The intermediate solution of quasilinear elliptic boundary value problems

THE INTERMEDIATE SOLUTION OF QUASILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

BONGSOO Ko*

1. Introduction

We study the existence of an intermediate solution of nonlinear el- liptic boundary value problems (BV P) of the form

(BVP) {

~u = I(x, u, Vu), Bu(x) = <j>(x),

10 n

on 8n,

where n is a smooth bounded domain in Rn, n ~ 1, and 8n ^{E C}

^,Ot,

(0 <

< 1), ~ is the Laplacian operator, Vu = (D

u,D

u,'" ,Dnu) denotes the gradient of u and

Bu(x) = p(x)u(x) + q(x) d)x), du

where ~~ denotes the outward normal derivative of u on an.

Suppose now, that v, v ^and w, tU are two pairs of subsolutions and supersolutions in the class C

(O) or in the usual Sobolev space W

,p(n), p > n of (BVP) such that vex) $ vex), w(x) $ w(x), vex) $ w(x) for all x E 0 ^and v(xo) < w(xo) for some Xo E O. Then there is a solution in the order interval [v, vJ = {u E C(O) : vex) $ u(x) $ vex), x EO} and a solution in lw, wJ. And furthermore Amann [lJ or Amann and Crandall [3] showed that there exists an intermediate solution in the set [v, wJ \ ([v, v] u lw, w]) under additional conditions.

The existence of a solution given a pair of quasisubsolution and quasisupersolution of (BV P), v and v, with v( x) $ v( x) for all x E n,

is well known (see [9]). Since these functions may have singular points

Received April 22, 1993.

* This paper was supported(in part) by NON DIRECTED RESEARCH FUND,

Korea Research Foundation, 1992.

in the interior of n, there arises the question, does there also exist an intermediate solution if there are pairs of quasisubsolutions and quasisupersolutions as in the preceding paragraph ?

Suppose now, in addition, that f is independent of 'Vu and that there are pairs of quasisubsolutions and quasisupersolutions as in the preceding paragraph~ Ko [6] proved the existence of an intermediate solution in [v, to] \ ([v, il] U lw, ^to]) under additional conditions. Hence there also arises the question, does there exist an intermediate solution if f depends nonlinearly on 'Vu ?

The author is able t~ solve the above problem which is the existence of an intermediate solution for (BVP) using Maximum Principles and the theorem on existence of several fixed points (see pp241, [4]). The multiplicity result is a generalization of Theorem(1.6) in [1] or Theorem 2 in [3].

As a simple application of our results, we prove the existence of sev- eral positive ordered solutions in a class of singularly perturbed quasi- linear elliptic Dirichlet boundary value problems with small positive parameter.

Throughout this paper we assume that p, q E C1,a(an) are non- negative real valued functions which either q( x) = 0 for all x E an or q( x) > 0 for all x E an, and f satisfies the following conditions:

0< a < 1,

(1) f: fi x I x R n ~ R is a a-Holder continuous function, such that

- af aj .

f (., e, ^{7J) E} ca ⁽ⁿ⁾ and such that ae and a7J are contmuous where (x, e, ^7J) denotes a generic point of fi x I x Rn and I is a fixed bounded and closed interval in R.

(2) There exists a continuous function c : R+ ~ R+ = [0, 00) such that

If(x,e,ry)1 _~ c(p)(l + 1171 ^{2 )}

for every p 2: 0 and (x,e,7J) E fi x [-p,p] x Rn.

(3) </J E C ² ,a(Q) and for the Dirichlet problem case, </J(x) E I for all x E an.

By a solution of (BV P) we mean a function u : fi ~ I such that

u E C ² (Q) and u satisfies (BVP) pointwise.

2. Main results

First of all, we state definitions of a quasisubsolution and a quasisu- persolution of (BV P).

DEFINITIONS. A function

n ~ R is a quasisupersolution of (BV P) in n, ^{if for any}

^E n, there exist a neighborhood N of

and a finite number of functions

E C

(N), k = 1,2" .. ,p such that

(I) 1O(X) = min 1Ok(X),

15,k5,p

for all x E N, where p may depend on

and

(11)

(Ill)

for all x E N n nand k = 1,2,' .. ,p. Furthermore, if

E an, p(XO)1Ok(XO) + q(xo) dl/ (:ro)

~ 4>(xo),

for all k.

A quasisubsolution

n ~ R is defined similarly, replacing min by max in (I) and reversing the inequalities (II) and (I II).

To state the theorem for the existence of an intermediate solution of (BV P), we need the following notations: Let u, v : n ~ R be functions. Then we write u ::; v if u( x) ::; v( x) for all x E n, ^{and u < v}

if U ::; v but u i= v. By [u, v] we mean the order interval between u and v, that is,

[u, v] = {1O : n ~ R: u ::; ^{tu ::;} v} . The following theorem is the main result.

THEOREM 1. Let f satisfy (1) and (2) and 4> satisfy (3). Suppose that Vj is a quasisubsolution a,nd Vj is a quasisupersolution of (BV P) for j = 1,2 such that VI ::; VI, V2 ::; V2, VI ::; V2 and VI(XO) < V2(XO) for some Xo E n. Assume moreover that ^['1 and V2 are not solutions of (BVP) and [ih(X),V2(X)] C I for all x E n. Then (BVP) has at least three distinct solu tions U

such tila t VI ::; U I < Uo < u2 ::; V2, Uj E [Vj, Vj] for j = 1,2 and Uo E [VI, £'2] \ ([VI, £'d U [V2' V2]).

Theorem 1 is a generalization of Theorem(1.6) in [1] or Theorem 2

in [3] and follows at once from the next proposition.

PROPOSITION. Let the hypotheses of Theorem 1 hold and let h : Rn -+ Rn be defined by hex) = (hl(x), h2(x),··· ,hn(x)) and bounded and of class Cl such that each partial derivatives for hi is bounded on Rn. Then the following elliptic boundary value problem

{

~u = I(x,u, h(\1u»

Bu =<P

ill

Q

on aQ

has at least three distinct solutions Uo, UI, U2 such that th S UI < Uo <

U2 S ih, Uj E [Vj, Vj] for j = 1,2, and Uo E [VI, V2] \ ([VI, VI] U [V2, V2]).

To prove Proposition, we first convert (BVPh) into an operator equation. Choose A > 0 large enough so that ~; ^(x, e, ^{h( 7J ») + A > 0}

for all (x,e,7J) E n x I x Rn. For any 9 E [VI,V2] n cO'(n) we assume that A satisfies

f(x,cI,h(O» + A(CI - g(x» < 0 < j(x,c2,h(O» + A(C2 - g(x»

for some constants Cl, C2 with Cl < 0 < C2 and for all x E n. Fur- thermore, if Btt = u, then <p : fi ^{-+ [CI,C2],} and if not, p(X)CI S

<p( x) S p( ^X )C2 for all x E an. Then it is known that there is a solution u E c ² ,0'(n) of the following boundary value problem

{

~U = f(x, u, h(\1u)) + A(U - g) Bu = <p on an.

in 0,

This solution is denoted by

= Tg below.

LEMMA 1. A function u E [VI,V2] n CO'(fi) is a solution of (BVPh)

if and only ifu = ^Tu.

LEMMA 2. Let Vj and Vj be a quasisubsolution and a quasisuperso- lution of (BVPh), respectively. Then

V· _{J -} < Tv' _J

Proof. To show TVj(x) S Vj(x) for all x E n, suppose that there is

a point Xo En ^{such that Vj(xo) < TVj(xo). Let a =} TVj(x) - Vj(x) be

positive maximum value of TVj - Vj. Then there exists a neighborhood Uz of x ^{such that} x E Uz and 0 < TVj(x) - Vj(x) $ a for all x E Uz.

Case 1. x ^{E n.}

By the definition of a quasisupersolution, there exist a neighborhood Nz and a finite number of functions Wk E C ² (N z), ^{k = 1,2,'" ,p such}

that x ^E Nz c Uz and

for all x E Nz n n. Let Vj(x) = ^Wk(X) for some k. Then 0 < TVj(x)- Wk(X) $ a for all x E Nz n nand TVj - Wk has the positive maximum value a at x in the neighborhood Nz n n.

On the other hand, in Nz n n, by Mean Value Theorem,

~(Wk - TVj)(x)

$f(x, Wk(X), h(~Wk(X))) - f(x, TVj(x), h(~Tvj(x)))

- A(Tvj - Vj)(x)

$[fe(x,C(x),h(~Wk(X))) + A](Wk - TVj)(x)

+ f,,(x,Tvj(x),h(7]*(x)))' dh· _~(Wk - TVj)(x).

Since f" . dh is bounded on n x I x Rn, we can choose a bounded function b : Rn -+ Rn such that

for all x E Nz n n. Hence, on Ni: n n,

By Maximum Principles, TVj(x) - Wk(X) = a for all x E Nz n n,

whence TVj(x) = ^Vj(x) + a for all x E Nz n n. By the continuation of the method on the boundary of Nz nn, we can conclude that TVj(x) =

Vj(x) + a for all x E n. And so, for any x E n,

~Vj(X)

=~Tvj(x)

=f(x, Vj(x), h(~vj(x))) + Ue(x, C(x), h(~f'j(x))) + A)a,

where C(x) lies between Vj(x) and Vj(x) + a. Hence, for any x E n,

~Vj{X) 2: f(x, Vj(x), h(\7vj(x))).

Since ~Vj(x) :s; f(x,vj(x),h(\7vj(x)) locally in n, so [fe(x, C(x), h(\7vj(x))) + >.]a :5 o.

This leads to a contradiction for a ^{> O.}

Case 2. x E an.

Since TVj(x) = Wk(X) + a and 0 < TVj(x) - Wk(X) < a for all x E Ni; n n, so

If p(x) > 0, then

</1(x) = p(x)Tv -(x) + q(x) dTvj(x)

J

dv

_ _ A

dWk(X)

2:p(X)[Wk(x)+a]+q(x) dv 2: 4>(x) + p(x)a.

This leads to a contradiction for p(x)a > o.

Let p(x) = O. Then q(x) > O. If

dTvj(x) dWk(X) dv > dv ' then

</1(x) = q(x) dT~~(x) ^> q(x) dw;:x) 2: </1(x).

This also leads to a contradiction. Let

For all ^x E Ni; n n,

~(TVj - Wk - a)(x)

?f(x, Tvj{x), h(\7Tvj(x))) - f(x, Wk(X), h(\7wk(X)))

+ >.(Tvj - Vj )(x).

By the Mean Value Theorem and choosing suitable bounded function b : Rn -+ Rn as before, we can also show that

t!J.(TVj - Wk - a)(x) - b(x)· "\l(Tvj - Wk - a)(x) 2:: o.

Since TVj - Wk - a has the zero maximum value of the boundary point

x ⁱⁿ Ni; n n, by Maximum Principles, for all x E Ni; n n,

This implies that TVj - Wk has the positive maximum value a at an interior point of n. By Case 1, this also leads to a contradiction.

Therefore, TVj :S Vj.

Similarly, we can show that TVj 2:: Vj' •

LEMMA 3. Let Vj and Vi be a quasisubsolution and a quasisuperso- lution of (BV Ph), respectively. Then T is an increa.<;ing operator from [vi, vi] n CQ:(f2) into itself, i.e. if u :S ^v, then ^Tu :S ^Tv.

Proof. Since [vi> vi] n CQ:(f2) is a bounded interval in C(f2), if we choose two constants

and

such that

< 0 <

g < 0,

C2 -

g > 0 for all g E [vj, Vj] n CQ:(f2), then there is u E

n CQ:(f2) such that u = Tg.

We first show that T is well-defined on [Vi, vi] n CQ:(f2). Let u = Tg and v = Tg. Then

t!J.( ^u-v )(x) = f( x, u( x), h("\lu( x») - f( x, v( x), h("\lv( x» )+A( u-v)( x) .

If we choose some bounded functions b _{i :} Rn

Rn, i = 1, 2 so that

~(u - v)(x) - br(x)· \7(u - v)(x) :S 0,

and

~(u - v)(x) - b

(x)· \7(u - v)(x) 20,

for all x E f2, then by Maximum Principles, u = ^v.

We secondly prove that T is increasing on [vj, vi] n cQ'(n). Let gI,g2 E [Vi, Vi] n cQ'(n) and gl :::; g2. Then

t:.(Tg2 - Tgl)(x)

=(J(x, Tg2(x), h(\7Tg2(x ») - f(x, Tg2(x), h(\7Tg 1 (x »)]

+ [f(x, Tg2(x), h(\7Tg

(x») -I(x, Tgl(x), h(\7Tg

(x »)]

+ )'(Tg2 - Tgd(x) + ).(gl - g2)(X).

We then choose two bounded functions b : Rn

Rn and fJ : Rn

R such that -). < fJ(x) < ). for all x E n ^and

t:.(Tg2 - Tgd(x) - b(x)· \7(Tg2 - Tgd(x) - (fJ(x) + )')(Tg2 - Tgd(x) :::; ).(gl - g2)(X) :::; ⁰

for all x E n. By Maximum Principles, (Tg

Tg

x) 2: 0, x E n.

By Lemma 2 and that T is increasing, we note that u = T 9 E

[vj, vi] n cQ'(n). •

LEMMA 4. Let vi and vi be a quasisubsolution and a quasisuper- solution of (BV Ph), respectively. Then T is continuous and compact from [vi, vi] n Ca(n) into itself.

Proof. We first show that T is continuous. Consider a sequence {gn}

in [vi, Vi] n cQ'(n) and suppose lim _{n _ co} gn ^{= 9 in} [VI, V2] n cQ'(n) and lim Tg n = y

n-co

in C ² ,a(n). Then lim _{n _ co} t:.Tgn ⁼ t:.y and lim _{n _ co} \7g n - \7y in C(n). Hence

t:.y(x) = f(x, y(x), h(\7y(x») + ).(y - g)(x)

for all x E f! and By(x) = <p( x) for all x E af!. By the uniqueness of solutions corresponding g, Tg = y. By the Closed Graph Theorem, T is continuous on [Vj, Vj] n cQ'(n).

We note that c ² ,Q'(n) is compactly embedded in cQ'(n). Hence T is compact on [vj, Vj] n ca(n). •

To extend the operator T to [Vi, Vj]nC(n) continuously, we will use

the following theorem. It can be found in Amann and Crandall [3].

THEOREM 2. Let f satisfy (/2). Then there is an increasing func- tion 'Y : [0,(0) ^-+ [0,(0) such that if u is a solution of (BV Ph) then

Moreover, 'Y depends only on b., B, n, n, p, and c.

Since CO'(Q) is dense in C(Q) and T is a continuous increasing com- pact operator from [vj, Vj] n CO'(Q) into itself, we can extend T to [Vj, Vj] n C(Q) continuously and compactly. To show that this is possi- ble, let u E [Vj, Vj]nC(Q). Then there exists a monotone sequence {Un}

in [vj,Vj] n CO'(Q) so that

-+

in C(Q) as n -+

Since {Tun}

is bounded in C(Q), by Theorem 2, {Tun} is bounded in W

,p(n), and if p > n, then {Tun} is bounded in ct,O'(n). By the Mean Value Theorem, {Tun} is equicontinuous on C(n). By Ascoli-Azela Theo- rem, {Tun} has a convergent subsequence in C(Q). Since {Tun} is monotone, we can define Tu by

Tu = lim TUn.

n-oo

Since TUn is bounded in ct,O'(n), so Tu E cO'(n). Therefore, we view T as a continuous extension to an operator (denoted again by T) mapping [vj,Vj] n C(n) into [Vj,Vj] n cO'(n). Since the imbedding of CO'(Q) in C(Q) is compact it follows that the operator T maps [Vj, Vj] n C(fi) compactly into [Vj, Vj] n C(fi).

To complete the proof of Proposition, we need the special ordered Banach space Ce(n) whose positive cone is normal and has nonempty interior. In defining Ce(n), e E C(n), e(x);::: 0 for all x E Q, e(x) of O.

Let Ce(Q) be the set of all functions u E C(fi) so that -ce(x) :::; u(x) :::; ce(x)

for some constant c 2 0 and for all x E n. ^{If u E Ce(n),} we define the norm

Ilulle ⁼ inf{c > 0: -ce(x):::; u(x):::; ce(x), x E n}.

It can be shown that the Minkowski functional 11· lie is a norm on Ce(U).

Furthermore, Ge(n) is a Banach space with respect to the norm.(see [2])

Now we state the theorem which will be use in proving the exis- tence of an intermediate solution of (BV Ph ) for Dirichlet boundary condition. The main idea of the proof for the following theorem can be found in Amann [2].

THEOREM 3. Let e be the unique solution of the boundary value problem

{

.6.e(x) = -1, x En e( x) = 0, x E an

and T be the operator induced by the boundary value problem

{

.6.Tu = f(x, Tu, h('\7Tu» + >..(Tu - u), x E n

Tu(x) = 0, x E an

with a quasisubsolution Vi and a quasisupersolution Vi of (BVPh) so that Vi < Vi. Then Ce(U) is continuously imbedded in C(U) and T is a compact operator from [vi, Vi} n C(n) into Ge(U).

Proof. By the previous sta.tements, T maps [vj, Vi] n {u E C(U) : ulao = O} compactly into [vj, Vi} n {u E Gl(U) : ulao = O}. Therefore it suffices to show that {u E Cl(U) : u!ao = O} is continuously imbed- ded in Ce(Q). ';Ye follows the proof in [2, Theorem 4.2]. Since, by the Maximum Principle, on every compact subset of n, e is bounded below by a positive constant and since, for every x E an, g~ < 0, it follows by continuity that,for every u E eMU), there exist a, f3 > 0 with ..

-ae :s; u :s; (3e ,

i.e. CJ(U) is a subset of Ce(U). Since convergence in the norm of Ce(U) as well ^as in the norm of CJ(U) implies pointwise convergence, it is easily seen that the injective ma.p from CJ(U) into Ce(U) is a closed linear operator. Hence, by the Closed Graph Theorem, CJ(U) is continuously imbedded in Ce(U) and the statement follows. •

Now, we obtain the conclusion.

THEOREM 4. Let Vj and Vj be a quasisubsolution and a quasisu- persolution of (BVPh), respectively. Then the operator T induced by the problem (BV Ph) is continuous, increasing and compact from [Vj, Vj] n C(O) into itself.

Finally, to prove Proposition, we will use the following theorem of existence of several fixed points, and we can find it in Deimling [4] or Amann [2].

THEOREM 5. Let X be a Banach space: S e X a retract and T : S -+ S compact; SI, S2 nonempty disjoint retracts of S; E j c ^{Sj open}

in 5 for j = 1,2. Suppose that T( Sj) c Sj and Fix(T) n (Sj \ Ej) = ⁰

for j = 1,2, where Fix(T) = {u E S : Tu = u}. Then T has fixed points Uj E Ej and a third fixed point Uo E 5 \ (SI U 52).

Proof of Proposition.

Case 1. q(x) > ^{0 for all x E} an.

Let

0

= {u E C(O): u(x) < Vl(X), X E O}, and

O 2 = {u E C(O) : u(x) > V2(X), x E 11},

5 = [VI, V2] n C(O), SI = [VI, VIJ n C(O), S2 = [V2, V2] n C(O), El = 5nOI, _{and E 2} = Sn02. Then El and E2 are open in 5. From Lemma 2,3,4, and Theorem 4, T : S -+ S is compact. Clearly, SI and S2 are disjoint retracts of S, Ej C Sj, T(Sj) c ^{Sj for j = 1,2.} To show that Fix(T) n (Sj \ Ej) = 0, we assume that there is u E Fix(T) n (Sj \ Ej) for some j. Then u E Sj \ Ej and Tu = u.

Let j = 1. We note that u is a solution of (BV Ph)' Since u E SI \E

so VI :::;

VI and there is a point Xo E 11 such that u(xo) = Vl(XO)' By the definition of a quasisupersolution, let

on some neighborhood U

of Xo and let u(xo) = Wk(XO) for some k.

For all x E U

n n, we can show that

~(u - Wk)(X) - b(x)· V(u - wk)(:r) - A(U - Wk)(X)

~[-Ge(x, C(x), h(Vu(x») + A](Wk - u)(x) ~ 0,

where b : Rn -+ Rn is some bounded function. Since VI is not a solution of (BVPh), by Maximum Principles, u(x) = Wk(X) for all x E U

n f! and Xo E af!. Since u - Wk has a zero maximum value at the boundary point Xo, either u(x) = Wk(X) and x E U

n n ^or

d;,}' (xo) _<~: (xo). We note that both cases lead to a contradiction.

Consequently, Fix(T) n (SI \ El) = 0.

Similarly, we can show that Fix(T) n (S2 \ ~) = 0. Therefore, by Theorem 5, T has at least three distinct fixed points

such that Uj E [Vj,Vj] for j = 1,2, and especially note that

Case 2. q(x) = 0 for all x E af!.

We assume that the Dirichlet boundary condition for (BVPh), i.e.

Bu = u = </J = 0 on af!. Let

and

Sj = Ce(n) n ^{[vj, Vj]}

for j = 1,2. 'Ve note that T : S -+ S is compact; Sj C S and nonempty; T(S) C S, T(Sj) C Sj for j = 1,2. Since S, SI and S2 are convex in Ce(n), these are retracts of S, and clearly SI n S2 = 0. Let

and

E 2 = S2 n ^{u E Ce(n): u(x) > V2(X),X E f!}.

We show that El and E

are open in S. Let v E El. Then v( x) < VI (

for all x E il, and there is constant c ~ 0 such that -ce(x) ~ vex) ~ ce(x)

for all x E n. Then we can choose 13 ~ 0 so that 13 ~ c and v( x) +

f3e(x) < VI (x) for all x E il. Let B( v, 13) be the open ball in S with respect to the norm 11 . lie, with center v and radius 13. Then for any u E B(v, 13),

-f3e(x) ~ u(x) - vex) ~ f3e(x)

for all x E O. Hence u(x) :::; (3e(x) + ^vex) < VI(X) for all x E n. Hence u E El' Therefore, B( v, (3) eEl.

Similarly, we can show that E ₂ is also open in S.

Next, we show that Fix(T) n (Sj \ Ej) = 0, j = 1,2. Suppose that there is u E Fix(T) n (Sj \ E j ) for some j. Then u E Sj \ Ej and Tu=u.

Let j = 1. We note that u is a solution of (BV Ph). Since u E SI \E

VI :::; U :::; VI and there is a point Xo E n such that u(xo) = VI(XO).

By Maximum Principles and the definition of a quasisupersolution of (BVPh), we can show that there is a neighborhood U

of Xo such that u( x) = VI (x) for all x E U

n n. By the continuation of this method on the boundary of the neighborhood U

n n, we can conclude that u( x) = VI (x) for all x E O. This implies that VI is a solution of the (BV Ph). This leads to a contradiction because VI is not a solution of (BVPh).

Similarly, we can prove that Fix(T) n (S2 \ E 2) ⁼ 0. Therefore, T satisfies all conditions of Theorem 5. So T has at least three distinct fixed points

UI,

such that

E [vj,VjJ, j = 1,2, and note that

To prove the main theorem, we will use the following well known theorem and it can be found in [7]:

THEOREM 6. Let! satisfy the condition (f2). For every constants P > 0 there exists a constant Q > 0 such that: if u is a solution of

.6.u=!(x,u,'Vu), xEn,

u E C

(0), lu(x)1 :::; P for all x E 0, then l'Vu(x)1 :::; Q for all x E O.

The constant Q only depends on P and the bounding function c.

Proof of Theorem 1.

Since we seek solutions of (BV Ph) on the order interval [VI, V2] n

C(O), we can choose Qo > 0 such that if u is a solution of .6.u =

f(x, u, VU), for all x En ^{and Vl ~ u ~ V2, then IVu(x)1 ~ Qo for all}

x EO. Since 0 is compact, we let

Qj = sup{any directional derivatives of Vj at x} < ⁰⁰

zEn

and

CJi ⁼ sup{any directional derivatives of ^Vj at x} < ⁰⁰

zEn

for j = 1,2. Furthermore, let Q = max{Qo,Ql,Ql,Q2,Q2}. Then we choose a bounded smooth function h : Rn ~ Rn such that h( T/) = T/ if IT/I < Q + ¹ and its differential dh is bounded on ^Rn. To get the main result, we solve the following boundary value problem

{

6u = f(x,u,h(\7u», x En

Bu(x) = </>(x), x E an.

Hence, Proposition implies the proof. •

REMARK. The above theorem is valid if we replace 6 by a uniformly elliptic operator

n n n

L = LLAij(x)Dij + LAi(X)D ⁱ +Ao(x),

i=l j=l i=l

where the coefficients of L and B are smooth.

3. A simple application

In this section we apply the main theorem to obtain the several positive ordered solutions in a class of singularly perturbed quasilinear elliptic Dirichlet problem. Let n be a bounded open domain in Rn with an ^{E C}

^,0l. We look for positive classical solutions of

{

1'?6u + g(x)IVuI

+ feu) = 0 in n

u = 0 on an,

where E is a small positive parameter, 9 : n ^---+ R is a-Holder contin- uous, and f : [0,00) ---+ R is required to be Cl. More assumptions on

f:

(1) There exist exactly ."fV.(?. 2) positive numbers ai such that al <

a2

< ... < aN, f(ai) = 0, and !'(ai) < O.

(2) There exists an n-dimensional open subdomain no ^of n ^such

that an o ^{E C}

^{,o:, g(x)?. 0 for all x E} no

(3) la; ^{f(u)du > 0}

for all r E [O,ai) and i = 1,2, .. · ,N.

The result of this application is the following multiplicity theorem.

THEOREM 7. With assumptions (1), (2), and (3), there exists EO > 0 such that for all E with 0 < E ~ EO, (Bll P

has 2N - 1 ordered positive solutions Ui( x; E), Ui+ 1 (X; E), Ui+l (X; E) ^for all x E nand Ui( x; E) ---+ ai

2

as

---+ 0 uniformly on every compact subsets of no.

Proof. We solve the following boundary value problem

{

E2tiu + feu) = 0

u=o

III

no

on an o .

By the results of Dancer[lO] and Ko[6]' there is

> 0 so that for all

10

with 0 <

~

the a.bove problem has 2N - 1 ordered positive solutions Vi(X;E), Vi+~(X;E), Vi+l(X;E), (i = 1,2, .. · ,N), such that Vi(X; E) _~ ai, Vi(X; E) _{~ Vi+~(X;} E) _~ Vi+l(X; E) for all Xo E no ^and

Vi(X; E) ---+ ai as

---+ 0 uniformly on every compact subsets of no. ^Let

_ { Vi(X; E) if x E no

Ui(X;f) = - -

o ^if n \ no

for i = 1,2,··· ,N. Since g(x) ?. 0 on no, ^{so Ui(X; E), ai and Ui+l(X; E),}

ai+I are two pairs of a quasisubsolution and a qua.sisupersolution of

(BllP

for 0 < E ~ ^EO, respectively and satisfy all conditions in the

main theorem. This completes the proof.

REMARK 1. In the Theorem 7, if we replace the condition fECI by the assumption that f E ca and there is a positive number

such that

f( ai - T) > 0 and f( ai + T) < 0

for all

with 0 <

~

we can also obtain the same multiplicity result by choosing a function j ^{E Cl} so that sup{ If(~) - le ^{u)1 : u E [O,oo)}}

is sufficiently small.

REMARK 2. If f(O) > 0, with assumption f'(ai) ~ 0 for i = 1,2, ... ,N, (BV PE) has 2N -1 ordered positive solutions as the ones of Theorem 7 by choosing a function h E Cl such that sup{ If( u) - Ji( u)1 : u E [O,ai]} is sufficiently small and fI(ai) < o.

References

1. Amann, H., Existence and multiplicity theorems for semi-linear elliptic bound- ary value problems, Math. Z. 150 (1976), 281-295.

2. Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18(4) (1976).

3. Amann, H. and Crandall, M., On some existence theorems for semi-linear el- liptic equations, Indiana Univ. Math. J. 27(5) (1978), 779-790.

4. Deimling, K., "Nonlinear Functional Analysis", Springer-Verlag, Berlin, 1985.

5. Inkmann, F., Existence and multiplicity theorems for semilinear elliptic equa- tions with nonlinear boundary conditions, Indiana Univ. Math. J. 31(2) (1982), 213-221.

6. Ko, B., The third solution of a class of semilinear elliptic boundary value problems and its applications to singular perturbation problems, J. Differen- 'tial Equations 101(1) (1993),. 1-14.

7. Ladyzhenskaya, O. and Ural'tseva, N~, Linear and Quasilinear Elliptic Equa- tions, Acad. Press, New York, 1968.

8. Protter, M. and Weinberger, H., Maximum Principles in Differential Equa- tions, Prentice-Hall, New Jersey, 1967.

9. Schmitt, K., Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. 3(2) (1978), 263-309.

10. Dancer, E., Multiple fixed points of positive mappings, J. Reine Angew. Math.

371 (1986), 46-66.

Department of Mathematics Education Cheju National University

1 Ara-dong, Cheju City 690-756, Korea