数理解析研究所講究録
第 1648 巻 2009 年 17-31

17Analytic singularities of solutions to a radial-Laplacian$p$

上智大学・理工学部
内山康一 (Koichi Uchiyama)
Faculty of Science and Technology,
Sophia University
Abstract

Analytic local description of $W^{1,p}(I)$ solutions to a radial p-Laplace
equation$r(|U_{r}|^{p-2}U_{r})_{r}+(n-1)|U_{r}|^{p-2}U_{r}+r|U|^{q-2}U=0$

on $I=[a, b]\subset(0, \infty)$ is given near singular points by a Briot-Bouquet
type theorem of two variables, where $1<p,$ $q<\infty$ .

1

Introduction

An n-dimensional p-elliptic PDE for $u(x)$ is$div(|\nabla u|^{p-2}\nabla u(x))+|u|^{q-2}u=0$,(1)where $x\in R^{n}$ and $1<p,$ $q<\infty$ .
If $x\in R^{1}$ , the equation reduces to$(|u_{x}|^{p-2}u_{x})_{x}+|u|^{q-2}u=0$.

(2)

L. Paredes and the present author, making use of a Briot-Bouquet
type theorem of one variable, gave analytic expression of solutions to
the equation (2) near the singularities ([8]). Our analytic expression
readily reproduces differentiability and analyticity obtained by M.
Otani [6], [7] and by M. \^Otani and T. Idogawa in [4].
If $r=|x|,$ $x\in R^{n}$ ,a radial solution $U(r)=u(x)$ satisfies$(r^{n-1}|U_{r}|^{p-2}U_{r})_{r}+r^{n-1}|U|^{q-2}U=0$

,

(3)

or
$r(|U_{r}|^{p-2}U_{r})_{r}+(n-1)|U_{r}|^{p-2}U_{r}+r|U|^{q-2}U=0$

$p$

.

(4)

The aim of this report is to extend the results for (2) to the radial
Laplacian (3) by a Briot-Bouquet type theorem of two variables.

18
Remark 0.1. R. Gerard and H. Tahara studied
and generalized it of many variables and of higher order case in[3]. Since our version of a Briot-Bouquet type theorem of two (orseveral) variables is not covered by theirs, a proof is given, inspiredby their work.
$t \frac{\partial u}{\partial t}=\Phi(t, x, u, \frac{\partial u}{\partial x})$A Briot-Bouquet type theorem

2

We recall a classical Briot-Bouquet type theorem of one variable incomplex domain. We assume$\bullet$$\Phi(t, h)$is holomorphic near $(0,0)\in C^{2}$ ,$\bullet\Phi(0,0)=0$$\bullet$$\frac{\partial\Phi}{\partial h}(0,0)$,

is not any positive integers.

Theorem 1 (Briot-Bouquet).
$t \frac{dh}{dt}=\Phi(t,$ $h)$

has a unique holomorphic solution near $t=0$ , satisfying $h(O)=0$ .
If $\Phi(t, h)$ is real analytic, so is $h(t)$ , too.

Proof. Set$a_{\alpha_{2}i}= \frac{1}{\alpha!i!}\frac{\partial^{\alpha+i}\Phi}{\partial t^{\alpha}\partial h^{i}}(0,0)$. Notice we can rewrite the equa-tion by$t \frac{dh}{dt}-a_{0,1}h=a_{1,0}t+\sum_{2\leq\alpha+i}a_{\alpha,i}t^{\alpha}h^{i}$.

Moreover, the left hand side satisfies the condition that there exists$\delta>0$ such that$|\alpha-a_{0,1}|\geq\delta$

for all $\alpha\in N^{*}=\{1,2,3, \cdots\}$ .
Formal solution:Let$\hat{h}(t)=\sum_{\alpha=1}^{\infty}h_{\alpha}t^{\alpha}$

be a formal solution. Then,

we have$(1-a_{0,1})h_{1}=a_{1,0}$ ,$(\alpha-a_{0,1})h_{\alpha}=Q_{\alpha}(a_{\alpha’,i},$ $h_{\alpha’’};\alpha’+i\leq\alpha,$ $\alpha’’\leq\alpha-1)$

19, wherefor allcoefficients.
$\alpha\geq 2$

Convergence:$Q_{a}$

is a polynomial with nonnegative integer

Then, we prove convergence of$\hat{h}(t)$through the in-plicit function theorem.
An auxiary equation of $H(t)$ is given by$\delta H=|a_{1,0}|t+\sum_{2\leq\alpha+i}|a_{\alpha_{\nu}i}|t^{\alpha}H^{i}$.

with $H(O)=0$ .
There exists a unique convergent series function
by the implicit function theorem. Since$\alpha’’\leq\alpha-1)$ ,$H(t)= \sum_{\alpha=1}^{\infty}H_{\alpha}t^{\alpha}$$\delta H_{\alpha}=Q_{\alpha}(|a_{\alpha’,i}|,$$H_{\alpha’}/;$$\alpha’+$$i\leq\alpha,$$|h_{1}|=|a_{1,0}|/|1-a_{0,1}|$$\leq|a_{1,0}|/\delta=H_{1}$,$|h_{\alpha}|=|Q_{\alpha}(a_{\alpha’}, h_{\alpha’’})/(\alpha-a_{0,1})$I$\leq Q_{\alpha}(|a_{\alpha}/|, H_{\alpha’’})/\delta=H_{\alpha}$for$\alpha\geq 2$

by induction.
$\square$We will make use of a Briot-Bouquet type theorem of two variables for our main results. We state it in a slightly more generalform for convenience.
,
Let $N=\{0,1,2, \cdots\}$ . $B=\{\beta\}$ is a fixed finite subset of.
is a d-dimensional multi-index with
where
$(t,
h,
\rho_{B})=(t_{1},\cdots,t_{d},h,\{\rho_{\beta};\beta\inB\})\inC^{d+1+|B|}$
be local variLet
ables near the origin, where $|B|$ is the number of the elements in $B$ .
Let $\xi=(\xi_{1}, \cdots, \xi_{d})\in C^{d}$ be global variables.
.
and denote d-dimensional power indices in$N^{d}$$|\beta|\geq 1$$\beta=(\beta_{1}, \cdots, \beta_{d})$$\alpha=(\alpha_{1}, \cdots, \alpha_{d}),$ $\beta$$N^{d}$$\gamma$

Theorem 2. We assume that a holomorphic function
and a polynomial$\phi(t, h, \rho_{B})$$L( \xi)=\sum_{0\leq|\gamma|\leq r}l_{\gamma}\xi^{\gamma}$

satisfy(i)$\phi(t, \rho_{B})$has a power series expansion near $(0,0)$ without linear20
parts;$\phi(t, \rho_{B})=\sum_{2\leq|\alpha|+|i_{B}|}a_{\alpha,i_{B}}t^{\alpha}\rho_{B}^{i_{B}}$$= \sum_{2\leq|\alpha|+|i_{B}|}a_{\alpha,i_{B}}t^{\alpha}\prod_{\beta\in B}\rho_{\beta}^{i_{\beta}}$

,

where
and
for a multi-index(ii) there exists a positive constant such that for all d-dimensional
multi indices with$|i_{B}|= \sum_{\beta\in B}i_{\beta}$$i_{B}=(i_{\beta})_{\beta\in B}$$\delta$$|\alpha|\geq 1_{f}$$\alpha$$| \sum_{0\leq|\gamma|\leq r}l_{\gamma}\alpha^{\gamma}|\geq\delta\max\{1, \alpha^{\beta};\beta\in B\}$

where
denotes the coefficient ofThen, a nonlinear equation$\alpha^{\beta}$$(t \frac{\partial}{\partial t})^{\beta}t^{\alpha}$

,

(5)

.

$\sum_{0\leq|\gamma|\leq r}l_{\gamma}(t\frac{\partial}{\partial t})^{\gamma}h(t)=a\cdot t+$(6)$+ \sum_{2\leq|\alpha|+|i_{B}|}a_{\alpha i_{B}\rangle}t^{\alpha}$.

$\prod_{\beta\in B}((t\frac{\partial}{\partial t})^{\beta}h(t))^{i_{\beta}}$

has a unique holomorphic solution $h(t)$ near the origin with $h(O)=$.

$0$

Proof. We will follow the previous proof.
Construction of$\hat{h}(t)$:We set$\hat{h}(t)=\sum_{|\alpha|\geq 1}h_{\alpha}t^{\alpha}$.

Substituting (7) into (6), we have$L(\alpha)h_{\alpha}=a_{\alpha}$

when $|\alpha|=1$ ,

(7)

21
and
$L(\alpha)h_{\alpha}=Q_{\alpha}(a_{\alpha’,i_{B}},$$(\alpha_{\beta})^{\beta}h_{\alpha_{\beta}};\beta\in B$$h_{\alpha’’},$

the indices at least satisfy$|\alpha’|+|i_{B}|\leq|\alpha|,$ $|\alpha^{//}|\leq|\alpha|-1$$|\alpha_{\beta}|\leq|\alpha|-1)$

where$\alpha’,$$\alpha’’,$$\alpha_{\beta}$

Convergence ofTahara [3],are copies of . Thus,$\alpha$$\hat{h}$:$h_{\alpha}$are determined succesively.

An auxiliary analytic equation (cf.

G\’erard-$)$$\delta H=|a_{1}|t_{1}+|a_{2}|t_{2}$$|a_{\alpha,i_{B}}|t^{\alpha}(H(t))^{|i_{B}|}$$\sum$$+$$2\leq|\alpha|+|i_{B}|$Solving this equation of $H$ by the implicit function theorem, we havea unique holomorphic solution near the origin $t=0$ with $H(O)=0$ .
We claim.
$H(t)= \sum_{\alpha}H_{\alpha}t^{\alpha}>>\hat{h}(t)$More strongly we claim,$|\alpha^{\beta}h_{\alpha}|;\beta\in B\}$

$H_{\alpha} \geq\max\{|h_{\alpha}|,$.

We notice$H_{\alpha}= \frac{1}{\delta}Q_{\alpha}(|a_{\alpha’,i_{B}}|,$$H_{\alpha};;,$$H_{\alpha_{\beta}}$; the indices satisfy(8)at least $|\alpha’|+|i_{B}|\leq|\alpha|,$ $|\alpha’’|\leq|\alpha|-1$ ,$|\alpha_{\beta}|\leq|\alpha|-1,$$\beta\in B)$.

We start with $|\alpha|=1$ :$|h_{\alpha}|=|a_{\alpha}/L(\alpha)|\leq|a_{\alpha}|/\delta=H_{\alpha}$.

Then, by induction, we have$\max\{1,$ $\alpha^{\beta};\beta\in B\}\cdot|h_{\alpha}|\leq\frac{1}{\delta}Q_{\alpha}(|a_{\alpha’,i_{B}}|,$ $|h_{\alpha’’}|,$$|(\alpha_{\beta})^{\beta}h_{\alpha_{\beta}}|$indices satisfy at least$|\alpha^{/}|+|i_{B}|\leq|\alpha|,$ $|\alpha^{//}|\leq|\alpha|-1$$|\alpha_{\beta}|\leq|\alpha|-1)$

,

;

22

$\leq\frac{1}{\delta}Q_{\alpha}(|a_{\alpha’,i,i_{B}}|,$$H_{\alpha’},$$H_{\alpha_{\beta^{1}}}\cdot$

the indices satisfy at least$|\alpha’|+|i_{B}|\leq|\alpha|,$ $|\alpha’’|\leq|\alpha|-1$,$|\alpha_{\beta}|\leq|\alpha|-1)$

Therefore, we have obtained$\max\{|h_{\alpha}|, |\alpha^{\beta}h_{\alpha}|\}\leq H_{\alpha}$.

$\square$

We need this Briot-Bouquet type theorem in case where $d=2$ ,$r=2,$ $\max\{|\beta|;\beta\in B\}=1$ .

3

Local uniquenessTo assure that weak solutions in $W^{1,p}(I)$ are filled locally by our analytical method, we need local uniqueness of solutions to the Cauchy
problem to (3). We adapt proofs of uniqueness in J. Benedikt [1]and P. Dr\’abek and M. \^Otani [2] for forth order p-elliptic equations,completing them with an energy inequality.
Let $I=[a, b]$ be a compact interval in $(0, \infty)$ .

Proposition 1 (Local uniqueness). Let
be an arbitrary positive
constant in I. Local solutions on I are uniquely determined nearby initial data $U(r_{0})$ and.
$r_{0}$$r_{0}$$U_{r}(r_{0})$Proof. Set $V(r)=r^{n-1}|U_{r}(r)|^{p-2}U_{r}(r),$ $p’=p/(p-1)$ and $f_{p}(X)=$. Notice $f_{p,X}(X)=(p-1)|X|^{p-2}$ . Then, we have from theequation,$|X|^{p-2}X$$\{\begin{array}{ll}V_{r}(r) =-r^{n-1}|U(r)|^{q-2}U(r)=-r^{n-1}f_{q}(U(r)),U_{r}(r) =r^{\frac{1-n}{p-1}}|V(r)|^{p’-2}V(r)=f_{p’}(r^{1-n}V(r)).\end{array}$(9)

Suppose$V_{1}(r_{0})=V_{2}(r_{0})$,$U_{1}(r_{0})=U_{2}(r_{0})$$|U_{0}|+|V_{0}|>0$.

We will show that there exists a positive constant such that $U_{1}(r)=$, as proved in Benedikt [1] in 4th orderonelliptic ordinary differential equation.
$\epsilon$$U_{2}(r)$$p$$J(\epsilon)=[r_{0}-\epsilon, r_{0}+\epsilon]$

23
Case (i): $1<p\leq 2$ and $2\leq q$ .
$V_{1}(r)-V_{2}(r)=r^{n-1}\{f_{p}(U_{1,r}(r))-f_{p}(U_{2,r}(r))\}$$=r^{n-1} \int_{U_{2,r}(r)}^{U_{1,r}(r)}f_{p,X}(\tau)d\tau$

We set $K_{1}= \max\{|U_{i,r}(r)|;r\in I, i=1,2\}$ . Noticing $f_{p,X}(X)$ is
positive decreasing on $(0, \infty)$ , when $1<p<2$ , we have$|r^{n-1}\{f_{p}(U_{1,r}(r))-f_{p}(U_{2,r}(r))\}|$$\geq(r_{0}-\epsilon)^{n-1}(p-1)K_{1}^{p-2}|U_{1,r}(r)-U_{2,r}(r)|$.

We set $K_{0}= \max\{|U_{i}(r)|;r\in I, i=1,2\}$ .
$V_{1}(r)-V_{2}(r)=- \int_{r_{0}}^{r}\tau^{n-1}\{f_{q}(U_{1}(\tau))-f_{q}(U_{2}(\tau))\}d\tau$.

On the other hand, we have$|f_{q}(U_{1}( \tau))-f_{q}(U_{2}(\tau))|=|\int_{U_{2}(\tau)}^{U_{1}(\tau)}f_{q,X}(\sigma)d\sigma|$$\leq(q-1)K_{2}^{q-2}|U_{1}(\tau)-U_{2}(\tau)|$$\leq(q-1)K_{2}^{q-2}|\int_{r_{0}}^{\tau}\{U_{1,r}(\sigma)-U_{2_{?}r}(\sigma)\}d\sigma|$$\leq(q-1)K_{2}^{q-2}\epsilon\Vert U_{1,r}-U_{2,r}\Vert_{J(\epsilon)}$where
Choosing$\Vert U\Vert_{J(\epsilon)}=\max_{|r-ro|\leq\epsilon}|U(r)|$$\epsilon$,.

sufficiently small, we conclude$\Vert U_{1,r}-U_{2,r}\Vert_{J(\epsilon)}=0$

Hence, $V_{1}=V_{2}$ on$J(\epsilon)$.

, therefore, $V_{1,r}=V_{2,r}$ , which gives $U_{1}(r)=$on
Since we can proceed the rest as in [1], we show only classificationof cases.
Case (ii): $1<p\leq 2$ and $1<q<2$.
Subcase (ii-l): We assume also.
Subcase (ii-2): $U_{0}=0,$$U_{2}(r)$$J(\epsilon)$$U_{0}\neq 0$$V_{0}\neq 0$

24Case (iii): $2<p$ , and $2\leq q$ .
Subcase (iii-l): $V_{1}(r_{0})=V_{2}(r_{0})\neq 0$ .
Subcase (iii-2): $V_{1}(r_{0})=V_{2}(r_{0})=0$ and $U_{1}(r_{0})=U_{2}(r_{0})\neq 0$ .
Case (iv): $2<p$ and $1<q<2$ .
Subcase (iv-l): $U_{0}=0$ , and.
Subcase (iv-2): $U_{0}\neq 0$ , and $V_{0}=0$ .
$V_{0}\neq 0$

We need different arguments, when $U_{0}=V_{0}=0$ . We assume$U_{0}=V_{0}=0$ .
To complete the proof, we use an energy inequality.
Proposition 2. (i) Every nonzero
regularity.
(ii) Then, it satisfies the energy equality$W^{1p}\rangle(I)$solution$U$

on I has$C^{1}(\overline{I})$$\frac{p-1}{p}|U_{r}(r)|^{p}+\frac{1}{q}|U(r)|^{q}$(10)$= \frac{p-1}{p}|U_{r}(c)|^{p}+\frac{1}{q}|U(c)|^{q}$$-(n-1) \int_{c}^{r}\frac{1}{\sigma}|U_{r}(\sigma)|^{p}d\sigma$

for all $c\in I$ .
(iii) If $U(r_{0})=U_{r}(r_{0})=0$ for someRemark 2.1. (i) is due to M. \^Otani [6].
$r,$$r_{0}\in I$, then, $U(r)=0$ on .
$I$

When $n=1$ , (ii) is the energy equality.
When $n=1$ , (iii) is trivial (for any $q>1$ ) in virtue of the energy
equality. When $n>1$ , this completes the proof of local uniqueness.
Proof of (iii). $U(r)=0$ for all $r\in[r_{0}, b]$ in virtue of the energy
inequality.
For all $r\in[a, r_{0}]$ we have$p,$,$\frac{p-1}{p}|U_{r}(r)|^{p}+\frac{1}{q}|U(r)|^{q}=(n-1)\int_{r}^{r_{0}}\frac{1}{\sigma}|U_{r}(\sigma)|^{p}d\sigma$

therefore,$\frac{p-1}{p}|U_{r}(r)|^{p}\leq(n-1)\int^{r0}\frac{1}{\sigma}|U_{r}(\sigma)|^{p}d\sigma$.

25
By Gronwall’s lemma, we have $U_{r}(r)=0$ on . Since $U(r_{0})=0$ , it
implies $U(r)=0$ on I.
$I$$\square$Remark 2.2. We note a different proof, when $q\geq p$ as in [2].
We note$r^{n-1}f_{p}(U_{r}(r))=V(r)= \int_{r0}^{r}V_{r}(\tau)d\tau=-\int_{r0}^{r}\tau^{n-1}f_{q}(U(\tau))d\tau$

We have$r_{0}^{n-1}\Vert U_{r}\Vert_{J(\epsilon)}^{p-1}\leq\epsilon$I$V_{r}\Vert_{J(\epsilon)}\leq\epsilon(r_{0}+\epsilon)^{n-1}$

1
4$U\Vert_{J(\epsilon)}=0$$U\Vert^{q-1}\leq$, we have. If we assume I
for any small positive . This gives contradiction, hence.
$U\Vert_{J(\epsilon)}>0$$\epsilon^{q}(r_{0}+\epsilon)^{n-1}\Vert U_{r}\Vert_{J(\epsilon)}^{q-1}$$\epsilon^{q}\Vert U_{r}\Vert_{J(\epsilon)}^{q-p}$I.

$r_{0}^{n-1}\leq$$\epsilon$

Analytic singularities

We shall now describe local analytic singularities of the solution$U(r)$ to (4).
When $n=1$ , a classical Briot-Bouquet type theorem of one variable was sufficient to obtain the unique existence of analytic solution
to the nonlinear ordinary differential equation ([8]).
is an analytic point of
When $U(r_{0})\neq 0$ and $U_{r}(r_{0})\neq 0,$
solution $U(r)$ . Hence, we consider two types of singularities:
where $U(\sigma)=0$ and $U_{r}(\sigma)=A\neq 0$ ,$r_{0}=\tau$ where $U(\tau)=A\neq 0$ and $U_{r}(\tau)=0$ .
CASE 1. where $U(\sigma)=0$ and $U_{r}(\sigma)=A\neq 0$ . We assume $A>0$
without loss of generality. We treat the case where $\sigma>0$ .
$r_{0}$$\bullet$$r_{0}=\sigma$$\bullet$$\sigma$

Theorem 3. For 1 $<p,$ $q<\infty$ , there exists a unique analytic
function $F(t, s)$ in a neigborhood of the origin such that we havenear $r=\sigma$$U(r)=(r-\sigma)F(r-\sigma, |r-\sigma|^{q})$ .
(11)$F(t, s)$is a holomorphic solution to$(p-1)(\sigma+t)\{F(t, s)+tF_{t}(r, s)+qsF_{s}(t, s)\}^{p-2}$$\{tF_{t}(t, s)+qsF_{s}(t, s)$$+t(tF_{t}(t, s))_{t}+2qtsF_{t,s}(t, s)+q^{2}s(sF_{s}(t, s))_{s}\}$$+(\sigma+t)s(F(t, s))^{q-1}$(continued)

(12)

26
$+(n-1)t\{F(t, s)+tF_{t}(t, s)+qsF_{s}(t, s)\}^{p-1}$
$=0$

(13)

with
$F(0,0)=A$ .

(14)Consequently, we can compute the expansion of $U(r)$ at $r=\sigma$ :$U(r)=(r-\sigma)\{$ $A- \frac{n-1}{2\sigma(p-1)}A(r-\sigma)$(15)$- \frac{A^{q-p+1}}{(p-1)q(q+1)}|r-\sigma|^{q}+\cdots\}$.

Proof. We reduce equation (13) by change of the unknown function$F(t, s)=A+h(t, s)$into$L(t \frac{\partial}{\partial t},$$s \frac{\partial}{\partial s})h(t, s)=-\frac{1}{(p-1)(\sigma+t)}$$\cross\{A+h(t, s)+th_{t}(t, s)+qsh_{s}(t, s)\}^{2-p}$(16)$\cross[(\sigma+t)s(A+h(t, s))^{q-1}+(n-1)t$$\{A+h(t, s)+th_{t}(t, s)+qsh_{s}(t, s)\}^{p-1}]$$=a_{1,0}t+a_{0,1^{S}}$$+ \sum_{2\leq p+q+i+j+k}a_{p,q,i,j,k}t^{p}s^{q}$(17)$\cross(h(t, s))^{i}(t\frac{\partial h}{\partial t}(t, s))^{j}(s\frac{\partial h}{\partial s}(t, s))^{k}$

,

where
$a_{1,0}=- \frac{(n-1)A}{(p-1)\sigma}$(18)$a_{0,1}=- \frac{1}{p-1}A^{q-p+1}$.

We have $L(\alpha, \beta)=\alpha+q\beta+\alpha^{2}+2q\alpha\beta+q^{2}\beta^{2}$ .

27
Since $L(1,0)=2$ and $L(O, 1)=q(q+1),$ $F_{t}(0,0)$ and $F_{s}(0,0)$ are
determined and so on. Thus, the unique existence of the solution is
obtained by the B-B type theorem of two variables.
Next, $(r-\sigma)F(r-\sigma, |r-\sigma|^{q})$ is afunction near . It satisfies(1) with the prescribed Cauchy data. By Proposition 1, it is equal
to the unique solution $U(r)$ with the same Cauchy data.
$C^{2}$$\sigma$Corollary 1 ([4], [7],[8]). (i) When is an even integer more than
1, the solution $U(r)$ is real analytic near .
(ii) When is not an even integer, the solution $U(r)$ is of class $c<q>$
at , where $<x>is$ the least integer greater than or equal to .
$q$$\sigma$$q$$x$$\sigma$

CASE 2. where $U(\tau)=A$ and $U_{r}(\tau)=0$ .
As in the case 1, we can assume without loss of generality that
$A>0$ .
Theorem 4. For any and satisfying $1<p,$ $q<\infty$ , there exists a
unique analytic function $G(t, s)$ in a neigborhood of the origin suchthat we have near $r=\tau$$\tau$$p$$q$$U(r)=A+|r-\tau|^{\underline{g}}\overline{p}\overline{1}G(r-\tau,$$|r-\tau|^{A}\overline{p}-\check{1})$

,

(19)

where $G(t, s)$ is a holomorphic solution to the nonlinear equation;$(p-1)(t+ \tau)\{-(\frac{p}{p-1}G(t, s)$$+tG_{t}(t, s)+ \frac{p}{p-1}sG_{s}(t, s))\}^{p-2}$(20)$\cross\{$$\frac{p}{(p-1)^{2}}G(t, s)+\frac{p+1}{p-1}tG_{t}(t, s)$$+ \frac{p(p+1)}{(p-1)^{2}}sG_{s}(t, s)+t(tG_{t})_{t}(t, s)$$+ \frac{2p}{p-1}tsG_{t,s}(t, s)+\frac{p^{2}}{(p-1)^{2}}s(sG_{s})_{s}(t, s)\}$$+(\tau+t)(A+sG(t, s))^{q-1}$$-(n-1)t\{$ $-( \frac{p}{p-1}G(t, s)$$+tG_{t}(t, s)+ \frac{p}{p-1}sG_{s}(t, s))\}^{p-1}=0$

(21)

28
with
$G(0,0)=- \frac{p-1}{p}A^{z}p^{\frac{-1}{-1}}$.

Consequently, we have a convergent expansion near $r=\tau$ :$U(r)=A+B|r-\tau|^{\overline{p}}\underline{R}_{\overline{1}}+C(r-\tau)|r-\tau|^{L}\overline{p}-\overline{1}$$+D|r-\tau|^{\frac{2}{p}2_{arrow}}-1+\cdots$

(22)

,

where
$B=- \frac{p-1}{p}$Agit and(23)$C= \frac{(n-1)}{2(2p-1)}A^{L_{\frac{1}{1}}^{-}}p-$(24)$D= \frac{q-1}{2(2p-1)}(\frac{p-1}{p})^{2}A^{1+_{p-1}^{2}}\lrcorner_{i^{-}}A$(25)

Proof. We show, at first, unique existence of the solution $G(t, s)$ .
We reduce the equation by change of unknown function by$G(t, s)=B+h(t, s)$ ,into$\frac{p}{(p-1)^{2}}h(t, s)+\frac{p+1}{p-1}th_{t}(t, s)+\frac{p(p+1)}{(p-1)^{2}}sh_{s}(t, s)$(26)$+t(th_{t})_{t}(t, s)+ \frac{2p}{p-1}tsh_{t,s}(t, s)$$+ \frac{p^{2}}{(p-1)^{2}}s(sh_{s})_{s}(t, s)$$=- \frac{p}{(p-1)^{2}}B-\frac{1}{(p-1)(t+\tau)}$$\cross\{$ $- \frac{pB}{p-1}-\frac{p}{p-1}h(t,

s)$

(27)
$-th_{t}(t, s)- \frac{p}{p-1}sh_{s}(t, s)\}^{2-p}$$\cross\{(\tau+t)(A+sB+sh(t, s))^{q-}1$29$-(n-1)t(- \frac{p}{p-1}B-\frac{p}{p-1}h(t, s)$(28)$-th_{t}(t, s)- \frac{p}{p-1}sh_{s}(t, s))^{p-1}\}$.

Developing the right hand side with respect ot $(t, s, h, \rho, \theta)$ , where$\rho=sh_{s}$ and, we, at first, obtain$\theta=sh_{s}$$B=- \frac{p-1}{p}A^{1}p^{\frac{-1}{-1}}$by the condition that the constant term vanishes:$- \frac{p}{(p-1)^{2}}B-\frac{1}{p-1}\{-\frac{p}{p-1}B\}^{2-p}A^{q-1}=0$.

Then, we have the development of the R.H.S. is

R.H.S. $=a_{1,0,0,0,0}t+a_{0,1,0,0,0}s$(29)$+ \frac{2p-p^{2}}{(p-1)^{2}}h(t, s)+\frac{2-p}{p-1}th_{t}(t, s)$$+ \frac{2p-p^{2}}{(p-1)^{2}}sh_{s}(t, s)$(30)$+ \sum_{2\leq\alpha+\beta+i+j+k}a_{\alpha,\beta,i,j,k}t^{\alpha}s^{\beta}$$\cross(h(t, s))^{i}(t\frac{\partial h}{\partial t}(t, s))^{j}(s\frac{\partial h}{\partial s}(t, s))^{k}$Set$l_{0,0}= \frac{p}{(p-1)^{2}}-\frac{2p-p^{2}}{(p-1)^{2}}=\frac{p}{p-1}$$l_{1,0}= \frac{p+1}{p-1}-\frac{2-p}{p-1}=\frac{2p-1}{p-1}$,,$l_{0,1}= \frac{p^{2}+p}{(p-1)^{2}}-\frac{2p-p^{2}}{(p-1)^{2}}=\frac{p(2p-1)}{(p-1)^{2}}$$l_{2,0}=1,$ $l_{1,1}= \frac{2p}{p-1},$ $l_{0,2}= \frac{p^{2}}{(p-1)^{2}}$

.

,

30
Thus, we have the reduced equation$L(t \frac{\partial}{\partial t},$$s \frac{\partial}{\partial s})h(t, s)$(31)$=a_{1,0,0,0_{2}0}t+a_{0,1,0,0,0^{S}}$$+ \sum_{2\leq\alpha+\beta+i+j+k}a_{\alpha,\beta,i_{1}j,k}t^{\alpha}s^{\beta}$(32)$\cross(h(t, s))^{i}(t\frac{\partial h}{\partial t}(t, s))^{j}(s\frac{\partial h}{\partial s}(t, s))^{k}$satisfies the condition (5), the unique existence of thesolution to (21) is obtained by our Briot-Bouquet type theorem.

Since

$L$

口

Corollary 2 ([4], [7],[8]). (i) If $p/(p-1)$ is an even integer, $i.e$ .
$p=(2m+2)/(2m+1)(m=0,1,2, \cdots),$ $u(x)$ is real analytic at .
(ii) If $p/(p-1)$ is not an even integer, the solution $U(r)$ is of class$C-1$
at where $<x>is$ the least integer greater than or equal
at . Whento . Especially, when $1<p\leq 2,$ $U(r)$ is of class$2<p,$ $U(r)$ is not of classat .
$\tau$$\tau_{\rangle}$$C^{2}$$x$$C^{2}$$\tau$$\tau$

References[1] J. BENEDIKT, Uniqueness theorem for p-biharmonic equations,Electronic J. Diff. Eqns., vol. 2002 (2002), No.53. 1-17

[2] P. DR\’ABEK and M. \^OTANI, Global bifurcation result for the
biharmonic operator, Electronic J. Diff. Eqns., Vol.2001 (2001),
No.481-19$\mathfrak{x}\succ$[3] R. GERARD and H. TAHARA, Singular Nonlinear Partial Differential Equations, Vieweg, Braunschweig/Wiesbaden, 1996

[4] T. IDOGAWA and M. \^OTANI, Analyticity and the best possible
constants for Sobolev-Poincar\’e inequalities, Advances in Math.
Sci. and Appl., 4 (1994), 71-78[5] P. LINDQVIST, Note on a Nonlinear Eigenvalue Problem, Rocky
Mountain Journal of Mathematics, 23 (1993), 281-288

31[6] M. \^OTANI, A Remark on Certain Nonlinear Elliptic Equations,
Proc. Fac. Sci. Tokai University, 19 (1984), 23-28[7] M. \^OTANI, On certain second order ordinary differential equations associated with Sobolev-Poincar\’e-type inequalities, Nonlinear Analysis, Theory and Applications, 8 (1984), 1255-1270[8] L. I. PAREDES and K. UCHIYAMA, Analytic Singularities of
Solutions to Certain Nonlinear Ordinary Differential Equations
Associated with p-Laplacian, Tokyo J. Math. 26 (2003) 229-240