\varepsilon}^{p}(x)\left(\varepsilon^{1-p}\kappa(1-\kappa)|x|^{\zeta}[\ln(\rho|x|)]^{\kappa-2-\kappa
p}\right),$

where

$\zeta=(-\mu-1)p-(-\mu+1+\tau)=(N+\mu-1)p-(N+\mu+1+\tau).$

Notice that due to (1.14), one has $\zeta>0$, which yields

$\lim_{s\to+\infty}\kappa(1-\kappa)s^{\zeta}[\ln(\rho s)]^{\kappa-2-\kappa
p}=+\infty.$

Consequently, there exists a constant $A>0$ (independent on $x$) such that

(3.17) $\kappa(1-\kappa)|x|^{\zeta}[\ln(\rho|x|)]^{\kappa-2-\kappa p}\geq
A,\quad x\in\Omega.$

Thus, taking

(3.18) $0<\varepsilon<A^{\frac{1}{p-1}},$

using (3.16) and (3.17), we obtain

$\mathcal{L}_{\lambda}u_{\kappa,\rho,\varepsilon}(x)\geq|x|^{\tau}u_{\kappa,\rho,\varepsilon}^{p}(x),\quad
x\in\Omega,$

which shows that for any $\kappa$, $\rho$ and $\varepsilon$ satisfying (3.13)
and (3.18), functions of the form (3.14) are stationary solutions to
(1.1)-(1.2) with

$w(x)=\varepsilon x_{N}(\ln\rho)^{\kappa},\quad x\in\Gamma_{1}.$

This completes the proof of part (II) of Theorem 1.2. ∎

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