数理解析研究所講究録
第 1901 巻 2014 年 116-120

116

Bifurcation diagram for interior single-peak solutions
in a Neumann problem for $u”+\lambda(-u+u^{p})=0$ with
$p\in \mathbb{R}$and $p>1^{*}$

Kazuyuki Yagasaki
Department of Mathematics,
Hiroshima University$\dagger$

1

Introduction

Letequation,$p\in \mathbb{R}$and $p>1$ . We study a Neumann problem for a second-order diﬀerential$u”+\lambda(-u+u^{p})=0$

in $(-1,1)$ ,(1)$u’(\pm 1)=0$ ,where $\lambda>0$ is a constant and represents a control parameter. Eq. (1) has a trivial
solution $u=1.$
We often encounter (1) in several situations. As an example, we consider the KellerSegel model for chemotaxis aggregation,$v_{t}=D_{2}v_{xx}-av-bu$$u_{t}=D_{1}u_{xx}-c(u(\log v)_{x})_{x},$$u_{x},$

where$v_{x}=0$$D_{1},$ $D_{2},$ $a,$ $b,$ $c$

in $(-1,1)$ ,at $x=\pm 1,$(2)

are constants. The stationary problem for (2) becomes(3)$D_{2}v_{xx}-av-b\mu v^{c/D_{1}}=0, v_{x}=0atx=\pm 1$

since $D_{1}u_{x}-cu(\log v)_{x}=0$ by the first equation, so that $u=\mu v^{c/D_{1}}$ for some constantEq. (3) is transformed to (1). Another example is related to the Gierer-Meinhardt model$\mu.$

for biological pattern formations,$u_{t}=D_{1}u_{xx}- \mu_{1}u+\rho_{1}(c_{1}\frac{u^{p_{1}}}{v^{q_{1}}}+\rho_{0})$$u_{x},$$v_{x}=0$,$v_{t}=D_{2}v_{xx}- \mu_{2}v+\rho_{2}c_{2}\frac{u^{p_{2}}}{v^{q_{2}}}$in $(-1,1)$ ,(4)at $x=\pm 1,$’This work was partially supported by the Japan Society for the Promotion of Science, Grant-in-Aidfor Scientific Research (C) (Subject Nos. 21540124 and 22540180). Some part of this article contains acollaboration with Yasuhito Miyamoto at the University of Tokyo.
Present address: Department of Applied Mathematics and Physics, Kyoto University, Yoshida Honmachi Sakyo-ku, Kyoto 606-8501$\dagger$

117$\frac{\pi^{2}}{p-1}$Figure 1: Bifurcation diagram for the Neumann problem (1)$i=1$ , 2, are constants. As
whereboundary conditions. Hence, in this limit, we have$D_{i},p_{i},$$q_{i},$$\mu_{i},$$D_{2}arrow\infty,$$\rho_{i},$$v_{xx}arrow 0$

so that$v_{x}arrow 0$

by the$\int_{0}^{1}(\mu_{2}v-\rho_{2}c_{2}\frac{u^{p_{2}}}{v^{q_{2}}})dx=0,$

so that$v^{q_{2}+1}= \frac{\rho_{2}c_{2}}{\mu_{2}}\int_{0}^{1}u^{p_{2}}dx$

by regarding?1 as a constant. Thus, for the stationary problem for (4), we obtain theshadow system,$D_{1}u_{xx}- \mu_{1}u+\rho_{1}(c_{1}\frac{u^{p_{1}}}{\xi^{q_{1}}}+\rho_{0})=0, u_{x}=0atx=\pm 1,$

which is transformed to (1) like (3).
The following theorem for (1) was proved for$p\in \mathbb{Z}$in [1] and for$p\in \mathbb{R}\backslash \mathbb{Z}$in [2].

Theorem 1. The branch of interior single-peak solutions emanates from $(\lambda, u)=(\pi^{2}/(p-$1), 1) and the bifurcation is a supercritical pitchfork one. The branch is a graph of andunbounded in . Moreover, each solution of the branch is non-degenerate and the Morseindex is two.
$\lambda$$\lambda$

Here the Morse index is the number of strictly positive eigenvalues for the associatedlinear problem$\phi"+\lambda(-1+pu_{\lambda}(y)^{p-1})\phi=\mu\phi$in $(-1,1)$ ,$\phi’(\pm 1)=0,$

whererepresents a solution of the Neumann problem. The last part of Theorem 1
is obvious from the other parts since $\mu=\lambda(p-1)$ ,are positive eigenvaluesof the linear problem for the trivial solution $u=1$ when $\lambda<\pi^{2}/(p-1)$ , and so is$\mu=\lambda(p-1)-\pi^{2}$ when $\lambda>\pi^{2}/(p-1)$ . The bifurcation diagram stated in Theorem 1 issketched in Fig. 1. The upper branch represents interior single-peak solutions.
In the rest of this article we outline the proof of Theorem 1.
$u_{\lambda}(y)$$\lambda(p-1)-\frac{1}{4}\pi^{2}$

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Figure 2: One-parameter family of periodic orbits in (5)2

Monotonicity of the period functions

Using a transformation$x\mapsto x/\sqrt{\lambda}$, we rewrite (1) as(5)$u”-u+u^{p}=0$ .

For any $u_{1}\in(0,1)$ Eq. (5) has a periodic solution satisfying $(u(O), u’(O))=(u_{1},0)$ . Let$T(u_{1})$ denote its period and let $u_{2}\in(1, u_{0})$ satisfy $F(u_{2})=F(u_{1})$ , where$F(u)=- \frac{1}{2}u^{2}+\frac{1}{p+1}u^{p+1}+\frac{p-1}{2(p+1)}, u_{0}=p-\sqrt[1]{\frac{1}{2}(p+1)}.$

Then we have $(u( \frac{1}{2}T(u_{1})), u’(\frac{1}{2}T(u_{1})))=(u_{2},0)$ . Note that
$F(u_{0})=F(0)= \frac{p-1}{2(p+1)}.$As shown in Fig. 2, there exists a one-parameter family of periodic orbits in (5).
The periodic solution $u(x)$ in (5) gives an interior single-peak solution in the Neumann. Hence, except the last part, Theorem 1 immediatelyproblem (1) when
follows from the following theorem.
$\frac{1}{2}T(u_{1})=2\sqrt{\lambda}$

Theorem 2. The period function $T(u_{1})$ in (5) is strictly decreasing on $(0,1)$ .
To prove this theorem, we use a result of Chicone [3]. We first recall his result. Letfunction satisfying:is a. Suppose thatand let$V(\xi_{1})=V(\xi_{2})$ and having a minimum $V(O)=0$ as its only extremum. Consider secondorder diﬀerential equations of the form$\xi_{1}<0<\xi_{2}$$V$$I=(\xi_{1}, \xi_{2})\subset \mathbb{R}$$\xi"+\frac{dV}{d\xi}(\xi)=0$.

$Iarrow \mathbb{R}$$C^{3}$(6)Eq. (6) has the trivial solution $\xi=0$ , and any solution $\xi=\xi(t)$ of (6) withand $\xi’(0)=0$ is periodic. Let $T(h)$ be its period with $h=V(\xi(0))$ , and define a function$\xi(0)\in I\backslash \{O\}$$\varphi(\xi)$as$\varphi(\xi)=\frac{V(\xi)}{V’(\xi)^{2}}$Chicone [3] essentially proved the following result.

.

(7)

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Proposition 3 (Chicone [3]). Suppose that(8)$\varphi"(\xi)\geq 0 for\xi\in I\backslash \{O\}$and the inequality holds in a punctured neighborhood of $\xi=$ O.
increasing on $(0, h_{0})$ , where $h_{0}=V(\xi_{1})(=V(\xi_{2}))$ .

3

Then $T(h)$ is strictly

Proof of Theorem 2

Using a transformation $u=\xi+1$ we rewrite (5) as the form of (6) with $\xi_{1}=-1,$$\xi_{2}=u_{0}-1>0$ and $V(\xi)=F(\xi+1)$ . We compute (7) as$\varphi"(\xi)=\frac{(p-1)g(\xi+1)}{(p+1)(\xi+1)^{4}((\xi+1)^{p-1}-1)^{4}}$(9),where$g(u)=pu^{3p-1}-(2p^{2}-3p+3)u^{2p}+p(2p+1)u^{2p-2}$$-p(p-2)u^{p+1}+p(p-7)u^{p-1}+3.$

We begin with the case ofwith $p>1$ . We easily see that the function $g(u)$ isdivisible by $(u-1)^{4}$ and define $(3p-5)$ -th order polynomial $\overline{9}(u)=g(u)/(u-1)^{4}$ . Aftersome highly nontrivial computations, we prove the following (see [1] for the proof).
$p\in \mathbb{Z}$$a$

Lemma 4. All coeﬃcients of $g(u)$ are positive.
From Lemma 4 and (9) we see that$\varphi"(u-1)=\frac{(p-1)\overline{g}(u)}{(p+1)u^{4}(\sum_{j=0}^{p-2}u^{j})^{4}}>0$

for $u\in(0, u_{0})$ ,i.e., condition (8) holds.
We next assume thatwith $p>1$ . Let $p=m/n>1$ , where $m,$ are relatively
prime integers and $n\geq 2$ . We set $v=u^{1/n},$ $k=m-n>0$ and $\psi(v)=n^{2}g(v^{n})$ to have$n$$p\in \mathbb{Q}\backslash \mathbb{Z}$$\psi(c))=n(n+k)v^{2n+3k}-(2k^{2}+kn+2n^{2})v^{2n+2k}+(n-k)(n+k)v^{2n+k}$$+(n+k)(3n+2k)v^{2k}-(n+k)(6n-k)v^{k}+3n^{2}$

We easily see that the polynomial
is factorized as $\psi(v)=(v-1)^{4}\overline{\psi}(v)$ , whereis $(2n+3k-4)$ -th order polynomial. We also prove the following (see [2] for the proof).
$\psi(v)$$\overline{\psi}(v)$$a$Lemma 5. All coeficients of$\overline{\psi}(v)$are positive.

From Lemma 5 and (9) we see that$\varphi"(\tau)^{n}-1)=\frac{(p-1)\overline{\psi}(v)}{(p+1)n^{2}v^{4n}(\sum_{j=0}^{k-1}v^{j})^{4}}>0$

for$v\in(0, \sqrt[n]{u_{0}})$i.e., condition (8) holds again.
We turn to the case ofwith $p>1$ . Take a sequenceand. We easily see that condition (8) holds for$p=p_{j}$ . This completes the proof of Theorem 2 by Proposition 3.
$p\in \mathbb{R}\backslash \mathbb{Q}$$\lim_{jarrow\infty}p_{j}=p$$\{p_{j}\}_{j=0}^{\infty}$$p\in \mathbb{R}\backslash \mathbb{Q}$.

such thatsince it does for$p_{j}\in \mathbb{Q}$120

References[1] Y. Miyamoto aIld K. Yagasaki, Monotonicity of the first eigenvalue and the global
bifurcation diagram for the branch of interior peak solutions, J. Diﬀerential Equations,
254 (2013), 342-367.

[2] K. Yagasaki, Monotonicity of the period function for $u”-u+u^{p}=0$ with pp 1, J. Diﬀerential Equations, 255 (2013), 1988-2001.

$\in \mathbb{R}$and$>$[3] C. Chicone, The monotonicity of the period function for planar Hamiltonian vector
fields, J. Diﬀerential Equations, 69 (1987), 310-321.