Caccioppoli-type inequalities for Dirac operators

Abstract

In this paper, we establish the Caccioppoli estimates for the nonlinear differential equation

$$ - \overline{D}\bigl( \vert Dv \vert ^{p-2}Dv\bigr) = \lambda \vert v \vert ^{p-2}v, \quad 1< p< \infty ,$$

where D is the Dirac operator. Moreover, we obtain general weighted versions of the Caccioppoli-type inequalities for the Dirac operators.

1 Introduction

In the Euclidean setting, we recall the Caccioppoli inequality

$$ \int _{\Omega } \phi ^{p} \vert \nabla v \vert ^{p}\,dx \leq p^{p} \int _{\Omega } v^{p} \vert \nabla \phi \vert ^{p}\,dx $$

(1.1)

for all nonnegative functions $\phi \in C_{0}^{\infty }(\Omega )$, where a positive function v is a subsolution of the Dirichlet boundary value problem for p-Laplacian

$$ \textstyle\begin{cases} \nabla \cdot ( \vert \nabla v \vert ^{p-2}\nabla v) = \lambda \vert v \vert ^{p-2}v & \text{in } \Omega , \\ v=0 &\text{on } \partial \Omega . \end{cases} $$

(1.2)

See, for example, [4] for the problem in the Euclidean setting. We also refer to [1, 5, 6, 8, 10] and references therein for discussions on the Caccioppoli-type estimates in different settings.

The main aim of this paper is to obtain the Caccioppoli-type inequality for the nonlinear equation

$$ \textstyle\begin{cases} - \overline{D}( \vert Dv \vert ^{p-2}Dv) = \lambda \vert v \vert ^{p-2}v & \text{in } \Omega , \\ v=0 & \text{on } \partial \Omega , \end{cases} $$

where v is the subsolution, D is the usual Dirac operator, and D̅ is its conjugate. Also, we obtain weighted versions of the Caccioppoli-type inequality for the Dirac operator.

In what follows, we will work in $\mathbb{H}$, the skew-field of the quaternion. This means that each element $x'\in \mathbb{H}$ has the following representation:

$$ x'=x_{0}+\sum_{i=1}^{n}e_{i} x_{i}, $$

where $1, e_{1},\ldots ,e_{n}$ are the basis elements of $\mathbb{H}$. For these elements, we have the multiplication rules

• $e_{1}^{2} =\cdots =e_{n}^{2} = - 1$,

• $e_{i}e_{j}+e_{j}e_{i}=-2\delta +{ij}$ for all $i,j=1,\ldots ,n$.

The conjugate element $\overline{x'}$ is given by $\overline{x'} = x_{0} -\sum_{i=1}^{n} e_{i} x_{i}$, and we have the properties

$$ \bigl\vert x' \bigr\vert ^{2}= x\overline{x'} = \overline{x'}x = x_{0}^{2}+\sum _{i=1}^{n} x_{i}^{2} $$

and

$$ \vert x \vert _{q}=\sqrt{\sum _{i=1}^{n} x_{i}^{2}} $$

(1.3)

for the norm on $\mathbb{H}$.

We recall the usual Dirac operator, which factorizes the n-dimensional Laplace operator,

$$ Df=\sum_{i=1}^{n} e_{i} \frac{\partial f}{\partial x_{i}} $$

and its conjugate operator

$$ \overline{D}f=-\sum_{i=1}^{n} e_{i}\frac{\partial f}{\partial x_{i}}. $$

The products of these operators

$$ D\overline{D} = \overline{D} D = \Delta _{n}, $$

where $\Delta _{n}$ is the Laplacian for functions defined over domains in $\mathbb{R}^{n}$. For further discussions in this direction, we refer, for example, to [9] (see also [3] for theory of QDEs).

In Sect. 2, we discuss Picone’s identity for the Dirac operator. The main results of this paper are presented in Sect. 3.

2 Picone’s identity for the Dirac operator

Lemma 2.1

Let u, v be a differentiable functions defined a.e. in $\Omega \subset \mathbb{H}$ such that $v>0$ a.e. in Ω and $u\geq 0$. Define

$$\begin{aligned} &R(u,v):= \vert Du \vert ^{p}- D \biggl(\frac{u^{p}}{v^{p-1}} \biggr) \vert Dv \vert ^{p-2}Dv, \\ &L(u,v):= \vert Du \vert ^{p}- p \biggl(\frac{u}{v} \biggr)^{p-1} \vert Dv \vert ^{p-2} DvDu \\ &\hphantom{L(u,v):=}{}+(p-1) \biggl(\frac{u}{v} \biggr)^{p} \vert Dv \vert ^{p}, \end{aligned}$$

(2.1)

where $p>1$. Then

$$ L(u,v)=R(u,v)\geq 0. $$

(2.2)

Also, $L(u,v)=0$ a.e. in Ω if and only if $u=cv$ a.e. in Ω with positive constant c.

Proof of Lemma 2.1

A direct computation gives

$$\begin{aligned} R(u,v) &= \vert D u \vert ^{p} - D \biggl( \frac{u^{p}}{v^{p-1}} \biggr) \vert D v \vert ^{p-2}D v \\ &= \vert D u \vert ^{p} - \frac{pu^{p-1}Du v^{p-1}-u^{p} (p-1)v^{p-2}Dv}{(v^{p-1})^{2}} \vert Dv \vert ^{p-2}Dv \\ &= \vert D u \vert ^{p} - p\frac{u^{p-1}}{v^{p-1}} \vert Dv \vert ^{p-2}D v Du + (p-1)\frac{u^{p}}{v^{p}} \vert Dv \vert ^{p} \\ & = L(u,v). \end{aligned}$$

This proves the equality in (2.2). Now we rewrite $L(u,v)$ to see that $L(u,v)\geq 0$:

$$\begin{aligned} L(u,v) = {}& \vert D u \vert ^{p} - p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-1} \vert D u \vert + (p-1) \frac{u^{p}}{v^{p}} \vert D v \vert ^{p} \\ &{}+p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-2} \bigl( \vert D v \vert \vert D u \vert -D v Du \bigr) \\ ={}& S_{1} +S_{2}, \end{aligned}$$

where

$$\begin{aligned} S_{1}:={}& p \biggl[ \frac{1}{p} \vert D u \vert ^{p} + \frac{p-1}{p} \biggl( \biggl( \frac{u}{v} \vert D v \vert \biggr)^{p-1} \biggr)^{\frac{p}{p-1}} \biggr] \\ &{}- p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-1} \vert D u \vert \end{aligned}$$

and

$$ S_{2}:= p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-2} \bigl( \vert D v \vert \vert D u \vert -D v Du \bigr). $$

We can see that $S_{2}\geq 0$ due to $|Dv||D u| \geq D v Du$. To check that $S_{1} \geq 0$, we need to use Young’s inequality

$$ ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q},\quad a\geq 0,b \geq 0, $$

(2.3)

where $p>1$, $q>1$, and $\frac{1}{p}+\frac{1}{q}=1$. The equality holds if and only if $a^{p}=b^{q}$, that is, if $a = b^{\frac{1}{p-1}}$. Let us take $a= |D u|$ and $b = (\frac{u}{v}|Dv| )^{p-1}$ in (2.3) to get

$$ p \vert D u \vert \biggl(\frac{u}{v} \vert D v \vert \biggr)^{p-1} \leq p \biggl[ \frac{1}{p} \vert D u \vert ^{p} + \frac{p-1}{p} \biggl( \biggl(\frac{u}{v} \vert D v \vert \biggr)^{p-1} \biggr)^{\frac{p}{p-1}} \biggr]. $$

(2.4)

From this we see that $S_{1} \geq 0$, which proves that $L(u,v)=S_{1}+S_{2} \geq 0$. It is easy to see that $u=cv$ implies $R(u,v)=0$. Now let us prove that $L(u,v)=0$ implies $u=cv$. Due to $u(x)\geq 0$ and $L(u,v)(x_{0})=0$, $x_{0}\in \Omega$, we consider the two cases $u(x_{0})>0$ and $u(x_{0}) =0$.

• (1) In the case $u(x_{0})>0$, from $L(u,v)(x_{0})=0$ it follows that $S_{1}=0$ and $S_{2}=0$. Then $S_{1}=0$ implies

  $$ \vert D u \vert = \frac{u}{v} \vert D v \vert , $$

  (2.5)

  and $S_{2}=0$ implies

  $$ \vert D v \vert \vert D u \vert - D v D u = 0, $$

  (2.6)

  Combination of (2.5) and (2.6) gives

  $$ \frac{D u}{D v} = \frac{u }{v}=c \quad \text{with } c\neq 0. $$

  (2.7)

• (2) Let us denote $\Omega ^{*}:=\{x \in \Omega | u(x)=0 \}$. If $\Omega ^{*} \neq \Omega$, then suppose that $x_{0} \in \partial \Omega ^{*}$. Then there exists a sequence $x_{k} \notin \Omega ^{*}$ such that $x_{k} \rightarrow x_{0}$. In particular, $u(x_{k}) \neq 0$, and hence by case (1) we have $u(x_{k})=cv(x_{k})$. Passing to the limit, we get $u(x_{0}) = c v(x_{0})$. Since $u(x_{0})= 0$ and $v(x_{0})\neq 0$, we get that $c=0$. Then by case (1) again, since $u =cv$ and $u\neq 0$ in $\Omega \backslash \Omega ^{*}$, it is impossible that $c=0$. This contradiction implies that $\Omega ^{*} = \Omega$.

This completes the proof of Lemma 2.1. □

3 Caccioppoli-type inequalities

Let us consider the Dirichlet boundary problem for the Dirac operator

$$ \textstyle\begin{cases} - \overline{D}( \vert Dv \vert ^{p-2}Dv) = \lambda \vert v \vert ^{p-2}v & \text{in } \Omega , \\ v=0 & \text{on } \partial \Omega . \end{cases} $$

(3.1)

We say that a weak solution of equation (3.1) means a function $v \in W_{\mathrm{loc}}^{1,p}(\Omega )$ such that

$$ \int _{\Omega } \vert D v \vert ^{p-2} Dv D\phi \,dx' - \lambda \int _{\Omega } \vert v \vert ^{p-2}v \phi \,dx' =0 $$

(3.2)

for all functions $\phi \in W_{0}^{1,p}(\Omega )\cap C(\Omega )$. The supsolution and subsolution of equation (3.1) mean a function $v \in W_{\mathrm{loc}}^{1,p}(\Omega )$ such that

$$ \int _{\Omega } \vert D v \vert ^{p-2} Dv \overline{D} \phi \,dx' - \lambda \int _{ \Omega } \vert v \vert ^{p-2}v\phi \,dx' \geq 0 $$

(3.3)

and

$$ \int _{\Omega } \vert D v \vert ^{p-2} Dv \overline{D} \phi \,dx' - \lambda \int _{ \Omega } \vert v \vert ^{p-2}v\phi \,dx' \leq 0 $$

(3.4)

for the test functions $\phi \in W_{0}^{1,p}(\Omega )\cap C(\Omega )$ with $\phi \geq 0$, respectively.

If we take the test function as $\phi =v$, then for the supsolution and subsolution, we have

$$ \int _{\Omega } \vert Dv \vert ^{p}\,dx' \geq \lambda \int _{\Omega } \vert v \vert ^{p}\,dx' $$

(3.5)

and

$$ \int _{\Omega } \vert Dv \vert ^{p}\,dx' \leq \lambda \int _{\Omega } \vert v \vert ^{p}\,dx'. $$

(3.6)

Now we are ready to establish a Caccioppoli-type inequality.

Theorem 3.1

Let $\Omega \in \mathbb{H}$. Let v be a positive subsolution of equation (3.1) in Ω. For any fixed $q>p-1$ and $q< p<\infty$, we have

$$\begin{aligned} \int _{\Omega } v^{q-p} \phi ^{p} \vert D v \vert ^{p}\,dx' \leq \biggl( \frac{p}{q-p+1} \biggr)^{p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' + \frac{\lambda p}{q-p+1} \int _{\Omega } v^{q} \phi ^{p} \,dx' \end{aligned}$$

(3.7)

for all nonnegative functions $\phi \in C^{\infty }_{0}(\Omega )$.

Remark 3.2

Note that Theorem 3.1 for the Finsler norm was obtained in [2].

• For the case $q=p$ and $\lambda =0$ in Theorem 3.1, we have

  $$ \int _{\Omega } \phi ^{p} \vert Dv \vert ^{p}\,dx' \leq p^{p} \int _{\Omega } v^{p} \vert D \phi \vert ^{p}\,dx'. $$

  (3.8)

• For the case $q=0$ in Theorem 3.1, we have

  $$ \int _{\Omega } \phi ^{p} \vert D\log v \vert ^{p}\,dx' \leq \biggl( \frac{p}{1-p} \biggr)^{p} \int _{\Omega } \vert D\phi \vert ^{p} \,dx' + \frac{\lambda p}{1-p} \int _{\Omega } \phi ^{p}\,dx'. $$

  (3.9)

Proof of Theorem 3.1

Let us begin the proof by replacing $u = v^{\frac{q}{p}}\phi$ in $L(u,v)$, which gives

$$\begin{aligned} \int _{\Omega } L\bigl(v^{q/p}\phi ,v\bigr) \,dx' = {}& \int _{\Omega } \bigl\vert D\bigl(v^{q/p} \phi \bigr) \bigr\vert ^{p}\,dx' + (p-1) \int _{\Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{} - p \int _{\Omega } \bigl(v^{\frac{q-p}{p}}\phi \bigr)^{p-1} \vert Dv \vert ^{p-2}D\bigl(v^{q/p} \phi \bigr) \,dx' \\ ={}& \int _{\Omega } \bigl\vert D\bigl(v^{q/p}\phi \bigr) \bigr\vert ^{p}\,dx' + (p-1) \int _{\Omega } v^{q-p} \phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{} -p \int _{\Omega } \bigl(v^{\frac{q-p}{p}}\phi \bigr)^{p-1} \vert Dv \vert ^{p-2} \biggl( \frac{q}{p} v^{\frac{q-p}{p}}\phi Dv+ v^{q/p} Dv D\phi \biggr)\,dx' \\ ={}& \int _{\Omega } \bigl\vert D\bigl(v^{q/p}\phi \bigr) \bigr\vert ^{p}\,dx' - (q-p+1) \int _{\Omega } v^{q-p} \phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{} +p \int _{\Omega } \bigl(v^{\frac{q-p}{p}}\phi \bigr)^{p-1} \vert Dv \vert ^{p-1} v^{q/p} \vert D \phi \vert \,dx'. \end{aligned}$$

In the last line, we have used the Schwarz inequality. Now we apply the Young inequality of the form

$$ a b^{p-1} \leq \frac{a^{p}}{p\tau ^{p-1}} + \frac{p-1}{p} \tau b^{p},\quad a,b\geq 0, \tau >0. $$

By choosing $a= v^{q/p}|D\phi |$ and $b=v^{\frac{q-p}{p}}\phi |Dv|$ and using inequality (3.6) we arrive at

$$\begin{aligned} 0\leq{ }& \int _{\Omega } \bigl\vert D\bigl(v^{q/p}\phi \bigr) \bigr\vert ^{p}\,dx' - (q-p+1) \int _{ \Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{}+ \tau ^{1-p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' + (p-1)\tau \int _{ \Omega } v^{q-p} \phi ^{p} \vert Dv \vert ^{p}\,dx' \\ \leq {}& \lambda \int _{\Omega } v^{q} \phi ^{p} \,dx'- \bigl(q-p+1- \tau (p-1)\bigr) \int _{\Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{}+\tau ^{1-p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx'. \end{aligned}$$

Thus we have the following inequality:

$$\begin{aligned} \int _{\Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx'\leq{}& \frac{\tau ^{1-p}}{(q-p+1- \tau (p-1))} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' \\ &{}+ \frac{\lambda }{(q-p+1- \tau (p-1))} \int _{\Omega } v^{q} \phi ^{p} \,dx'. \end{aligned}$$

(3.10)

Taking a suitable constant $\tau = \frac{q-p+1}{p}$ leads to

$$\begin{aligned} \int _{\Omega } v^{q-p} \phi ^{p} \vert D v \vert ^{p}\,dx' \leq \biggl( \frac{p}{q-p+1} \biggr)^{p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' + \frac{\lambda p}{q-p+1} \int _{\Omega } v^{q} \phi ^{p} \,dx'. \end{aligned}$$

This proves the theorem. □

4 Weighted versions

Let us consider the following weighted operator:

$$ \Delta _{p,w}f = \overline{D}\bigl(w(x) \vert Df \vert ^{p-2}Df\bigr),\quad 1< p< \infty , $$

(4.1)

where $0\leq w \in C^{1}(\mathbb{H})$.

Theorem 4.1

Let $2\leq p< \infty$. Let $0 \leq F \in C^{\infty }(\mathbb{H})$ and $0\leq \eta \in L_{\mathrm{loc}}^{1}(\mathbb{H})$ be such that

$$ \eta F^{p-1}\leq - \Delta _{p,w}F\quad \textit{a.e. in } \mathbb{H}. $$

(4.2)

Then we have

$$\begin{aligned} \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{p}\,dx' + C_{p} \int _{\mathbb{H}} w(x) \bigl\vert F(x) \bigr\vert ^{p} \bigl\vert D(f/F) \bigr\vert ^{p}\,dx' \leq \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx' \end{aligned}$$

(4.3)

for all real-valued functions $f \in C_{0}^{\infty }(\mathbb{H})$. Here $C_{p}$ is a positive constant.

Note that the Carnot group version of Theorem 4.1 was obtained in [7].

Proof of Theorem 4.1

For all $a,b \in \mathbb{R}^{n}$, there exists a positive number $C_{p}$ such that

$$ \vert a \vert ^{p} + C_{p} \vert b \vert ^{p} + p \vert a \vert ^{p-2} a\cdot b \leq \vert a+b \vert ^{p},\quad 2 \leq p < \infty . $$

(4.4)

Using this by taking $a=g(x)DF(x)$ and $b=F(x)Dg(x)$, we get

$$\begin{aligned} &\bigl\vert g(x) \bigr\vert ^{p} \bigl\vert DF(x) \bigr\vert ^{p} + C_{p} \bigl\vert F(x) \bigr\vert ^{p} \bigl\vert Dg(x) \bigr\vert ^{p} +F(x) \bigl\vert DF(x) \bigr\vert ^{p-2}DF(x) \cdot D \bigl\vert g(x) \bigr\vert ^{p} \\ &\quad \leq \bigl\vert g(x) DF(x) +F(x) Dg(x) \bigr\vert ^{p} = \bigl\vert Df(x) \bigr\vert ^{p}, \end{aligned}$$

(4.5)

where $g = f/F$. It follows that

$$\begin{aligned} \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx' \geq {}& \int _{\mathbb{H}} w(x) \bigl\vert DF(x) \bigr\vert ^{p} \bigl\vert g(x) \bigr\vert ^{p}\,dx' \\ & {}+ C_{p} \int _{\mathbb{H}} w(x) \bigl\vert Dg(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' \\ & {}- \int _{\mathbb{H}} \overline{D} \bigl(w(x)F(x) \bigl\vert DF(x) \bigr\vert ^{p-2}DF(x)\bigr) \bigl\vert g(x) \bigr\vert ^{p} \,dx' \\ \geq {}& C_{p} \int _{\mathbb{H}} w(x) \bigl\vert Dg(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' \\ & {}+ \int _{\mathbb{H}} - \overline{D}\bigl(w(x) \bigl\vert DF(x) \bigr\vert ^{p-2}DF(x)\bigr) F(x) \bigl\vert g(x) \bigr\vert ^{p}\,dx'. \end{aligned}$$

Using (4.2), this implies that

$$\begin{aligned}& \int _{\mathbb{H}} \eta (x) \bigl\vert g(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' + C_{p} \int _{ \mathbb{H}} w(x) \bigl\vert Dg(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' \\& \quad \leq \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx'. \end{aligned}$$

(4.6)

Since $g =f/F$, we arrive at

$$ \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{p}\,dx' + C_{p} \int _{\mathbb{H}} w(x) \bigl\vert F(x) \bigr\vert ^{p} \bigl\vert D(f/F) \bigr\vert ^{p}\,dx' \leq \int _{\Omega } w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx', $$

(4.7)

which proves (4.3). □

Remark 4.2

For $p=2$, we have equality in inequality (4.4) with $C_{2}=1$, that is, the above proof gives the following remainder formula:

$$\begin{aligned} \int _{\mathbb{H}} w(x) \bigl\vert F(x) \bigr\vert ^{2} \bigl\vert D(f/F) \bigr\vert ^{2}\,dx' = \int _{ \mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{2} \,dx' - \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{2}\,dx'. \end{aligned}$$

(4.8)

Remark 4.3

For $1< p<2$, inequality (4.4) can be stated as for all $a,b \in \mathbb{R}^{n}$, there exists a positive number $C_{p}$ such that

$$ \vert a \vert ^{p} +C_{p} \frac{ \vert b \vert ^{p}}{( \vert a \vert + \vert b \vert )^{2-p}} + p \vert a \vert ^{p-2}a\cdot b \leq \vert a+b \vert ^{p},\quad 1< p< 2. $$

(4.9)

In turn, from the proof it follows that

$$\begin{aligned} & \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx' \\ &\quad \geq \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{p}\,dx' \\ &\qquad {}+ C_{p} \int _{\mathbb{H}} w(x) \biggl( \biggl\vert \frac{f(x)}{F(x)}DF(x) \biggr\vert + F \biggl\vert D\frac{f(x)}{F(x)} \biggr\vert \biggr)^{p-2} \bigl\vert F(x) \bigr\vert ^{2} |D (f(x)/F(x)|^{2}\,dx' \end{aligned}$$

(4.10)

for all real-valued functions $f \in C_{0}^{\infty }(\mathbb{H})$.

Proposition 4.4

For $f \in C_{0}^{\infty }(\mathbb{H})$, we have

$$ \int _{\Omega } \frac{ \vert f(x) \vert ^{p}}{ \vert x \vert _{q}^{p}}\,dx'\leq \biggl( \frac{p}{\gamma -p-2} \biggr)^{p} \int _{\Omega } \bigl\vert Df(x) \bigr\vert ^{p} \,dx', $$

(4.11)

where $1< p<\gamma -2$ and $\gamma \leq 2+n$ with $\gamma \in \mathbb{R}$.

Proof of Proposition 4.4

In Theorem 4.1, we take $w=1$ and

$$ F_{\varepsilon } = \vert x_{\varepsilon } \vert _{q}^{-\frac{\gamma -p-2}{p}} = \bigl( (x_{1}+\varepsilon )^{2} + \cdots + (x_{n}+\varepsilon )^{2} \bigr)^{-\frac{\gamma -p-2}{2p}} $$

for a given $\varepsilon >0$. A direct computation gives

$$\begin{aligned}& D \vert x_{\varepsilon } \vert _{q}^{\alpha } = \alpha \vert x_{\varepsilon } \vert _{q}^{ \alpha -2}\sum _{i=1}^{n} e_{i}x_{i}, \\& \vert D \vert x_{\varepsilon }|_{q}^{\alpha } |_{q}^{p-2} = \vert \alpha \vert ^{p-2} \vert x_{ \varepsilon } \vert _{q}^{(\alpha -2)(p-2)+(p-2)}, \\& \overline{D} \bigl( \vert D \vert x_{\varepsilon }|_{q}^{\alpha } |_{q}^{p-2}D \vert x_{ \varepsilon }|_{q}^{\alpha } \bigr) =\overline{D} \Biggl(\alpha \vert \alpha \vert ^{p-2} \vert x_{ \varepsilon } \vert _{q}^{\alpha p - \alpha -p}\sum _{i=1}^{n}e_{i}x_{i}\Biggr) \\& \hphantom{\overline{D} \bigl( \vert D \vert x_{\varepsilon }|_{q}^{\alpha } |_{q}^{p-2}D \vert x_{ \varepsilon }|_{q}^{\alpha } \bigr) } = \alpha \vert \alpha \vert ^{p-2}(\alpha p- \alpha -p+n) \vert x_{\varepsilon } \vert _{q}^{ \alpha p -\alpha -p}. \end{aligned}$$

By taking $\alpha =-\frac{\gamma -p-2}{2p}$ we compute

$$\begin{aligned} - \Delta _{p,1} F_{\varepsilon } &=- \overline{D} \bigl( \vert DF_{ \varepsilon } \vert ^{p-2}DF_{\varepsilon } \bigr) \\ & =- \overline{D} \bigl( \vert D \vert x_{\varepsilon }|_{q}^{- \frac{\gamma -p-2}{p}}|_{q}^{p-2}D \vert x_{\varepsilon }|_{q}^{- \frac{\gamma -p-2}{p}} \bigr) \\ & = \frac{\gamma -p-2}{p} \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p-2} \biggl( \frac{\gamma -p-2}{p} - \gamma +2 +n \biggr) \vert x_{\varepsilon } \vert _{q}^{- \frac{(\gamma -p-2)(p-1)}{p}-p} \\ &= \biggl( \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p} + \frac{\gamma -p-2}{p} \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p-2}(- \gamma +2+n) \biggr) \vert x_{\varepsilon } \vert _{q}^{- \frac{(\gamma -p-2)(p-1)}{p}-p}. \end{aligned}$$

If $1< p<\gamma -2$ and $\gamma \leq 2+n$, then the above expression gives

$$ - \Delta _{p,1} F_{\varepsilon } \geq \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p} \frac{1}{ \vert x_{\varepsilon } \vert _{q}^{p}}F_{\varepsilon }^{p-1}, $$

(4.12)

that is, according to the assumption in Theorem 4.1, we can put

$$ \eta (x) = \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p} \frac{1}{ \vert x_{\varepsilon } \vert _{q}^{p}}, $$

(4.13)

which shows that (4.3) implies (4.11). □