Singular solutions of elliptic equations with iterated exponentials

TL;DR

This paper constructs positive singular solutions for a class of elliptic equations involving iterated exponentials, extending previous work to more complex nonlinearities and analyzing their behavior near singularities.

Contribution

It introduces a method to construct and analyze singular solutions for elliptic equations with iterated exponential nonlinearities, expanding the understanding of such equations.

Findings

Constructed positive singular solutions with prescribed behavior.

Extended previous results to iterated exponential nonlinearities.

Provided a framework for analyzing solutions near singularities.

Abstract

We construct positive singular solutions for the problem $-\Delta u=\lambda \exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$, having a prescribed behaviour around the origin. Our study extends the one in Y. Miyamoto [Y. Miyamoto, A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth. J. Differential Equations \textbf{264} (2018), 2684--2707] for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials.

Equations

$$\{\begin{array}{rlrl}-\Delta u& =\lambda \exp (e^u)& & \text{mbox}in{B}_1\setminus \{0\},\\ u& =0& & \text{mbox}on\partial B_1,\end{array}$$

$$\exp (e^u)=e^{e^u}.$$

$$\{\begin{array}{rlrl}-\Delta u& =\lambda e^u& & \text{mbox}in{B}_1\setminus \{0\},\\ u& =0& & \text{mbox}on\partial B_1,\end{array}$$

$$u^{*}\Big{(}\frac{x}{\sqrt{\lambda^{*}}}\Big{)}=\ln\left[2\ln\frac{1}{|x|}+\ln\frac{n-2}{\ln\frac{1}{|x|}}+\ln\left(1+\frac{\ln\ln\frac{1}{|x|}}{2\ln\frac{1}{|x|}}\right)\right]+O\Big{(}\ln^{-2}\frac{1}{|x|}\Big{)},$$

$$\frac{1}{\sqrt{\lambda^{*}}}\big{|}\nabla u^{*}\big{|}\Big{(}\frac{x}{\sqrt{\lambda^{*}}}\Big{)}=\frac{1}{|x|\ln\frac{1}{|x|}}+\frac{\ln\Big{(}\ln\frac{1}{|x|}\Big{)}}{2|x|\ln^{2}\frac{1}{|x|}}+O\Big{(}\frac{1}{|x|\ln^{2}\frac{1}{|x|}}\Big{)}.$$

$$u^{*}\Big{(}\frac{x}{\sqrt{\lambda^{*}}}\Big{)}=\ln(2\rho)+\frac{1}{2\rho}\ln\frac{n-2}{\rho}-\frac{1}{8\rho^{2}}\ln^{2}\frac{1}{\rho}+\frac{1}{4\rho^{2}}\ln\rho+O\Big{(}\frac{1}{\rho^{2}}\Big{)}$$

$$\{\begin{array}{rlrl}-\Delta u& =\lambda \exp (u^p)& & \text{mbox}in{B}_1\setminus \{0\},\\ u& =0& & \text{mbox}on\partial B_1,\end{array}$$

$$u^{*}\Big{(}\frac{x}{\sqrt{\lambda^{*}}}\Big{)}=\left(2\ln\frac{1}{|x|}-\big{(}1-\frac{1}{p}\big{)}\ln\ln\frac{1}{|x|}\right)^{1/p}+o\Big{(}\ln^{-1+\frac{1}{p}}\frac{1}{|x|}\Big{)}\mbox{ as }|x|\to 0.$$

$$\{\begin{array}{rlrl}-\Delta U& =\exp (e^U)& & \text{mbox}in{B}_R\setminus \{0\},R>0,\\ U& =0& & \text{mbox}on\partial B_R,\end{array}$$

$$U(r)=F^{-1}\Big{(}\frac{r^{2}}{2(n-2)}(1+o(1))\Big{)}\quad\mbox{ as }r\to 0,$$

$$F(t)={\int }_t^{\infty }\exp (-e^s)ds.$$

$$\{\begin{array}{rlrl}-\Delta u& =\lambda \exp (e^u)& & \text{mbox}in{B}_1,\\ u& >0& & \text{mbox}in{B}_1,\\ u& =0& & \text{mbox}on\partial B_1.\end{array}$$

$$\{\begin{array}{rlrl}-\Delta u& =\lambda \exp (G_m(u))& & \text{mbox}in{B}_1\setminus \{0\},\\ u& =0& & \text{mbox}on\partial B_1.\end{array}$$

$$\begin{split}u^{*}\Big{(}\frac{x}{\sqrt{\lambda^{*}}}\Big{)}=H_{m}(2\rho)+H^{\prime}_{m}(2\rho)\left(\ln(2(n-2))-\sum_{j=1}^{m}H_{j}(2\rho)\right)+\\ -\frac{1}{4\rho}H^{\prime}_{m}(2\rho)(\ln\rho)^{2}+O\Big{(}\rho^{2}\Big{)},\end{split}$$

$$\frac{1}{\sqrt{\lambda^{*}}}\big{|}\nabla u^{*}\big{|}\Big{(}\frac{x}{\sqrt{\lambda^{*}}}\Big{)}=2\frac{H^{\prime}_{m}(2\ln(\frac{1}{|x|}))}{|x|}+O\Big{(}\frac{1}{|x|\ln^{2}(\frac{1}{|x|})}\Big{)}.$$

$$\{\begin{array}{rl}& v_{rr}+\frac{n-1}{r}{v}_r+\exp (e^v)=0\text{mbox}forall0<r<\sqrt{\lambda },\\ & v(\sqrt{\lambda })=0.\end{array}$$

$$\{\begin{array}{rl}& w_{tt}-(n-2)w_t+\exp (-2t+e^w)=0,w>0\text{mbox}forall-\ln \sqrt{\lambda }<t<\infty ,\\ & w(-\ln \sqrt{\lambda })=0.\end{array}$$

$$w(t)=\ln\big{(}2t+\varphi(t)\big{)}+\eta(t)$$

$$\varphi(t)=\ln\frac{n-2}{t}+\ln\Big{(}1+\frac{\ln t}{2t}\Big{)}.$$

$${\phi }_t(t)$$

$${\phi }_{tt}(t)$$

$$f_t$$

$$f_{tt}$$

$$e^{\phi }-(n-2)f_t$$

$$=\frac{n-2}{t}\cdot \frac{2t(\phi +\ln t)-2t^2{\phi }_t+\phi \ln t}{2t(2t+\phi )}.$$

$$\phi (t)+\ln t=O(1)\text{mbox}ast\to \infty ,$$

$$e^{\varphi}-(n-2)f_{t}=O\Big{(}\frac{1}{t^{2}}\Big{)}\quad\mbox{ as }t\to\infty.$$

$$\exp (-2t+e^w)=$$

$$=$$

$$+e^{\phi }(e^{\eta }-1)(2t+\phi )+e^{\phi }$$

$$=$$

$$+e^{\phi }(e^{\eta }-\eta -1)(2t+\phi )+e^{\phi }\eta (2t+\phi )+e^{\phi },$$

$$\exp (-2t+e^w)=F_1(t)\eta +F_2(t,\eta )+F_3(t,\eta )+2(n-2)\eta +e^{\phi },$$

Full text

1. Introduction and the main results

Consider the problem

[TABLE]

where $B_{1}\subset\mathbb{R}^{n}$ $(n\geq 3)$ is the open unit ball, $\lambda>0$ is a real number and

[TABLE]

The related problem, also known as the Gelfand problem, namely

[TABLE]

has been long investigated starting with J. Liouville since 1853 (see [16]). One particular feature of (1.2) is that for $\lambda=2(n-2)>0$ it has the explicit singular solution $u_{s}(x)=-2\ln|x|$. Joseph and Lundgren [14] completely determined the structure of the radial solutions of (1.2) emphasizing the role of the singular solution $u_{s}$ in the global picture of the solution set to (1.2). Thanks to the standard Hardy inequality, the explicit singular solution $u_{s}$ is stable for all space dimensions $n\geq 10$. Further studies related to (1.2) are contained in [4, 11, 20, 17, 18, 21] and in the monograph [6]. Problems with exponential nonlinearities also appear in other contexts involving higher order operators [1, 2, 3, 7], $p$-Laplace operators [8] or $k$-Hessian operators [12, 13] or even systems of coupled equations [5, 10].

Returning to (1.1) we point out that such a problem does not possess an explicit singular solution. However, we are able to construct a radial singular solution $u^{*}$ with a prescribed behaviour around the origin. We prove:

Theorem 1.1**.**

There exists a unique $\lambda^{*}>0$ such that (1.1) has a singular solution $u^{*}$ such that, as $|x|\to 0$ we have

[TABLE]

and

[TABLE]

Letting $\rho=\ln\frac{1}{|x|}$ and using the Maclaurin series approximation we may re-write (1.3) as

[TABLE]

as $\rho\to\infty$.

The related problem

[TABLE]

was recently studied in [15]. It is proved in [15] that (1.6) has a singular solution $(\lambda^{*},u^{*})$ that satisfies

[TABLE]

Also the Morse index of $u^{*}$ is infinite (resp finite) provided $3\leq n\leq 9$ (resp. $n\geq 11$).

We would like to point out that in [19] a positive radial singular solution $U$ of

[TABLE]

is constructed. Such a singular solution $U$ has the property that

[TABLE]

where

[TABLE]

We are able to show that the solution $U(r)$ of (1.7) coincides with $u^{*}(\sqrt{\lambda^{*}}r)$ in a neighbourhood of the origin. Thus, we may further investigate the bifurcation problem

[TABLE]

By the classical result of Gidas, Ni and Nirenberg [9] all solutions of (1.10) are radially symmetric. Furthermore (see [19]) the solution set of (1.10) can be described as $\{(\lambda(\rho),u(\rho))\}$ where $\rho=\|u(\rho)\|_{L^{\infty}(B_{1})}$ and $\lambda(0)=0$. Hence, the solution set $(\lambda,u)$ is a curve emanating from $(\lambda,u)=(0,0)$. Using [19, Theorem 1.1, Corollary 1.2, Corollary 1.3] we have:

Theorem 1.2**.**

Let $u^{*}$ be the solution obtained in Theorem 1.1.

1. (i)

If $n\geq 11$, then the Morse index of $u^{*}$ is finite; 2. (ii)

If $3\leq n\leq 9$, then the Morse index of $u^{*}$ is infinite. Furthermore:

1. (ii1)

The curve $(\lambda(\rho),u(\rho))$ has infinitely many turning points around $\lambda^{*}$. In particular problem (1.10) has infinitely many solutions for $\lambda=\lambda^{*}$; 2. (ii2)

The number of intersection points between $u(\rho)$ and the singular solution $u^{*}$ tends to infinity as $\rho\to\infty$.

We also address in this article the similar problem with iterared exponential, that reads for $m\geq 2$ and $G_{0}(y)=y$ and $G_{m}(y)=\exp(G_{m-1}(y))$

[TABLE]

For problem (1.11) we prove

Theorem 1.3**.**

Let $m\geq 2$ and let $H_{m}(y)=\ln(H_{m-1}(y))$ be the iterated logarithm ($H_{0}(y)=y$). There exists a unique $\lambda^{*}>0$ such that (1.11) has a singular solution $u^{*}$ such that, as $|x|\to 0$ we have, for $\rho=\ln\frac{1}{|x|}$,

[TABLE]

and

[TABLE]

The next sections contain the proofs of the main results. Throughout this paper for any functions $f(t)$, $g(t)$ defined in a neighbourhood of infinity, we use the notation $f(t)=O(g(t))$ (resp. $f(t)=o(g(t))$ as $t\to\infty$ to indicate that $|\frac{f(t)}{g(t)}|$ is bounded (resp. tends to zero) as $t\to\infty$. A similar notation is used for $t\to 0$. Also, the symbols $C$, $c$ stand for generic positive constants whose values may be different on each occurence.

2. Proof of Theorem 1.1

Let $u$ be a radial solution of (1.1). Letting v(x)=u\big{(}\frac{x}{\sqrt{\lambda}}\big{)} we find

[TABLE]

Letting $t=-\ln r$ and $w(t)=v(r)$ we find that $w\in C^{2}(-\ln\sqrt{\lambda},\infty)$ satisfies

[TABLE]

We next look for a solution $w(t)$ of (2.2) in the form

[TABLE]

where

[TABLE]

Let us observe that, as $t\to\infty$, we have

[TABLE]

Letting f=\ln\big{(}2t+\varphi(t)\big{)} we have

[TABLE]

Also,

[TABLE]

Observe that

[TABLE]

so by (2.4) and the above calculations we find

[TABLE]

Using equation (2.3) we have

[TABLE]

so

[TABLE]

where

[TABLE]

Let also $F_{0}(t)=e^{\varphi}+f_{tt}-(n-2)f_{t}$.

Using the first equation of (2.2) together with (2.3), (2.8)-(2.11) we deduce that $\eta$ satisfies

[TABLE]

where

[TABLE]

We shall show that equation (2.12) has a solution $\eta\in X_{T}$ where $T>0$ is a real number and

[TABLE]

equipped with the norm $\|\eta\|_{X_{T}}=\sup_{t>T}t^{2}|\eta(t)|$. As in [15], we discuss in the following the case $3\leq n\leq 9$, the case $n\geq 10$ being similar. We transform (2.12) into the fixed point equation

[TABLE]

where the integral operator $\Psi[\eta]$ is given by

[TABLE]

and $\mu=\sqrt{(n-2)(10-n)}\geq 1$. The existence of a solution to (2.14) will be derived by means of the contraction principle; to this end, for $M>0$ set

[TABLE]

Lemma 2.1**.**

There exist $M,T>0$ such that $\Psi(\Sigma_{M})\subset\Sigma_{M}$ and $\Psi:\Sigma_{M}\to\Sigma_{M}$ is a contraction.

Proof.

Using (2.6) and (2.7) we have

[TABLE]

where $A>0$. Let now $\eta\in\Sigma_{M}$. Then, for $t>0$ large, we estimate

[TABLE]

Thus, by taking $M=2A$ and $T>0$ large enough we have

[TABLE]

Using this fact we have

[TABLE]

since $(n-2)\mu\geq 2$. This shows that $\Psi(\Sigma_{M})\subset\Sigma_{M}$.

To prove that $\Psi:\Sigma_{M}\to\Sigma_{M}$ is a contraction, let $\eta_{1},\eta_{2}\in\Sigma_{M}$. Then

[TABLE]

From (2.11) we have

[TABLE]

Using the Mean Value Theorem and (2.10) we estimate

[TABLE]

for some $\eta(s)$ between $\eta_{1}(s)$ and $\eta_{2}(s)$. Thus,

[TABLE]

Also, leting $\theta_{j}(s)=(e^{\eta_{j}(s)}-1)(2s+\eta_{j}(s))$, by (2.9) and the Mean Value Theorem we estimate

[TABLE]

for some $\theta(s)$ between $\theta_{1}(s)$ and $\theta_{2}(s)$. By the Mean Value Theorem we further estimate

[TABLE]

where $\eta(s)$ lies between $\eta_{1}(s)$ and $\eta_{2}(s)$. Hence,

[TABLE]

Now, using the above estimates in (2.17) we deduce

[TABLE]

By taking now $T>0$ large enough it follows that $\Psi:\Sigma_{M}\to\Sigma_{M}$ is a contraction. ∎

We are now in a position to prove the result in Theorem 1.1. First, there exists $\eta\in\Sigma_{M}$ a a solution of (2.14), that is, $\eta\in C^{2}(T,\infty)$ satisfies

[TABLE]

Thus, the function $w$ given by (2.3) is positive in a neighbourhood of infinity and satisfies

[TABLE]

We claim that $w$ vanish at some point $T_{0}>T$. Otherwise, $w>0$ in $(T,\infty)$ and by the continuation principle $w$ satisfies (2.19) on the whole real line. We claim that this implies that $w$ is monotone increasing. Indeed, assuming the contrary, there would exist a minimum point $t_{0}\in\mathbb{R}$ at which $w_{t}(t_{0})=0$ and $w_{tt}(t_{0})\geq 0$ which contradicts (2.19). Hence, $w$ is monotone increasing and there exists $L:=\lim_{t\to-\infty}w(t)\in[0,\infty)$.

Multiply in (2.19) by $e^{-(n-2)t}$ and using the fact that $w$ in increasing, we find

[TABLE]

Integrating in the above equality over the interval $[t,t_{0}]$, $-\infty<t<t_{0}$, we find

[TABLE]

where $C=\frac{1}{n}\exp(e^{L})>0$. This implies further that $\lim_{t\to-\infty}w_{t}(t)=\infty$ which contradicts the fact that $\lim_{t\to-\infty}w(t)$ is finite. This shows that $w$ vanishes at some point $T^{*}\in\mathbb{R}$ and $w\in C^{2}(T^{*},\infty)$ satisfies

[TABLE]

Letting $\lambda^{*}=e^{-2T^{*}}$ and u^{*}(x)=v^{*}\big{(}\sqrt{\lambda^{*}}x\big{)} (where $v$ is the solution of (2.1) with $\lambda=\lambda^{*}$) we obtain that $u^{*}$ is a solution of (1.1) with $\lambda=\lambda^{*}$ which satisfies (1.3). Concerning the proof of the asymptotic behaviour in (1.5) we have

[TABLE]

as $t\to\infty$. Since

[TABLE]

we have

[TABLE]

Using (2.22)-(2.23) in (2.21) we obtain

[TABLE]

which proves (1.5).

We next focus on the expansion of the gradient $\nabla u^{*}$ around the origin.

Lemma 2.2**.**

The solution $\eta$ of (2.18) satisfies

[TABLE]

Proof.

From the proof of Lemma 2.1 we infer that

[TABLE]

Then $\eta_{t}$ is solution to the integral equation

[TABLE]

The result follows promptly. ∎

Now, from (2.3), (2.7) and (2.24) we have

[TABLE]

Recall that the singular solution $u^{*}$ is given by u^{*}\big{(}\frac{r}{\sqrt{\lambda^{*}}}\big{)}=v(r)=w(t) where $v$ and $w$ are solutions of (2.1) and (2.2) (with $\lambda=\lambda^{*}$). Since $|w_{t}|=|v_{r}||\frac{dr}{dt}|=r|v_{r}|$, from (2.25) we find

[TABLE]

3. Proof of Theorem 1.2

Let us recall that the solution $U$ of (1.7) is obtained in [19, Lemma 3.1] as

[TABLE]

Further, the unknowns $x(t)$ and $y(t)=\frac{dx}{dt}(t)$ are found as a unique fixed point through a contraction mapping in the set

[TABLE]

From (3.1) we find

[TABLE]

Let now $u^{*}$ be the singular solution of (1.1) for $\lambda^{*}$ constructed in Theorem 1.1. Let v^{*}(x)=u^{*}\big{(}\frac{x}{\sqrt{\lambda^{*}}}\big{)} which satisfies (2.1), that is the same equation as (1.7) in $B_{\sqrt{\lambda^{*}}}$. Define

[TABLE]

In order to prove that $v^{*}\equiv U$ in a neighbourhood of infinity it is enough to show that $(x^{*}(t),y^{*}(t))$ belongs to $B_{\varepsilon}$.

Letting $w^{*}(t)=v^{*}(r)$ where $t=-\ln r$, we have that $w^{*}$ satisfies (2.2). Thus, (3.3) reads

[TABLE]

In order to conclude the proof it suffices to show that

[TABLE]

Then, for large $T>0$ we have $(x^{*},y^{*})\in B_{\varepsilon}$ so, by the uniqueness of the fixed point $(x^{*},y^{*})=(x,y)$ in a neighbourhood of infinity, that is $v^{*}\equiv U$ in a neighbourhood of the origin and then we conclude from Theorem 1.1, Corollary 1.2, Corollary 1.3 from [19]. Let us now turn to (3.5).

From the definition of $F$ in (1.8) and L’Hospital’s rule we find

[TABLE]

Thus,

[TABLE]

Now, using (2.3) we find

[TABLE]

Since $\eta(t)=O(t^{-2})$ as $t\to\infty$ it follows that

[TABLE]

which proves the first part of (3.5). For the second part, we first note that

[TABLE]

Thus, from the above arguments we find

[TABLE]

Using (2.3) and Lemma 2.2 we have

[TABLE]

We also have

[TABLE]

Since $\eta(t)=O(t^{-2})$ as $t\to\infty$, we find

[TABLE]

Combining (3.7), (3.8) and (3.9) we deduce $\lim_{t\to\infty}y^{*}(t)=0$ which finishes our proof. ∎

4. Proof of Theorem 1.3

We follow the lines of the proof of Theorem 1.1. For $m\geq 2$, we plan to solve here the equation

[TABLE]

We look for a solution $w(t)$ of (4.1) in the form

[TABLE]

where

[TABLE]

Here $H_{m}$ is the iterated logarithm function defined by $H_{m}(G_{m}(y))=y$. We then have to solve the equation

[TABLE]

where

[TABLE]

with

[TABLE]

Here $\rho(\eta)$ is defined thanks to Taylor’s formula as

[TABLE]

The next result provides the estimates we need to construct our solution $\eta$ in the space $\Sigma_{M}$ defined in (2.16).

Lemma 4.1**.**

Let $m\geq 2$.

1. (a)

For $k=1,2,3$ we have

[TABLE]

and

[TABLE] 2. (b)

Also

[TABLE] 3. (c)

For $k=1,2,3$ there holds

[TABLE]

Proof.

(a) The estimates (4.8)-(4.9) follow from the identities

[TABLE]

(b) We have

[TABLE]

From (4.9) we find

[TABLE]

Combining the above equality with (4.9) we find

[TABLE]

(c) The estimate (4.11) follows from the following computations

[TABLE]

∎

We seek a solution $\eta\in\Sigma_{M}$ by Lemma 2.1 as in the proof of Theorem 1.1.

We begin with an upper bound for $F_{1}(t)$ defined in (4.5). By Mean Value theorem we have that

[TABLE]

for some $s$ between $2t+\varphi$ and $2t$. Since $s\mapsto|H^{\prime\prime}_{m}(s)|$ is decreasing and $\varphi(t)$ is negative for large $t>0$, from (4.16) and (4.9)-(4.10) we find

[TABLE]

We now bound $F_{0}(t)$ defined in (4.4). First, by the estimates (4.8) we have

[TABLE]

Also, by the Mean Value Theorem and (4.8) we have

[TABLE]

for some $s$ between $2t+\varphi(t)$ and $\varphi(t)$. Hence, from (4.18)-(4.19) we find

[TABLE]

We now handle the nonlinear term $F_{2}(t,\eta)$ defined in (4.6). Observe first that $\rho(\eta)$ given in (4.6)-(4.7) can be written as

[TABLE]

Since $\varphi(r)\simeq-\ln t$ and $\eta\in\Sigma_{M}$, a further adjustment of $M$ and Mean Value Theorem together with (4.8) yields

[TABLE]

Since $G_{m}^{\prime\prime}$ is increasing, using (4.11) we deduce

[TABLE]

Let now $\eta_{1},\eta_{2}\in\sigma_{M}$ and denote $\theta_{j}(t)=\frac{\eta_{j}(t)}{H^{\prime}_{m}(2t+\varphi(t))}$. Using that

[TABLE]

we have

[TABLE]

Since $G_{m}$ and $H_{m}$ are inverse each other, for $j=1,2$ we may write

[TABLE]

Thus

[TABLE]

By the Mean Value Theorem, (4.23) and (4.24), for some $\theta(t)$ between $\theta_{1}(t)$ and $\theta_{2}(t)$, we have

[TABLE]

Now, the second term in (4.25) is bounded from above by

[TABLE]

Let us observe first that from (4.22)-(4.24) and the Mean Value Theorem we have

[TABLE]

To bound from above the first term in (4.27) we have

[TABLE]

For $\rho\in\Sigma_{M}$, from (4.21) we have

[TABLE]

Then

[TABLE]

As before,

[TABLE]

Also, by the Mean Value Theorem, for some $\eta$ between $\eta_{1}$ and $\eta_{2}$ we have

[TABLE]

The last two estimates yield

[TABLE]

Finally, combining (4.25)-(4.30) we deduce

[TABLE]

Setting $\eta_{2}=0$ gives $F_{2}(t,\eta_{1})\leq O(t^{-2})$ and we can appply the Lemma 2.1. We then have a unique solution of (4.3) in $\Sigma_{M}$.

Proceeding as in the proof of Theorem 1.1, we have that there exists a $T^{*}$ such that $w(T^{*})=0$ and hence we have completed the proof of the existence and uniqueness result.

We next derive the asymptotic expansion (1.12). We begin with

[TABLE]

We have, using (4.9), that

[TABLE]

Then

[TABLE]

Finally, we use use the expansion (4.15) to deduce (1.12). We now prove the estimate for the gradient. Exactly as in the proof of Theorem 1.1, we have that $\eta_{t}=O(\frac{1}{t^{2}})$. Then

[TABLE]

since $|\varphi_{t}H^{\prime}_{m}(2t+\varphi)|\leq ct^{-2}$. We use the expansion

[TABLE]

since $|H^{\prime\prime}_{m}(2t)\varphi|\leq ct^{-2}$. This completes the proof of Theorem 1.3, using that $|v_{r}|=|w_{t}\frac{dt}{dr}|$.