数理解析研究所講究録
第 1542 巻 2007 年 77-91

77Singular perturbation problem

for nonlinear-diffusive logistic equations
工学院大学工学部

竹内慎吾(Shingo Takeuchi)

Department of General Education
Kogakuin University

1

Introduction

In this paper we consider the following boundary value problem(P)$\{\begin{array}{ll}-\epsilon\Delta_{p}u=f(x, u), x\in\Omega,u=0, x\in\partial\Omega\end{array}$for small$\epsilon>0$. Here$\partial\Omega(0<\omega<1),$$\Delta_{p}$$\Omega$

is a bounded domain in$\mathbb{R}^{N}(N\geq 2)$

with$C^{2,\omega}$-boundaryis the p-Laplace operator $\Delta_{p}u=div(|\nabla u|^{p-2}\nabla u)(p>1)$ , andis assumed to satisfy the following conditions:

$f$

(F1)

$f(x,u)\in C(\overline{\Omega}\cross[0, \infty))$;(F2) There exist $\xi>0$ and $\sigma>0$ such that $f(x, 0)=0$ , the map $u\vdasharrow f(x, u)$ isnondecreasing in$\Omega$;$[0, \xi]$for all$x\in\Omega$

and $\lim\inf_{uarrow+0}f(x,u)/u^{p-1}>\sigma$ uniformly in78(F3) There exists a positive function$a(x)\in C(\overline{\Omega})$such that$f(x, u)\{\begin{array}{ll}>0 when 0<u<a(x),<0 when u>a(x);\end{array}$(F4) There exists a strictly increasing function $g(x)\in C([0, \infty))$ such that $g(O)=$and the map$0$$u-\rangle$$f(x, u)+g(u)$ is nondecreasing in $[0, \infty$ ) for all $x\in\Omega$ .

We shall describe the case where $f(x, u)$ is independent of , e.g., $f(x,u)=$$f(u)=u^{p-1}|1-u|^{q-1}(1-u)(p>1, q>0)$ , which satisfies (F3) with $a(x)\equiv 1$ . For$x$$\epsilon>0$

small enough, the boundary value problem$\{\begin{array}{ll}-\epsilon\Delta_{p}u=f(u), x\in\Omega,u=0, x\in\partial\Omega\end{array}$has the unique positive solution$u_{\epsilon}$, which converges the value 1 uniformly in any. Guedda and V\’eron [5] in l-dimensional case andKamin and V\’eron [7] in N-dimensional case investigated that when $q<p-1$ andcompact subset of$\epsilon$$\Omega$as$\epsilonarrow 0$

is sufficiently small, the coincidence set of$u_{\epsilon}$$u_{\epsilon}$with the value 1, or the flat core of, defined by$\mathcal{O}_{\epsilon}=\{x\in\Omega|u_{\epsilon}(x)=1\}$

is not empty and that there exists a constant $C>0$ such that$\{x\in\Omega|dist(x, \partial\Omega)>C\epsilon^{1/p}\}\subset O_{\epsilon}$

If $q\geq p-1$ , then$\mathcal{O}_{\epsilon}$is empty for any because$\epsilon$$u_{\epsilon}$.

is strictly less than 1 by the strong

maximum principle of V\’azquez [9]. After their works, Garc\’ia-Meli\’an and Sabina de

Lis [4] gave the precise speed of expansion of$\mathcal{O}_{\epsilon}$as$\epsilonarrow 0$, namely, the estimate. In the results above, they all assume that$f(u)/u^{p-1}$ is decreasing in order to assure the uniqueness of positive solutions. Guoof width of the boundary layer of$u_{\epsilon}$

79[6] eliminated this as sumption and showed that the positive solution is neverthelessunique for small$\epsilon$(cf. Theorem 2 of Dancer [3] for $p=2$ ).

This paper deals with the case where $f(x, u)$ depends on , particularly, $a(x)$ is$x$

not constant. In the semilinear case $p=2$ , Angenent [1] described that for small$\epsilon>0$, the positive solution of (P) is unique and converges to $a(x)$ uniformly in any

compact subset of$\Omega$as$\epsilonarrow 0$. For the quasilinear case $p\neq 2$ , however, there is

no preceding study on singular perturbation problems for (P). Our purpose is toextend the results of Angenent [1] and of Kamin and V\’eron [7], respectively, to thex-dependent case: we give the proof that any positive solution of (P) converges to, and we show that forassmall enough, the solutions coincide with $a(x)$ on the domain where $a(x)$ is constant$a(x)$

uniformly in any compact subset of$\epsilon>0$$\epsilonarrow 0$$\Omega$

an$df(x, u)$ tends to zero as $uarrow a(x)$ with the order less than $p-1$ .
To state theorems, we give the following notation, which will be in force throughthe paper:$A= \max$ $a(x)|x\in$ St},{$\alpha=\min\{a(x)|x\in\overline{\Omega}\}$,$D(\Omega, R)=\{x\in\Omega|dist(x, \partial\Omega)>R\}$.

Theorem 1.1. Suppose $(F1)-(F4)$ . All nontrivial nonnegative solutions are positivein$\Omega$. Moreover, for sufficiently small.
of (P) with some$\epsilon>0$, there exists a positive solution $u\in$$\tilde{\omega}\in(0,1)$$C^{1,\tilde{\omega}}(\overline{\Omega})$

Theorem 1.2. Suppose $(F1)-(F4)$ . For any$\epsilon_{*}>0$

such thatsolution$u_{e}$$D(\Omega, K\epsilon_{*}^{1/p})$$\delta\in(0, \alpha)$

is not empty and that if, there exist $K>0$ and$\epsilon\in(0,\epsilon_{*})$of (P) satisfies

1$u_{\epsilon}(x)-a(x)|<\delta$

for all$x\in D(\Omega, K\epsilon^{1/p})$.

then every positive

80

Theorem 1.3. Suppose $(F1)-(F4)$ and(F5) $a(x)\equiv a$ for some $a\in[\alpha, A]$ in a nonempty subdomain

exist $q\in(O,p-1)$ and$\lambda>0$

uniforrrely in

Then, for sufficiently small $\eta>0$ , there exists$u_{\epsilon}$of$\Omega$and there

such that$\lim_{uarrow}\sup_{a}\frac{f(x,u)-f(x,a)}{|u-a|^{q-1}(u-a)}<-\lambda$

then every positive solution$\Omega_{0}$$\Omega_{0}$.

such that if$\epsilon_{0}\in(0, \epsilon_{*})$(1.1)$\epsilon\in(0, \epsilon_{0})$of (P) satisfies$u_{\epsilon}(x)=a=a(x)$for all$x\in D(\Omega_{0}, \eta)$.

Sections 2 and 3 are devoted to proofs of the theorems. In Section 4, we shallannounce that when $p=2$ , the condition (F5) in Theorem 1.3 can be weaker. In
Section 5, we give a few remark on the theorems.

2

Preliminaries

In this section, we shall define solutions, super- and subsolutions of (P), and show aweak comparison principle for the p-Laplace operator with monotone perturbation.
-G\’amez
We also acquaint the reader with an existence result given by[2] and the strong maximum principle given by V\’azquez [9]. Finally, we prove a$Ca\tilde{n}ada_{r}\cdot Dr\acute{a}bek$

generaiization of Serrin’s sweeping principle to the p-Laplace operator.

Definition 2.1. A function$u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$

is called a solution of (P) when$\epsilon\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi dx=\int_{\Omega}f(x, u)\varphi dx$

for any$\varphi\in W_{0}^{1,p}(\Omega)$.

For any function , define the positive part $v+ofv$ by $v+= \max\{v, 0\}$ . We
ifsay that a function $v\in W^{1,p}(\Omega)$ is less than or equal to $w\in W^{1,p}(\Omega)$ on$v$$\partial\Omega$$(v-w)_{+}\in W_{0}^{1,p}(\Omega)$, which is denoted by $v\leq w$ on$\partial\Omega$.

81

Definition 2.2. A function

when $u\leq 0$ on$\underline{u}\in W^{1,p}(\Omega)\cap L^{\infty}(\Omega)$$\partial\Omega and-\epsilon\triangle_{p}\underline{u}\leq f(x,\underline{u})$in$\Omega$

is called a subsolution of (P), i.e.,$\epsilon\int_{\Omega}|\nabla\underline{u}|^{p-2}\nabla\underline{u}\cdot\nabla\varphi dx\leq\int_{\Omega}f(x, \underline{u})\varphi dx$

for any$L^{\infty}(\Omega)$$\Omega$a.e. in . In the same way, a
is called a supersolution of (P) when $u\geq 0$ on
with$\varphi\in W_{0}^{1,p}(\Omega)$$\Omega$$\varphi\geq 0$$function\overline{u}\in W^{1,p}(\Omega)\cap$$\partial\Omega and-\epsilon\Delta_{p}\overline{u}\geq f(x,\overline{u})$in, i.e.,$\epsilon\int_{\Omega}|\nabla\overline{u}|^{p-2}\nabla\overline{u}\cdot\nabla\varphi dx\geq\int_{\Omega}f(x,\overline{u})\varphi dx$

for any$\varphi\in W_{0}^{1,p}(\Omega)$

Lemma 2.1. Letfunctions$\underline{u},$

$h$

with

$\varphi\geq 0$

a.e. in .
$\Omega$

be a strictly increasing continuous function, and assume that$\overline{u}\in W^{1,p}(\Omega)\cap L^{\infty}(\Omega)$satish$\{\begin{array}{l}-\triangle_{p}\underline{u}+h(\underline{u})\leq-\Delta_{p}\overline{u}h(\overline{u})\Omega\underline{u}\leq\overline{u}\partial\Omega\end{array}$Then$\underline{u}\leq\overline{u}a.e$. in$\Omega$,

Proof. We use an inequality for$a,$$b\in \mathbb{R}^{N}$: There exist positive numbers$C_{1}$

and$C_{2}$

such that$(|a|^{p-2}a-|b|^{p-2}b)\cdot(a-b)\geq\{\begin{array}{ll}C_{1}|a-b|^{p} (p\geq 2),C_{2} (1 <p<2).\end{array}$Choosing$\varphi=(\underline{u}-\overline{u})_{+}\in W_{0}^{1,p}(\Omega)$, we have$0 \geq\int_{\Omega}(|\nabla\underline{u}|^{p-2}\nabla\underline{u}-|\nabla\overline{u}|^{p-2}\nabla\overline{u})\cdot\nabla(\underline{u}-\overline{u})_{+}dx+\int_{\Omega}(h(\underline{u})-h(\overline{u}))(\underline{u}-\overline{u})_{+}dx$$\geq\int_{\Omega}(h(\underline{u})-h(\overline{u}))(\underline{u}-\overline{u})_{+}dx+\{\begin{array}{ll}C_{1}\int_{\Omega}|\nabla(\underline{u}-\overline{u})_{+}|^{p}dx (p\geq 2),C_{2}\int_{\{|\nabla\underline{u}|+|\nabla\varpi|\neq 0\}} dx (1<p<2)\end{array}$$\geq\int_{\Omega}(h(\underline{u})-h(\overline{u}))(\underline{u}-\overline{u})_{+}dx$.

82

The last expression is nonnegative, and hence$(\underline{u}-\overline{u})_{+}=0$

a.e. in . Thus$\Omega$$\underline{u}\leq\overline{u}$a.e. in .

口$\Omega$Lemma 2.2 ([2]). Suppose (F1) and (F4). Let$\underline{u},$$\overline{u}\in W^{1,p}(\Omega)\cap L^{\infty}(\Omega)$

tivdy, a subsolution and a supersolution of (P), withexists a minimal (resp. a maximal) solution$[\underline{u},\overline{u}]=$

{

. in

$\Omega$(resp.

$u\in[\underline{u},\overline{u}]$. in$\Omega$. Then there) for (P) in the interval. in }.
$\Omega$of (P) satisfies also $u_{*}(x)\leq u(x)\leq u^{*}(x)$.

Lemma 2.3 ([9]). Let $u\in C^{1}(\Omega)$ be such that$\Omega,$$u^{*}$$u\in L^{\infty}(\Omega)|\underline{u}(x)\leq u(x)\leq\overline{u}(x)a.e$

In particular, every solution$a.e$$u_{*}$$\underline{u}\leq\overline{u}a.e$be, respec-$-\Delta_{p}u+\beta(u)\geq 0a.e$$\beta(0)=0$. in$\Omega$with$\beta$:$[0, \infty$)$u\geq 0a.e$$\Delta_{p}u\in L_{1oc}^{2}(\Omega),$$arrow \mathbb{R}$. incontinuous, nondecreasing,and either $\beta(s)=0$ for some $s>0$ or $\beta(s)>0$ for all $s>0$ but$\int_{0}^{1}(s\beta(s))^{-1/p}ds=+\infty$

positive everywhere in.

$\Omega$

Then if$u$does not vanish identically on. Moreover, if $u\in C^{1}(\Omega\cup\{x_{0}\})$ for ansatisfies an interior sphere condition and $u(x_{0})=0$ , thenat .
the outer normal unit vector to$\partial\Omega$$\Omega$, then it is$x_{0}\in\partial\Omega$$\frac{\partial u}{\partial n}(x_{0})<0$, where

that
$n$

is

$x_{0}$

Finally in this section, we generalize Serrin’s sweeping principle for uniformly
elliptic operators to the p-Laplace operator. When $f(x,u)$ is independent of , a$x$generalized principle has been already given by Guo [6].

Proposition 2.1. Let (F1), (F4), $I=[a, b](a<b)$ and$u\in W^{1,p}(\Omega)\cap C(\overline{\Omega})$a solution $of-\Delta_{p}u=f(x, u)$ . Suppose that a family of functions$C( \prod)|t\in I\}$and that there exists $c>0$ such
satisfies $v_{t}<u$ onbe$\{v_{t}\in W^{1,p}(\Omega)\cap$$\partial\Omega$$f(x, v_{t})-c$of$C(\overline{\Omega})$

for alland$t\in I$$v_{a}\leq u$. If the map$t\mapsto v_{t}$$that-\Delta_{p}v_{t}\leq$

is continuous with respect to the topology

in S2, then $v_{t}<u$ in Sr2 for all $t\in I$ .

83

Proof. Set $E=$ { $t\in I|u\geq v_{t}$ in }. By the as sumption of the proposition,$\overline{\Omega}$

$E$

is

nonempty and closed. It suffices to show that $E$ is also open in , which means that$I$

$E=I$ .

Fix $t\in E$ . Since $u>v_{t}$ on$u>v_{t}$on . Let$\Gamma$$\Omega_{*}$$\partial\Omega$, there exists a neighborhood

be a subset of$\Omega$with$\partial\Omega_{*}\subset\Gamma$$\Gamma$of$\partial\Omega$. Then $u>v_{t}$ onsuch that$\partial\Omega_{*}$. Thereexists $\tau_{*}>0$ such that if $0<\tau<\tau_{*}$ , then $g(u)-g(u-\tau)<c$ , and we choose. From (F4), we havesuch that $u-\tau>v_{t}$ on$\partial\Omega_{*}$$\tau\in(0, \tau_{*})$$-\triangle_{p}(u-\tau)+g(u-\tau)=f(x,u)+g(u-\tau)$$>f(x, u)+g(u)-c$$\geq f(x,v_{t})+g(v_{t})-c$$\geq-\Delta_{p}v_{t}+g(v_{t})$in$\Omega_{*}$. It follows from Lemma 2.1 that $u>u-\tau\geq v_{t}$ in

conclude that $u>v_{t}$ in$\overline{\Omega}$$\Omega_{*}$. Since so is in, and hence $E$ is open.

Remark 2.1. Suppose that a family of functions$\Gamma$, we$\square$$\{w_{t}\in W^{1,p}(\Omega)\cap C(\overline{\Omega})|t\in I\}$and that there exists $c>0$ such $that-\Delta_{p}w_{t}\geq f(x,w_{t})+c$
is continuous
for all $t\in I$ . In the same way, we can prove that if the mapsatisfies $w_{t}>u$ on$\partial\Omega$$trightarrow w_{t}$with respect to the topology of$t\in I$

3

$C(\overline{\Omega})$and $w_{a}\geq u$ in St, then $w_{t}>u$ in St for all.

ProofsWe devote the rest of this paper to the proofs of Theorems 1.1, 1.2 and 1.3. Alongthe way, we prepare Lemma 3.1, which is needed for proving Theorem 1.3.

Proof of Theorem 1.1. First we shall show the existence of nonnegative solutions of

84(P). Since$\overline{u}=A$of a subsolution
Take any$\underline{u}$which is less than or equal to Of.

$x_{0}\in\Omega$. Rom (F2), there exist $r>0$ and $\delta\in(0, A)$ such that

if $|x-x_{0}|<r$ and $0<u<\delta$ . Let$f(x, u)>\sigma u^{p-1}$$-\Delta_{p}$is a supersolution, by Lemma 2.2, it suffices to show the existence$\lambda_{0}$be the principal eigenvalue ofwith Dirichlet boundary condition on the unit ball $B(O, 1)$ in

principal eigenfunction corresponding to$\lambda_{0}$$\mathbb{R}^{N}$and$\phi_{0}$thesuch that $\max\{\phi_{0}(x)|x\in B(0,1)\}=1$ :$\{\begin{array}{ll}-\Delta_{p}\phi_{0}=\lambda_{0}|\phi_{0}|^{p-2}\phi_{0}, x\in B(0,1),\phi_{0}=0, x\in\partial B(0,1).\end{array}$

It is $wen$-known that$\lambda_{0}>0$and$\phi_{0}$function is a subsolution of (P) foris positive. Then we can show that the following$\epsilon<\sigma/\lambda_{0}$:$\underline{u}(x)=\{\begin{array}{ll}\gamma\phi_{0}(\frac{x-xo}{r}) in B,0 in \Omega\backslash B,\end{array}$where$\gamma\in(0, \delta)$

Indeed, for any

and $B=B(x_{0}, r)$ is the ball in$\varphi\in W_{0}^{1,p}(\Omega_{0})$with$\mathbb{R}^{N}$

with center$x_{0}$

and radius .
$r$$\varphi\geq 0$$\epsilon\int_{\Omega}|\nabla\underline{u}|^{p-2}\nabla\underline{u}\cdot\nabla\varphi dx=\epsilon\gamma^{p-1}\int_{B}|\nabla\phi_{0}|^{p-2}\nabla\phi_{0}\cdot\nabla\varphi dx$$= \epsilon\gamma^{p-1}\int_{\partial B}|\nabla\phi_{0}|^{p-2}\frac{\partial\phi_{0}}{\partial n}\varphi ds$

一$\epsilon\gamma^{p-1}/B\Delta_{p}\phi_{0}\varphi dx$$\leq\lambda_{0}\epsilon\gamma^{p-1}\int_{B}\phi_{0}^{p-1}\varphi dx$$< \frac{\lambda_{0}\epsilon}{\sigma}\int_{B}f(x,\gamma\phi_{0})\varphi,$$< \int_{\Omega}f(x,\underline{u})\varphi dx$

where

$n$

is the outer normal unit vector to

$dx$

,

$\partial B$

at

$s$

. Since$\underline{u}\leq\delta<A=\overline{u}$from (F4) and Lemma 2.2 that there exists a nonnegative solution, it follows$u\in[\underline{u},\overline{u}]$of (P).

85

By virtue of Theorem 1 of Lieberman [8] combined with the use of boundedness ofthe solution, we have that$u\in C^{1,\tilde{\omega}}$(S2) for some$\tilde{\omega}\in(0,1)$.

Next we shall show the positivity of nonnegative solutions of (P). From (F2),there exists $\xi>0$ such that the map $u\mapsto f(x_{f}u)$ is nondecreasing in$x\in\Omega$$[0, \xi]$for all. Let$\beta(u)=\{\begin{array}{ll}0, u\in[0, \xi],g(u)-g(\xi), u\in(\xi, \infty),\end{array}$

where

$g$

is an increasing continuous function in (F4). Then, by (F5),$\beta$is nonde-creasing $and-\epsilon\Delta_{p}u+\beta(u)=f(x, u)+\beta(u)\geq f(x, 0)+\beta(0)=0$ for any nonnegative

solution$\Omega$$u$of (P). By Lemma 2.3 with $\beta(\xi)=0$ , we conclude that$u$is positive in.

口Remark 3.1. For the positivity, we assumed in (F2) that $f(x, u)$ is nondecreasingin$[0, \xi]$for ome $\xi>0$ . Or alternatively, we may as sume in (F4) that$s$$\int_{0}^{1}(sg(s))^{-1/p}ds=+\infty$$\lambda_{0}$and$\phi_{0}$be the same as those in the proof of Theorem

1.1. From (F2) and (F3), there exist $r>0$ and$\underline{K}$$f(x, u)\geq\underline{\sigma}u^{p-1}$

be a constant satisfyingLet$-\epsilon_{*}>0$$D(\Omega,\underline{K}\epsilon^{1/p})$$\underline{K}^{p}>\lambda_{0}/(q\underline{\sigma})$$\epsilon=-\epsilon_{*}has$$\underline{\sigma}\in(0, \sigma)$such that for any$x_{0}\in\Omega$,for all $x\in B(x_{0}, r)\cap\Omega$ and all $u\in[0,$ $a(x_{0})-\delta)$ . Let

be a number such that

Problem (P) withsatisfies, for Lemma 2.3.

Proof of Theorem 1.2. Let
we have that$g$with $c_{p}= \min\{2^{p-2},1\}$ .

$D(\Omega,\underline{K}\epsilon_{*}^{1/p})-\neq\emptyset,$a positive solution. Take any. Changing scaling as$\underline{K}\epsilon_{*}^{1/p}-<r$$\epsilon<-\epsilon_{*}and$$\underline{\phi}(x)=\phi_{0}((x-x_{0})/(\underline{K}\epsilon^{1/p}))$and thatany$x_{0}\in$, we have$\{\begin{array}{ll}-\epsilon\Delta_{p}\underline{\phi}=\frac{\lambda_{0}}{\underline K^{p}}\underline{\phi}^{p-1}, x\in B(x_{0},\underline{K}\epsilon^{1/p}),\underline{\phi}=0, x\in\partial B(x_{0},\underline{K}\epsilon^{1/p}).\end{array}$Taking a constant$\underline{\eta}\in(0, a(x_{0})-\delta)$

so thatshall show that the family of functions$\underline{\eta}<\min\{u_{\epsilon}(x)|x\in B(x_{0},\underline{K}\epsilon^{1/p})\}$$\{t\underline{\phi}(x)+\underline{\eta}|0\leq t\leq a(x_{0})-\delta-\underline{\eta}\}$, wesatisfies

86

the assumption foron$v_{t}$of Proposition 2.1. Indeed, set, and since $0\leq v_{t}\leq a(x_{0})-\delta$ in$\partial B(x_{0}, \underline{K}\epsilon^{1/p})$$v_{t}=t\underline{\phi}+\underline{\eta}$$B(x_{0},\underline{K}\epsilon^{1/p})$. Then$v_{t}=\underline{\eta}<u_{\epsilon}$, we have$-\epsilon\Delta_{p}v_{t}-f(x, v_{t})\leq-\epsilon t^{p-1}\Delta_{p}\underline{\phi}-\underline{\sigma}(t\underline{\phi}+\underline{\eta})^{p-1}$$\leq t^{p-1}\frac{\lambda_{0}}{\underline K^{p}}\underline{\phi}^{p-1}-q\underline{\sigma}(t^{p-1}\underline{\phi}^{p-1}+\underline{\eta}^{p-1})$$=( \frac{\lambda_{0}}{\underline K^{p}}-q\underline{\sigma})t^{p-1}\underline{\phi}^{p-1}-q\underline{\sigma}\underline{\eta}^{p-1}$$\leq-q\underline{\sigma}\underline{\eta}^{p-1}$

infor all$B(x_{0},\underline{K}\epsilon^{1/p})$$t\in[0, a(x_{0})-\delta-\underline{\eta}]$. Since$v_{0}\leq u_{\epsilon}$in$B(x_{0},\underline{K}\epsilon^{1/p})$$hom$ Proposition 2.1 that $u_{\epsilon}(x)>(a(x_{0})-\delta-\underline{\eta})\underline{\phi}(x)+\underline{\eta}$ in$u_{e}(x_{0})>a(x_{0})-\delta$

for all$\epsilon<-\epsilon_{*}$

and all$x_{0}\in D(\Omega,\underline{K}\epsilon^{1/p})$, it follows$B(x_{0},\underline{K}\epsilon^{1/p})$. Thus.

Next we show the inverse inequality in a similar way. From (F3), there exist$r>0$ andsuch that for any$\overline{\sigma}>0$$x_{0}\in\Omega$, we have that $f(x, u)\leq-\overline{\sigma}(3A-u)^{p-1}$

for $aUx\in B(x_{0}, r)\cap\Omega$ and all $u\in(a(x_{0})+\delta, 3A$]. Let$\overline{K}$be a constant satisfying$\overline{K}>\lambda_{0}/(q\overline{\sigma}).\cdot$

Let

with

be a number such that$\Xi_{*}^{-}>0$$\epsilon=\overline{\epsilon_{*}}$$D(\Omega,\overline{K}\overline{\epsilon_{*}})\neq\emptyset,$

has a positive solution. Take any

Changing scaling as$\epsilon<\overline{\epsilon_{*}}$$\overline{\phi}(x)=\phi_{0}((x-x_{0})/(\overline{K}\epsilon^{1/p}))$$\overline{K}\overline{\epsilon_{*}}^{1/p}<r$

and any

and that (P)$x_{0}\in D(\Omega,\overline{K}\epsilon^{1/p})$.

, we have$\{\begin{array}{ll}-\epsilon\Delta_{p}\overline{\phi}=\frac{\lambda_{0}}{\overline,K^{p}}\overline{\phi}^{\varphi-1}, x\in B(x_{0}, \overline{K}\epsilon^{1/p}),\overline{\phi}=0, x\in\partial B(x_{0},\overline{K}\epsilon^{1/p}).\end{array}$Taking a constant $\overline{\eta}\in(0,2A-a(x_{0})-\delta)$ , we shall show that the family of functions$\{3A-t\overline{\phi}(x)-\overline{\eta}|0\leq t\leq 3A-a(x_{0})-\delta-\overline{\eta}\}$2.1. Note that$A$

gives $u_{e}\leq A$ insatisfies the assumption for$w_{t}$

of Remarkis a supersolution of (P) and that Lemma 2.1 with (F3) and (F4)$\Omega$. Set$w_{t}=3A-t\overline{\phi}(x)-\overline{\eta}$. Then $w_{t}=3A-\overline{\eta}>A+a(x_{0})+\delta>$

87$A\geq u_{\epsilon}$on$\partial B(x_{0}, \overline{K}\epsilon^{1/p})$, and since $a(x_{0})+\delta\leq w_{t}\leq 3A$ in$B(x_{0}, \overline{K}\epsilon^{1/p})$, we have$-\epsilon\Delta_{p}w_{t}-f(x, w_{t})\geq\epsilon t^{p-1}\triangle_{p}\overline{\phi}+\overline{\sigma}(t\overline{\phi}+\overline{\eta})^{p-1}$$\geq-t^{p-1}\frac{\lambda_{0}}{\overline,K^{r}}\overline{\phi}^{p-1}+q\overline{\sigma}(t^{p-1}\overline{\phi}^{\varphi-1}+\pi^{-1})$$=(c_{p} \overline{\sigma}-\frac{\lambda_{0}}{\overline,K^{\varphi}})t^{p-1}\overline{\phi}^{\varphi-1}+q\overline{\sigma}\pi^{-1}$$\geq\%^{\sigma}V^{-1}$in$B(x_{0},\overline{K}\epsilon^{1/p})$for all $t\in[0,3A-a(x_{0})-\delta-\overline{\eta}]$ . Since$w_{0}>A\geq u_{\epsilon}$in$B(x_{0},\overline{K}\epsilon^{1/p})$,it follows from Remark 2.1 that $u_{\epsilon}(x)<3A-(3A-a(x_{0})-\delta-\overline{\eta})\overline{\phi}(x)$ –fi in$B(x_{0},\overline{K}\epsilon^{1/p})$

Settingwhen. Thus$u_{\epsilon}(x_{0})<a(x_{0})+\delta$$\epsilon_{*}=\min\{\underline{\epsilon_{*}},\overline{\epsilon_{*}}\}$

andfor all$K= \min\{\underline{K},\overline{K}\}$$\epsilon<\epsilon_{*}\bm{t}dx_{0}\in D(\Omega, K\epsilon^{1/p})$$\epsilon<\overline{\epsilon_{*}}\bm{t}d$all, we conclude that$x_{0}\in D(\Omega,\overline{K}\epsilon^{1/p})$.

$|u_{\epsilon}(x_{0})-a(x_{0})|<\delta$.

$\square$

To show Theorem 1.3, we prepare
Lemma 3.1. Let$\lambda,$

$q,$

$R$

and

$\delta$be positive constants and$h$the unique solution of(3.2)$\{\begin{array}{ll}-\epsilon\triangle_{p}h+\lambda h^{q}=0, x\in B(O, R),h=\delta, x\in\partial B(0, R).\end{array}$

If $q<p-1$ and$0< \epsilon<\frac{\lambda\theta^{p}R^{p}}{(pq+N\theta)p^{p-1}\delta^{\theta}}$$(\theta :=p-1-q>0)$ ,(3.3)

then $h(O)=0$ .

Proof. Due to the uniqueness of the solution of (3.2), it is easy to see that thesolution$h$must be radially symmetric. Writing $h=h(r)$ with $r=|x|$ , we have$\{\begin{array}{ll}-\frac{\epsilon}{r^{N-1}}(r^{N-1}|h_{r}|^{p-2}h_{r})_{r}+\lambda h^{q}=0, r\in(O, R),h’(0)=0, h(R)=\delta.

\end{array}$

(3.4)

88

It follows from direct computation that the following function satisfies the equationof (3.4) when $q<p-1$ :$\overline{h}(r)=Cr^{p/\theta}$,

where $\theta=p-1-q>0$ and$C=( \frac{\lambda\theta^{p}}{\epsilon(pq+N\theta)p^{p-1}})^{1/\theta}$

Since (3.3) implies$r\in[0, R]$. Since$\delta<\overline{h}(R)=CR^{p/\theta}$$\overline{h}(0)=0$.

, Lemma 2.1 gives $0\leq h(r)\leq\overline{h}(r)$ for, we conclude that $h(O)=0$ .

Proof of Theorem 1.3. By (F5), there exists$f(x, a+v)\leq-\lambda v^{q}$$f(x, a-v)\geq\lambda v^{q}$$\square$$\delta_{0}\in(0, \alpha)$

for all
for allsuch that for any$x\in\Omega_{0}$$x\in\Omega_{0}$$v\in[0, \delta_{0}$

,

(3.5)

.

(3.6)

Since the function $v=u_{\epsilon}-a$ satisfies $-\epsilon\Delta_{p}v=f(x, a+v)$ a.e. inpositive part $v+\in W^{1,p}(\Omega_{0})$ of$v$$\varphi\in W_{0}^{1,p}(\Omega_{0})$$\Omega_{0}$, thesatisfies$-\epsilon\Delta_{p}v_{+}\leq f(x,a+v_{+})$

Indeed, for any

)

with

in

$\Omega_{0}$

.

(3.7)

$\varphi\geq 0$$\epsilon\int_{\Omega_{0}}|\nabla v_{+}|^{p-2}\nabla v_{+}\cdot\nabla\varphi dx=\epsilon\int_{\{v>0\}\cap\Omega_{0}}|\nabla v|^{p-2}\nabla v\cdot\nabla\varphi dx$$= \epsilon\int_{\partial(\{v>0\}\cap\Omega_{0})}|\nabla v|^{p-2}\frac{\partial v}{\partial n}\varphi ds$

一$\epsilon\int_{\{v>0\}\cap\Omega_{0}}\Delta_{p}v\varphi dx$$\leq-\int_{\{v>0\}\cap\Omega_{0}}f(x, a+v)\varphi dx$$=- \int_{\Omega_{O}}f(x, a+v_{+})\varphi dx$

where

$n$

is the outer normal unit vector to$v_{-}=(a-u_{\epsilon})_{+}$,$\partial(\{v>0\}\cap\Omega_{0})$

at

$s$

. In a similar way,satisfies$-\epsilon\Delta_{p}v_{-}\leq-f(x, a-v_{-})$

in$\Omega_{0}$

.

(3.8)

89

Take so small $\eta>0$ thatexists$\epsilon(\delta)>0$$D(\Omega_{0}, \eta)\neq\emptyset$

such that for any. Fix$\epsilon\in(0,\epsilon(\delta)),$$\delta\in(0, \delta_{0})$. By Theorem 1.2, there$\max\{v_{\pm}(x)|x\in D(\Omega_{0}, \eta/2)\}<\delta$.

Applying (3.5) and (3.6) to $v=v_{\pm}\in[0, \delta$), we have, respectively,$f(x, a+v_{+})\leq-\lambda v_{+}^{q}$$f(x, a-v_{-})\geq\lambda v_{-}^{q}$

for all $x\in D(\Omega_{0}, \eta/2)$ ,for all $x\in D(\Omega_{0}, \eta/2)$ .

(3.9)(3.10)

Combining these inequalities $(3.7)-(3.10)$ , we obtain$-\epsilon\triangle_{p}v\pm+\lambda v_{\pm}^{q}\leq 0$Let$\epsilon_{0}\in(0, \epsilon(\delta))$be an$q\in(O,p-1)$ , for any$\epsilon$in$D(\Omega_{0}, \eta/2)$.

(3.11)

satisfying (3.3) with $R=\eta/2$ . Then, by Lemma 3.1 with$\epsilon\in(0, \epsilon_{0})$, the unique solution of the boundary value problem$\{\begin{array}{ll}-\epsilon\Delta_{p}h+\lambda h^{q}=0, x\in B(O, \eta/2),h=\delta, x\in\partial B(0, \eta/2)\end{array}$(3.12)

satisfies $h(O)=0$ .

Take any$x_{0}\in D(\Omega_{0}, \eta)$$v_{\pm}(x)\leq h(x-x_{0})$. It follows from (3.11), (3.12) and Lemma 2.1 thatfor all $x\in B(x_{0}, \eta/2)$ . Since $h(O)=0$ , we have $v_{\pm}(x_{0})=0$ , andfor all.

hence$u_{\epsilon}(x)=a$4

Announcement: Semilinear Case$x\in D(\Omega_{0}, \eta)$

Theorem 1.3 says that if$\Omega|u_{e}(x)=a(x)\}$$\epsilon$

is sufficiently smaf, then the coincidence set$\square$$O_{\epsilon}=\{x\in$has an interior point in a subdomain where $a(x)$ is constant.

However, if we as sume that$O_{e}$

equation in (P) on the interior ofhas an interior point, then $a(x)$ also satisfies the$O_{\epsilon}$, and hence $a(x)$ has to be p-harmonic. Thus itis natural to expect that the coincidence set has an interior point if and only if $a(x)$is p-harmonic.

90

We shall announce that Theorem 1.3 with $p=2$ will be extended to the casewhere $a(x)$ is harmonic on a subdomain.
Theorem 4.1. Let $p=2$ . Suppose $(F1)-(F4)$ and(F6)$a(x)\in C(\overline{\Omega})\cap H^{2}(\Omega),$and there enist $q\in(O, 1)$ and$\triangle a(x)=0a.e$$\lambda>0$. in a nonempty subdomain

Then, for sufficiently small $\eta>0_{f}$ there exists$u_{\epsilon}$of$\Omega$such that$\lim_{uarrow a}\sup_{(x)}\frac{f(x,u)-f(x,a(x))}{|u-a(x)|^{q-1}(u-a(x))}<-\lambda$

then every positive solution$\Omega_{0}$uniformly in$\epsilon_{0}\in(0, \epsilon_{*})$$\Omega_{0}$.

such that if(4.13)$\epsilon\in(0, \epsilon_{0})$of (P) satisfies$u.(x)=a(x)$for all$x\in D(\Omega_{0}, \eta)$.

Theorem 4.1 can be shown in the similar way as Theorem 1.3. The corresponding

result to the p-Laplace operator has not been obtained because the proof strongly
relies on the linearity of Laplace operator.

5

Remarks

We give a few remark on the theorems.
(1) For all the results of this paper, it is sufficient for (F4) to be ass umed onlyin the interval$[\xi, A]$with $g(\xi)=0$ , instead of$[0, \infty$) with $g(O)=0$ .

(2) It is easy to extend Theorem 1.3 to the case where $a(x)$ is constant on morethan one subdomain.
(3) Theorem 1.1 does not assure the uniqueness of positive solutions. It is aninteresting problem whether the positive solutions will be unique (when$\epsilon$

is suffi-

ciently small). We have positive answers under some cases: $p=2$ by Angenent [1];$p>1$ and $f(x, u)=f(u)$ by Guo [6].

91References[1] S. Angenent, Uniqueness of the solutions of a semilinear boundary value prob-lem, Math. Ann. 272 (1985), 129-138.
[2] A.

$Ca\tilde{n}ada$, P. Dr\’abek and J. L. G\’amez, Existence of positive solutions forsome problems with nonlinear diffusion, Rans. Amer. Math. Soc. 349 (1997),
4231-4249.
[3] E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic
equations when a parameter is large, Proc. London Math. Soc. 53 (1986), 429-

452.
[4] J. $Garc’1a_{r}Meli\acute{a}n$ and J. Sabina de Lis, Stationary profiles of degenerate prob-

lems when a parameter is large, Differential Integral Equations 13 (2000), 12011232.
[5] M. Guedda and L. V\’eron, Bifurcation phenomena as sociated to the p-Laplace
operator, Rans. Amer. Math. Soc. 310 (1988), 419-431.

[6] Z. Guo, Uniqueness and flat core of positive solutions for quasilinear elliptic
eigenvalue problems in general smooth domains, Math. Nachr. 243 (2002), 43-

74.
[7] S. Kamin and L. V\’eron, Flat core properties as sociated to the p-Laplace operator, Proc. Amer. Math. Soc. 118 (1993), 1079-1085.

[8] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219.

[9] J. L. V\’azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202.