Blow-up solutions to the Monge–Ampère equation with a gradient term: sharp conditions for the existence and asymptotic estimates

Abstract

We consider the boundary blow-up Monge–Ampère problem with a gradient term

$$\begin{aligned} M[u]=K(x)f(u)|\nabla u|^q \text{ for } x \in \Omega ,\; u(x)\rightarrow +\infty \text{ as } \mathrm{dist}(x,\partial \Omega )\rightarrow 0, \end{aligned}$$

where $M[u]=\det \, (u_{x_{i}x_{j}})$ is the Monge–Ampère operator, $q\ge 0$, $\Omega$ is a smooth, bounded, strictly convex domain in $\mathbb {R}^N \, (N\ge 2)$, and K, f are smooth positive functions. Under K and q satisfying suitable conditions, we first prove that the above boundary blow-up problem admits a strictly convex solution if and only if f satisfies a Keller–Osserman type condition. Then we show the asymptotic behavior of strictly convex solutions to the boundary blow-up Monge–Ampère problem under a new weaker condition than previous references. Finally, we also show the existence of strictly convex solutions under appropriate assumptions on K and q, without assuming that f satisfies a Keller–Osserman type condition. On the technical level, we adopt the sub-supersolution method and the Karamata regular variation theory.

Appendix

We present here some basic facts of Karamata regular variation theory, for example, see references [47, 48].

Definition AP.1

A positive measurable function f defined on $[A,\infty )$, for some $A>0$, is called regularly varying at infinity with index $\rho \in \mathbb {R}$, written $f \in RV_{\rho }$, if for all $\xi >0$,

$$\begin{aligned} \lim _{s\rightarrow \infty } \frac{f(\xi s)}{f(s)}=\xi ^\rho . \end{aligned}$$

(7.1)

In particular, when $\rho =0$, f is called slowly varying at infinity.

Clearly, if $f \in RV_\rho$, then $L(s)=\frac{f(s)}{s^\rho }$ is slowly varying at infinity.

Definition AP.$1'$. A positive measurable function f defined on (0, a), for some constant $a>0$, is called regularly varying at zero with index $\rho$, written $f \in RVZ_{\rho }$, if for each $\xi >0$ and some $\rho \in \mathbb {R}$,

$$\begin{aligned} \lim _{s\rightarrow 0^+} \frac{f(\xi s)}{f(s)}=\xi ^\rho . \end{aligned}$$

Clearly, if $f \in RVZ_\rho$, then $L(s)=\frac{f(s)}{s^\rho }$ is slowly varying at zero.

Definition AP.2

A positive measurable function f defined on $[A,\infty )$, for some $A>0$, is called rapidly varying at infinity if for each $\rho >1$

$$\begin{aligned} \lim _{s\rightarrow \infty }\frac{f(s)}{s^\rho }=\infty , \end{aligned}$$

(7.2)

or, equivalently, if for each $\xi >1$

$$\begin{aligned} \lim _{s\rightarrow \infty }\frac{f(\xi s)}{f(s)}=\infty . \end{aligned}$$

Definition AP$2'$. A positive measurable function f defined on (0, a), for some constant $a>0$, is called rapidly varying at zero,

$$\begin{aligned}&if \lim _{s\rightarrow 0^+}f(s)=\infty ,and\ for\ each\ \rho>1,\lim _{s\rightarrow 0^+}f(s)s^\rho =\infty \\&or\ if \lim _{s\rightarrow 0^+}f(s)=0,and\ for\ each\ \rho >1,\lim _{s\rightarrow 0^+}f(s)s^{-\rho }=0 \end{aligned}$$

In the following we only state the properties of regularly (slowly or rapidly) varying at infinity.

Proposition AP.1

(Uniform convergence theorem). If $f \in RV_\rho$, then (7.1) holds uniformly for $\xi \in [c_1,c_2]$ with $0<c_1<c_2$. Moreover, if $\rho < 0$, then uniform convergence holds on intervals $(c_{1}, \infty )$ with $c_{1} > 0$; if $\rho > 0$, then uniform convergence holds on intervals $(0, c_{2}]$ provided f is bounded on $(0, c_{2}]$ with $c_{2} > 0$.

Proposition AP.2

(Representation theorem). A function L is slowly varying at infinity if and only if it may be written in the form

$$\begin{aligned} L(s)=\psi (s)exp\left( \int _{A_1}^s\frac{y(\tau )}{\tau }d\tau \right) ,s\ge A_1, \end{aligned}$$

for some $A_1\ge A$, where the function $\psi$ and y are continuous and for $s \rightarrow \infty ,\ y(s)\rightarrow 0$ and $\psi (s) \rightarrow c_0$, with $c_0>0$.

We say that

$$\begin{aligned} \hat{L}(s)=c_0exp\left( \int _{A_1}^s\frac{y(\tau )}{\tau }d\tau \right) \end{aligned}$$

is normalized slowly varying at infinity and

$$\begin{aligned} f(s)=s^\rho \hat{L}(s),\ s\ge A_{1}, \end{aligned}$$

is normalized regularly varying at infinity with index $\rho$ and write $f\in NRV_\rho$.

Proposition AP.3

A function $f\in RV_\rho$ belongs to $NRV_\rho$ if and only if

$$\begin{aligned} f\in C^1[A_1,\infty ),\ for\ some\ A_1>0\ and\ \lim _{s\rightarrow \infty } \frac{sf'(s)}{f(s)}=\rho . \end{aligned}$$

Proposition AP.4

If the functions f, g, L are slowly varying at infinity,then

• (1) $f^p$ for every $p \in \mathbb {R}$, $c_1f+c_2g(c_1,c_2 \ge 0)$,$f\circ g(if\ g(s)\rightarrow 0\ as \ s \rightarrow 0^+)$ are also slowly varying at infinity.

• (2) For every $\rho >0$ and $s\rightarrow \infty$,

  $$\begin{aligned} s^{-\rho } L(s)\rightarrow 0,\ s^{\rho }L(s)\rightarrow \infty . \end{aligned}$$

• (3) For $\rho \in \mathbb {R}$ and $s\rightarrow \infty$,$\frac{\ln (L(s))}{\ln s} \rightarrow 0$ and $\frac{\ln (s^\rho L(s))}{\ln s} \rightarrow \rho$.

Proposition AP.5

If $f_1\in RV_{\rho _1}, f_2\in RV_{\rho _2}$, then $f_1f_2\in RV_{\rho _1+\rho _2}$ and $f_1 \circ f_2 \in RV_{\rho _1 \rho _2}$.

Proposition AP.6

(Asymptotic behavior) If a function L is slowly varying at infinity, then for $a\ge 0$ and $t \rightarrow \infty$,

• (1) $\int _a^{t}s^\rho L(s)ds \cong (1+\rho )^{-1}t^{1+\rho }L(t)$, for $\rho >-1$;

• (2) $\int _t^{\infty } s^\rho L(s)ds \cong (-1-\rho )^{-1}t^{1+\rho }L(t)$, for $\rho <-1$.

Remark AP.1

The result of Proposition AP.6 remains true for $\rho =-1$ in the sense that

$$\begin{aligned} \frac{\int _a^{t}s^{-1} L(s)ds}{L(t)}\rightarrow \infty \ \text {as}\ \ t\rightarrow \infty . \end{aligned}$$

When $\rho =-1$, let $z(s)=s^{-1}L(s)$.

Proposition AP.7

(Asymptotic behavior)(See Karamata’s Theorem 1.5.9b in [47].) If $z\in RV_{-1}$ and $\int _{s}^{\infty }z(\tau )d\tau <\infty , s>0,$ then $\int _{s}^{\infty }z(\tau )d\tau$ is slowly varying at infinity and

$$\begin{aligned} \lim \limits _{s\rightarrow \infty }\frac{sz(s)}{\int _{s}^{\infty }z(\tau )d\tau }=0. \end{aligned}$$

By (1.6) we have

$$\begin{aligned} \Psi '(s)=-[(N+1-q)F(s)]^{-\frac{1}{N+1-q}},\ \Psi ''(s)=[(N+1-q)F(s)]^{-\frac{1}{N+1-q}-1}f(s). \end{aligned}$$

It follows that

$$\begin{aligned} -\frac{1}{\Psi '(s)}=[(N+1-q)F(s)]^{\frac{1}{N+1-q}}. \end{aligned}$$

(7.3)

By the definition of I(s) we have

$$\begin{aligned} I(s)=\frac{f(s)\Psi (s)}{[(N+1-q)F(s)]^{\frac{N-q}{N+1-q}}}. \end{aligned}$$

Lemma 7.1

Let $0\le q<N+1.$ Suppose that f satisfies $\mathbf{(f)}$ and (1.6). Assume that $I_{0}, I_{\infty }$ exist. Then $I_{0}\ge 1, I_{\infty }\ge 1.$

Lemma 7.2

Let $0\le q<N+1.$ Suppose that f satisfies $\mathbf{(f)}$ and (1.6). Assume that $I_{\infty }$ exists. We have

• (1) $I_{\infty }\in (1,\infty )$ if and only if $F\in NRV_{p+1}$ with $p>N-q$;

• (2) If $I_{\infty }=1$, then F is rapidly varying at infinity;

• (3) If $F\in NRV_{N+1-q}$, then $I_{\infty }=\infty$.

Proof

(1) Necessity. By Lemma 7.1, we see that

$$\begin{aligned} \begin{array}{ll} \lim \limits _{s\rightarrow \infty }\frac{-\frac{1}{\Psi '(s)}}{s(-\frac{1}{\Psi '(s)})'}=-\lim \limits _{s\rightarrow \infty }\frac{\frac{1}{\Psi '(s)}\Psi (s)}{s\frac{\Psi ''(s)}{(\Psi '(s))^2}\Psi (s)}=-\frac{1}{I_{\infty }}\lim \limits _{s\rightarrow \infty }\frac{\frac{\Psi (s)}{\Psi '(s)}}{s}\\ =-\frac{1}{I_{\infty }}\lim \limits _{s\rightarrow \infty }\frac{(\Psi '(s))^2-\Psi (s)\Psi ''(s)}{(\Psi '(s))^2}=\frac{I_{\infty }-1}{I_{\infty }}. \end{array} \end{aligned}$$

(7.4)

By Proposition AP.3 we have $-\frac{1}{\Psi '(s)}\in NRV_{I_{\infty }/(I_{\infty }-1)}.$ By (7.3) $F\in NRV_{(N+1-q)I_{\infty }/(I_{\infty }-1)}$. Denote $(N+1-q)I_{\infty }/(I_{\infty }-1)$ by $p+1$, then $p+1>N+1-q$, i.e. $p>N-q$.

Sufficiency. If $F\in NRV_{p+1}$ with $p>N-q$, then $-\frac{1}{\Psi '}\in NRV_{(p+1)/(N+1-q)}$, and

$$\begin{aligned} \lim \limits _{s\rightarrow \infty }\frac{s(-\frac{1}{\Psi '(s)})'}{-\frac{1}{\Psi '(s)}}=\frac{p+1}{N+1-q},\ \ -\frac{1}{\Psi '(s)}=s^{\frac{p+1}{N+1-q}}\hat{L}(s), \ \ \forall s\ge S_{0}, \end{aligned}$$

where $S_{0}$ is large enough, $\hat{L}(s)$ is normalized slowly varying at infinity. By Proposition AP.6,

$$\begin{aligned} \begin{array}{ll} \lim \limits _{s\rightarrow \infty }(-\frac{1}{\Psi '(s)})'\Psi (s)\\ =\lim \limits _{s\rightarrow \infty }\frac{s(-\frac{1}{\Psi '(s)})'}{-\frac{1}{\Psi '(s)}}\lim \limits _{s\rightarrow \infty }\frac{-\frac{1}{\Psi '(s)}}{s}\Psi (s)\\ =\frac{p+1}{N+1-q}\lim \limits _{s\rightarrow \infty }s^{\frac{p-N}{N+1-q}}\hat{L}(s)\int _{s}^{+\infty }\tau ^{-\frac{p+1}{N+1-q}}(\hat{L}(\tau ))^{-1}d\tau \\ =\frac{p+1}{N+1-q}\lim \limits _{s\rightarrow \infty }s^{\frac{p-N}{N+1-q}}\hat{L}(s)(\frac{p+1}{N+1-q}-1)^{-1}s^{1-\frac{p+1}{N+1-q}}(\hat{L}(s))^{-1}. \end{array} \end{aligned}$$

It follows that $I_{\infty }=\frac{p+1}{p+q-N}>1$.

In this case,

$$\begin{aligned} F(s)=s^{p+1}\hat{L}(s), \forall s\ge S_{0}. \end{aligned}$$

Then by Proposition AP.2 we obtain

$$\begin{aligned} f(s)=s^p[(p+1)+y(s)]\hat{L}(s)\in RV_{p}, \forall s\ge S_{0}, \end{aligned}$$

where $\lim \limits _{s\rightarrow \infty }y(s)=0$.

(2) If $I_{\infty }=1$, by (7.4) we see

$$\begin{aligned} \lim \limits _{s\rightarrow \infty }\frac{s(-\frac{1}{\Psi '(s)})'}{-\frac{1}{\Psi '(s)}}=\infty . \end{aligned}$$

Let

$$\begin{aligned} \frac{s(-\frac{1}{\Psi '(s)})'}{-\frac{1}{\Psi '(s)}}:=y(s),\forall s>0, \end{aligned}$$

i.e.

$$\begin{aligned} \frac{(-\frac{1}{\Psi '(s)})'}{-\frac{1}{\Psi '(s)}}=\frac{y(s)}{s},\forall s>0. \end{aligned}$$

(7.5)

Integrating (7.5) from $S_{0}$ to s, we have

$$\begin{aligned} -\frac{1}{\Psi '(s)}={\bar{c}}_{1}\text {exp}\ \bigg (\int _{S_{0}}^{s}\frac{y(\tau )}{\tau }d\tau \bigg ),\ s>S_{0}, \end{aligned}$$

where ${\bar{c}}_{1}=-\frac{1}{\Psi '(S_{0})}.$

Since $\lim \limits _{s\rightarrow \infty }y(s)=\infty ,$ we have that for each $\xi >1,$

$$\begin{aligned} -\frac{1}{\Psi '(\xi s)}\bigg /-\frac{1}{\Psi '(s)}=\text {exp}\ \bigg (\int _{s}^{\xi s} \frac{y(\tau )}{\tau }d\tau \bigg )=\text {exp}\ \bigg (\int _{1}^{\xi }\frac{y(sv)}{v}dv\bigg )\rightarrow \infty \ \text {as}\ s\rightarrow \infty . \end{aligned}$$

(7.6)

Then $-\frac{1}{\Psi '(s)}$ is rapidly varying at infinity. It follows that F is rapidly varying at infinity.

(3) If $F\in NRV_{N+1-q}$, by Proposition AP.3 we see that $-\frac{1}{\Psi '(s)}\in NRV_{1}, -\Psi '(s)\in NRV_{-1}$. Then

$$\begin{aligned} \lim \limits _{s\rightarrow \infty }\frac{s(-\frac{1}{\Psi '(s)})'}{-\frac{1}{\Psi '(s)}}=1. \end{aligned}$$

It follows from Proposition AP.7 that

$$\begin{aligned} \lim \limits _{s\rightarrow \infty }\frac{s\Psi '(s)}{\int _{s}^{\infty }\Psi '(\tau )d\tau }=0. \end{aligned}$$

Thus

$$\begin{aligned} I_{\infty }=\lim \limits _{s\rightarrow \infty }\frac{\Psi ''(s)\Psi (s)}{(\Psi '(s))^2} =\lim \limits _{s\rightarrow \infty }\frac{s(\frac{1}{\Psi '(s)})'}{\frac{1}{\Psi '(s)}} \frac{\int _{s}^{\infty }(-\Psi '(\tau ))d\tau }{-s\Psi '(s)}=\infty . \ \ \ \end{aligned}$$

$\square$

Similarly we have

Lemma 7.3

Let $0\le q<N+1.$ Suppose that f satisfies $\mathbf{(f)}$ and (1.6). Assume that $I_{0}$ exists. We have

• (1) $I_{0}\in (1,\infty )$ if and only if $F\in NRVZ_{p+1}$ with $p>N-q$;

• (2) If $I_{0}=1$, then F is rapidly varying at zero;

• (3) If $F\in NRVZ_{N+1-q}$, then $I_{0}=\infty$.

Lemma 7.4

Let $0\le q<N.$ Suppose f be a positive measurable function defined on $(0,a_{1})$ for some $a_{1}>0$ and $\lim \limits _{s\rightarrow 0^{+}}f(s)=0$.

• (1) If f is rapidly varying at 0 or $f\in RVZ_{p}$ with $p>N-q$, then (1.7) holds;

• (2) If f is slow varying at 0 or $f\in RVZ_{p}$ with $p<N-q$, then (1.7) does not hold.

Proof

(1) If f is rapidly varying at 0, then $\lim \limits _{s\rightarrow 0^+}\frac{f(s)}{s^{N-q}}=0.$ Then there exists $\delta >0$ such that

$$\begin{aligned} f(s)<s^{N-q}, \ \forall \ 0<s<\delta . \end{aligned}$$

It follows that

$$\begin{aligned} F(s)<\frac{1}{N+1-q}s^{N+1-q}, \forall 0<s<\delta . \end{aligned}$$

We can see

$$\begin{aligned} \int _{0}^{\delta }F(s)^{-\frac{1}{N+1-q}}ds=\infty . \end{aligned}$$

Then (1.7) holds.

If $f\in RVZ_{p}$ with $p>N-q$, then $F\in RVZ_{p+1}$ with $p+1>N+1-q$. It follows that $F(s)=s^{p+1}L(s)$, where L(s) is slowly varying at 0. By Proposition AP.6 we have

$$\begin{aligned} \begin{array}{l} \int _{t}^{\infty }[(N+1-q)F(s)]^{-\frac{1}{N+1-q}}ds=\int _{t}^{\infty }(N+1-q)^{-\frac{1}{N+1-q}}s^{-\frac{p+1}{N+1-q}}(L(s))^{-\frac{1}{N+1-q}}ds\\ =(N+1-q)^{-\frac{1}{N+1-q}}(\frac{q+p-N}{N+1-q})^{-1}t^{\frac{N-q-p}{N+1-q}}(L(t))^{-\frac{1}{N+1-q}}\rightarrow \infty \ \text {as}\ t\rightarrow 0. \end{array} \end{aligned}$$

Then (1.7) holds.

(2) It is similar to the proof above. So we omit it. $\square$

Lemma 7.5

• (1) Suppose that $f\in RVZ_p,p>N-q$ or f is rapidly varying at zero. Then

  $$\begin{aligned} \int _{0^+}[f(\tau )]^{-1/N-q}d\tau =\infty \ \text {and}\ \int _{0^+} [F(\tau )]^{-1/(N+1-q)}d\tau =\infty . \end{aligned}$$

• (2) Suppose that $f\in RVZ_p(p<N-q)$ or f is slowly varying at zero. Then

  $$\begin{aligned} \int _{0^+} [f(\tau )]^{-1/N-q}d\tau<\infty \ \text {and}\ \int _{0^+} [F(\tau )]^{-1/(N+1-q)}d\tau <\infty . \end{aligned}$$

• (3) Suppose that $f\in RVZ_{N-q}$. Then

  $$\begin{aligned} \int _{0^+} [f(\tau )]^{-1/N-q}d\tau =\infty \ \text {implies}\ \int _{0^+} [F(\tau )]^{-1/(N+1-q)}d\tau =\infty , \end{aligned}$$

  however, it is not applicable in reverse.

Proof

Letting $f\in RVZ_p(p> N-q)$, then

$$\begin{aligned} F\in RVZ_{p+1}, f(s)=s^pL(s),F(s)=s^{p+1}L(s), \end{aligned}$$

where L(s) is slowly varying at infinity. From Proposition AP.6 we derive

$$\begin{aligned} \int _{0^+} [F(\tau )]^{-1/(N+1-q)}d\tau =\infty ,\int _{0^+}[f(\tau )]^{-1/(N-q)}d\tau =\infty . \end{aligned}$$

Suppose that f is rapidly varying at infinity. Then for $\rho >N-q$ we have

$$\begin{aligned} \lim \limits _{s\rightarrow 0^+}\frac{f(s)}{s^{\rho }}=0. \end{aligned}$$

It so follows that there exist $\varepsilon>0,\ S>0$ so that $f(s)<\varepsilon s^{\rho }$ for $0<s<S$. This shows that

$$\begin{aligned} f(s)^{-\frac{1}{N-q}}>[\varepsilon s^{\rho }]^{-\frac{1}{N-q}}. \end{aligned}$$

By integrating from $\epsilon$ to $t\ (t<S)$, we derive

$$\begin{aligned} \int _{\epsilon }^{t}[f(\tau )]^{-1/(N-q)}d\tau \rightarrow \infty \ (\epsilon \rightarrow 0). \end{aligned}$$

Similarly one can prove that

$$\begin{aligned} \int _{0^{+}} [F(\tau )]^{-1/(N+1-q)}d\tau =\infty . \end{aligned}$$

For $f\in RVZ_p\ (p<N-q)$, from Proposition AP.6, we get

$$\begin{aligned} \int _{0^+}[F(\tau )]^{-1/(N+1-q)}d\tau<\infty ,\ \ \int _{0^+} [f(\tau )]^{-1/(N-q)}d\tau <\infty . \end{aligned}$$

On the other hand, if f is slowly varying at infinity, then by Proposition AP.4, we have for $\rho <N-q$

$$\begin{aligned} \lim \limits _{s\rightarrow 0^+}\frac{f(s)}{s^\rho }=\infty . \end{aligned}$$

It hence follows that there exist $M>0,\ S>0$ so that $f(s)>Ms^\rho$ for $0<s<S$, which shows that $f(s)^{-\frac{1}{N-q}}<[Ms^\rho ]^{-\frac{1}{N-q}}$. By integrating from 0 to $t\ (t>S)$, we derive that

$$\begin{aligned} \int _{0}^t [f(\tau )]^{-1/(N-q)}d\tau <\infty . \end{aligned}$$

Similarly one can get

$$\begin{aligned} \int _{0}^t [F(\tau )]^{-1/(N+1-q)}d\tau <\infty . \end{aligned}$$

Assume that

$$\begin{aligned} f(s)=(N+1-q)s^{N-q}L(s)+s^{N+1-q}L'(s) \end{aligned}$$

satisfying

$$\begin{aligned} \int _{0^{+}}[f(\tau )]^{-1/(N-q)}d\tau =\infty , \end{aligned}$$

where $L(s)\in C^1(0,a)(a>0)$ is normalised slowly varying at infinity. Then $f\in RVZ_{N-q}$ and $F(s)=s^{N+1-q}L(s)$.

Letting $\lim \limits _{s\rightarrow 0^+}L(s)=c\ge 0$, then we get that

$$\begin{aligned} \int _{0^+} [F(\tau )]^{-1/(N+1-q)}d\tau =\int _{0^+}\tau ^{-1}(L(\tau ))^{-1}d\tau >M\int _{0^+}\tau ^{-1}d\tau =\infty . \end{aligned}$$

Letting $\lim \limits _{s\rightarrow 0^+}L(s)=\infty$, then we obtain that

$$\begin{aligned} \lim \limits _{s\rightarrow 0^+}\frac{[F(s)]^{\frac{N-q}{N+1-q}}}{f(s)}&=\lim \limits _{s\rightarrow 0^+}\frac{[L(s)]^{\frac{N-q}{N+1-q}}}{(N+1-q)L(s)+sL'(s)}\\&=\lim \limits _{s\rightarrow 0^+}\frac{1}{(N+1-q)[L(s)]^{\frac{1}{N+1-q}}+\frac{sL'(s)}{[L(s)]^{\frac{N-q}{N+1-q}}}}. \end{aligned}$$

From the definition of normalised slowly varying function, we derive that

$$\begin{aligned} L(s)=c_0exp\left( \int _{A_1}^s\frac{y(\tau )}{\tau }d\tau \right) . \end{aligned}$$

It so yields that

$$\begin{aligned} \lim \limits _{s\rightarrow 0^+}\frac{sL'(s)}{[L(s)]^{\frac{N-q}{N+1-q}}}=0. \end{aligned}$$

We thus derive

$$\begin{aligned} \lim \limits _{s\rightarrow 0^+}\frac{[F(s)]^{\frac{N-q}{N+1-q}}}{f(s)}=0. \end{aligned}$$

It hence follows that $\int _{0^+} [F(\tau )]^{-1/(N+1-q)}d\tau =\infty .$

Conversely, letting

$$\begin{aligned} F(s)=s^{N+1-q}(\ln \frac{1}{s})^\beta ,N-q<\beta \le N+1-q, \end{aligned}$$

then by Remark AP.1 we get that

$$\begin{aligned} \int _{0^+} [F(\tau )]^{-1/(N+1-q)}d\tau =\infty ,\int _{0^+} [f(\tau )]^{-1/(N-q)}d\tau <\infty . \end{aligned}$$

The proof of Lemma 7.5 is finished. $\square$

Remark AP.2

From Lemma 7.5 we see that (1.7) is a weaker condition than (1.8) in this kind of classification. But we can not conclude whether or not it is true for arbitrary f.