Symmetry breaking and existence of many positive nonsymmetric solutions for semi-linear elliptic equations on finite cylindrical domains

Pergamon

Vonlmear I nalt:~ts. Theory: .~h.th~nl~ & .4pphcatl,ms. M*I 31. No 3'4. pp 465-474. 19'4g m+ 1998 Elsevier Science Lid Printed I n (irt-at Britain+ All nghts reserved 0362-546X.08 $19 O0 • [lOO Pll: SO362-546X(96)O0324-O S Y M M E T R Y B R E A K I N G A N D E X I S T E N C E O F M A N Y P O S I T I V E N O N S Y M M E T R I C S O L U T I O N S F O R S E M I - L I N E A R E L L I P T I C E Q U A T I O N S O N F I N I T E C Y L I N D R I C A L D O M A I N S YI-WEN SHIH

Department of Applied Mathematics. National Chiao Tung University, Hsinchu 30050. Taiwan. Republic of China

(Received 20 ,hdv 1996: received lor publwation 27 November 1996) Key words and phrases: Symmetry breaking, cylinders.

I. I N T R O D U C T I ( ) N

Let oJ be a bounded smooth domain in R", n -< 1, a > o, and f~, = ( - a , a) × co be a finite cylindrical domain in R "*~ = {(x,y): x E R ~ and v E R"}. Taking the length a as a bifurcation parameter, we consider the symmetry-breaking problem o f the following semilinear elliptic equation with mixed-type boundary conditions.

{/~u 0

This problem was posed by shall study the problem when We consider the equation

[

Au + f(u) = 0 l t = 0 Ou 0 Z = Au + f ( u ) = 0 in fit,, u = 0 on [ - a , a] × i~¢u ( I . I ) on l - a , a ) × o~.

Professor H. Berestycki and studied by Lin [1]. In this paper, we n = l .

in ~u = ( - a , a) X (0, b)

on [ - a , a ] × {0, b} (1.2)

on { - a , a } × (0. b).

where]" satisfies the following conditions: (H-0) f E C ' ( R ~ ) , f ( u ) > 0 for u > 0.

( H - l ) there exists a > 0 such that u]"'(u) >- (I + e ) f ( u ) . For example, f ( u ) = u p, p > 1. satisfies (H-0) and (H-I).

Definition 1.1. A solution u E C 2 ( ~ , ) of (I.2) is said to be symmetric (with respect to the

x-axis, i.e. u = u(y)) when it satisfies

u" + f ( u ) = 0 in (0, b)

u(O) = u(b) = 0, (1.3)

otherwise, u is called an asymmetric solution. 465

4 6 6 Y I - W E N SHIH

Remark 1.2. Assume conditions (H-0), (H-l) arc satisfied. Then the nontrivial solution o f ( l . 3 )

is unique and exists for all b > 0. The existence result can be found in Lin [2], and the uniqueness is proven in Remark 2.3 below.

From Remark 1.2, we can let b = I. Our main results are as follows.

THEOREM 1.3. Assume conditions (H-0), (H-1) are satisfied. Let uo be the nontrivial symmetric solution. Then there must exist an increasing sequence ak ~ ~, as k---, ~, such that (1.2) possesses symmetry-breaking from uo at a~.

In addition, if f (u) satisfies (H-2) J E C2(R'),

then the bifurcation in Theorem 1.3 is global. We express the theorem precisely below. THEOREM 1.4. Assume condition ( H - 0 ) - (H-2) are satisfied. Let uo be the nontrivial sym- metric solution. Then there must exist an increasing sequence ak ---' zc, as k --. zc. such that (a~, uo) is a bifurcation point o f a global, unbounded branch o f solutions o f (1.2), and the branches must be globally separated.

In Lin [1], using variational method, Lin proved that an increasing sequence ak ---' ~¢, as k ~ z¢ exists, such that ( 1. I ) has at least k + 1 solutions for a > ak, for all n -> 1, provided that (H-0), (H-I) are satisfied a n d f i s sub-critical. We use the bifurcation method to prove that there are at least 2k + 1 solutions for a > a~, n = 1, and provide a global bifurcation diagram. Our result follows.

THEOREM 1.5. A s s u m e f s a t i s f i e s (H-0) - (H-2) and (H-3) lim . . . . f(u)/u p = C > 0 for 1 < p < zc.

Then there exists an increasing sequence a~ ~ ~c, as k ~ z¢, such that (1.2) has at least 2k + 1 solutions for a > a~.

The paper is organized as follows. In Section 2 we study the linear eigenvalue problems and give the Proof o f Theorem 1.3. In Section 3 we prove Theorems 1.4 and 1.5 and give the bifurcation diagram.

2. L O C A L B I F U R C A T I O N

We begin with the linearized eigenvalue problems o f (1.2) at uo. It is clear that the linearized eigenvalue o f (1.2) at uo is A W + f(uo(y))W = - t l W W = 0 ~W m = 0 8x in ( - a , a) × (O, 1) on [ - a , a ] × {0, 1} on { - a , a} × (0, 1). By separation o f variables, (2.1) is equivalent to the problems

{

~b" + v~b = 0 in ( - a , a)

~ ' ( - a ) = ~'(a) = 0

(2.1)

Symmetry breaking 467

and

with

{ - ~ , " - J"(u,)~u = (/a - v)~u

~,(0) = ,,(1 ) = 0.

in (0, 1)

{2.3)

W(.v..v) = ~b(x)tu(y). (2.4)

It is clear that the e i g e n v a l u e s in (2.2) are given by v~(a) = - ( k n / 2 a ) 2, k = 0. 1, 2 . . . . and the associated e i g e n f u n c t i o n s are

{ sin(kTt/2a)x i f k is odd

q~(x) = c o s ( k n / 2 a ) x i f k is even.

Let r = ,u - v, then there is a sequence {r~, r~, rs . . . . } o f e i g e n v a l u e s for (2.3) that satisfies r~ < r, < rs . . . where each distinct e i g e n v a l u e is multiplicity one. Therefore, u is an e i g e n v a - lue o f (2.1) if an o n l y if r~ = u - vk for some k and 1. Hence, we denote

/z~.l = g e l ( a ) = rl + l~ = rl + (kJz:'2a):.

w h e r e k = 0 , 1 , 2 . . . . a n d l = 1 . 2 . 3 . . .

LV.MMA 2.1. For a n y s y m m e t r i c positive solution uo we have r~ < 0.

P r o o £ It is well k n o w n that r~ can be characterized by

{o/(:

}

rl = i n f (~)) t): ~) E H,~(O. 1)\10} , I where O(,)) ₌ IV,~] z _{- . /} ., (t,,,)t, . )

Since uo is a solution o f ( l . 3 ) , we have

lu,;l:

=

uo.liu.).

) )

Hence, by ( H - I ) , we have

O(uo) = lu;I 2 - f'(u,,)uo = u , , J t u , , ) - ! (u,,)ua i

I o

Therefore, we have ~t < 0. •

By m o d i f y i n g an a r g u m e n t in N i - N u s s b a u m [3], we can prove the following lemma. LEMMA 2.2. For a n y s y m m e t r i c positive solution uo, we have t2 > 0.

ProoJ: Let u(r, d ) be the solution o f the initial-value p r o b l e m

u"(r) + f ( u ( r ) ) = 0 for r > 0

468 YI-WEN SHIH

Since u'(r, d) = d - f[,f(u(s, d)) (is, (H-O) and ( H - l ) implies there are 0 < r/(d) < R(d) such that

u'(rl(d), d) = 0

and

u'(r, d ) > 0 i f r < q(d)

u'(r, d) < 0 i f r E (r/(d), R(d));

i.e. R(d) be the first zero o f it(r,

d).

Set ~ ( r , d) = 3u(r, d)/ad, ~ then satisfies 0 " + f'(u)¢l) = 0

@(0) = 0, ~ ' ( 0 ) = I. (2.6)

We will claim that ¢b has exactly one zero in (0, R(d)). First, we prove that • has at most one zero. Suppose • has zeros in (0, R(d)). Denote the first zero o f • by ~(d).

Let X = r u ' ( r , d ) Y = u'(r, d). We then have d (X'(I) - (1)'X) = - 2 / ( u ) ( I ) (2.7) and d ~ ( Y ' ( I ) - O ' Y ) = 0. (2.8) From (2.7) we have X ' e - * ' X l i = - 2 f ( u ) ~ , which implies X'(~)O(~) - ¢)'(~)X(~) - X'(0)O(0) + ~'(0)X(0) < 0.

Since ($)(~) = (I)(0) = X(0) = 0 and q)'(~) -< 0. We thus have (I)(~) < 0 and X(~) < 0 (note: X(~) < 0 implies q(d) < ~(d)). If there exists a point ~ E (~(d),

R(d))

such that W'(~) = 0, let

be the first point. By (2.8), we have

Y ' ~ - ~ ' YI! = 0, which implies

Y'(()q)(() - O ' ( ~ ) Y ( ( ) + O'(~)Y(~) - Y'(~)O(~) = 0.

Since (I)'(~) = 0, (1)(~) = 0, O ( ( ) < O, O ' ( ~ ) < 0, Y(~) < 0 and Y'((,) = u"(~) = - f ( u ( ( ) ) < 0, the left-hand side is negative, a contradiction. Therefore, (I) has at most one zero in (0, R(d)).

Now, compare these two equations:

~"(r, d) + J"(u(r, d))~(r, d ) = 0 in (0, R(d))

Symmetry breaking 46q Since r~ < 0, by the Sturm comparison principle, • must have at least one zero in (0, R(d)). Hence, • has exactly one zero in (0, R(d)). Now, if r2 <- 0, by the Sturm comparison principle again, • must have at least two zeros in (0, R(d)], a contradiction, so r2 > 0. •

Remark 2.3. In Lemma 2.2, • has exactly one zero in (0, R(d)), so we have ~ ( R ( d ) , d) < 0. By u(R(d), d) =- 0 for all d > 0, we have R'(d) = -¢~(R(d), d)/u'(R(d), d), so R'(d) < 0 for all d. Hence, there is a unique solution for (I.3) (see N i - N u s s b a u m [3]).

LEMMA 2.4. There exists an increasing sequence a~, a: . . . such that p,.~(a,) = 0 and p,.~ is a simple eigenvalue.

Proq[:

p,.l(a) = rt + vk = rl + (kn/2a) 2. So p,.t(a) = 0, unless rt <- 0. By Lemmas 2.1, 2.2, only r~ < 0, so

p , , l ( a ) = 0 if and only i f a = krc/(2~/-rl).

Let a,, = mz/(2"~Lrl), n = 1, 2 . . . where r~ is the first eigenvalue of (2.3). We have pk.;(a,) = 0 i f k = n and n = 1

pk.~(a,) ¢ 0 otherwise.

Since there are no generalized eigenfunctions (by the symmetry o f the operator, see H e a l e y - Kielh6fer [4]), the p r o o f o f Lemma 2.3 is complete. •

We now prove Theorem 1.3.

Pmol: We first let x = @.t, 2 = a 2, ~(t,y) = u ( x , y ) - uo(x,y). (1.2) is then equivalent to

1 iJ2r) i)2~) 2 ~t ~" + ~ + liu,, + ~)) - f t u , , ) = 0 ~ i = 0 - - = 0 #t inf2L = ( - 1 , I) × (0, I) on [ - 1 , 1] × {0, 1} on { - 1 , 1) × (0, 1). (2.9) Let F()., ,,): R" x X ~ Y. be defined by F ( 2 . t~) = - - - + + f ( u , + ~) - f ( u , ) . /. i~t 2 ~ where - - = 0 o n { - 1 , 1 } × ( 0 , 1 ) . X = t~ ~ C - " " ( ~ ) : *~ = 0 on [ - 1, I] × {0, 1/, Ot y = C'°."(~l~.

C k " ( ~ ) is the usual space o f all k-times differentiable functions u in ~ such that u and its derivatives are HSlder-continuous with exponent or. We will use the bifurcation result from the simple eigenvalues in Crandall--Rabinowitz [5], we reproduce the theorem as follows.

470 YI-WEN SHIH

THEOREM 2.5 (Crandall-Rabinowitz). Let X, Y be Banach space, V a neighborhood o f O in X

and F: ( - 1, 1 ) × V -* Y have the properties:

(a)

F(t,

0) = 0 for ]t] < 1;

(b) the partial derivatives

F,, F,

and

F,,

exist and are continuous: (c) N(k',(0, 0)) and

Y/R(F,(O,

0)) are one-dimensional;

(d) F,,(0, 0)xo ~

R(F,(O,

0)), where

N(FdO,

0)) = span{xo}.

l f Z is any complement

of N(F,(O,

0)) in X, then there is a neighborhood U o f ( 0 , 0) in R x X, and interval ( - a , a), and continuous functions 4: ( - a , a ) - * R, ~: ( - a , a ) - , Z such that ~b(0) = 0, ~'(0) = 0 and F - ' ( 0 ) CI U = {(~b(a). ax,, + a~(o~)): la[ < a} U {(t, 0): (t, 0) E U}.

We check whether F satisfies all properties o f Theorem 2.4. (a) F(2, 0) = 0 for all ). > 0.

(b) The smoothness o f f ensures that F,.,

F,,, F,.,.

exist and are continuous.

(C) /",,(~, 0)h =

(l/~'.)(02h/3t 2) + (32h/c3y 2) + f'(uo)h,

for all h E X, where ~ _{a = a , is a simple} 2 eigenvalue, n = 1, 2 . . . and since the dimension o f

N(F,,(~,

0)) is equal to the co-dimensions o f R(F,.(~, 0)) (see Remark 3.5), N(F,,(~, 0)) and

Y/R(F,,(~,

0)) are one-dimensional.

(d) Let N(F,,(,[, 0 ) ) = span(t). If w E R(F,,(~, 0)), then there exists t)0 E X, such that (1/fO(O2t)o/Ftt

2) + (~12~)0/~y2) +

f'(uo)t)o = w, and since

(1/;.)(326/i112) + (fl2tj/ay 2)

+ f ' ( u o ) ~ = 0, f~, ~vt, = 0. Hence,

F;,;(~,

0)5 ~ R(F,.(,~)) if f~, fiF,.,.(~, 0)6 ~ 0.

- 1

32fi

; 2

~tt 2

- 1 a 2

_

- ;2 at2

~(x)~(y) [by (2.4)]

- 1 _ l & ~ \ 2 a ~

- ; l")lV, )

1

= -~ V~(y)~(x)

[by (2.2)] A

F~.,,(~, 0)~;

- So, F = _ 5 . A 6F:, (~, 0) = 6 - ~ = - I~12 ~ 0,

where ~ =

- ( k n / 2 a ) 2

for k = 1, 2 . . . Hence, by Theorem 2.4, it will bifurcate at ~ = a~ for all n = 1,2 . . . . ; i.e. problem (1.2) will bifurcate from uo at )7 = a~,. •

3. GLOBAL BIFURCATION

We prove Theorem 1.4 by modifying an argument in Healey-KielhOfer [4]. We only sketch the difference here. If v is a solution o f (2.9), then ~) has an even 4-periodic extension on R~; see, for example, Lin [1] or Healey-KielhOfer [4]. Therefore, we define the function spaces

Symmetry breaking 471 and the Banach spaces

D = C:'~(D.~) and E = c : ~ ( g z ~ ) Let G(),, 0): R + x D --* E, be defined by 1

820 820

= - - + - - + f ( u o + 0) - f ( u o ) . g(2, v) 2 Ot 2 c3v 2

Clearly, the smoothness o f f ( f E C2(R)) ensures that G is at least twice continuously Frechet- differentiable. We use the global result in KielhSfer [6], which is a generalization o f Rabinowitz's result [7]. First, we give some definitions.

with HSlder norm

11"112.~

with HSIder norm

I1"11o.~.

Definition 3.1. Let D C E be both separable Banach spaces. A linear operator A: D ---, E is called admissible if it satisfies

Ca) A is a Fredhoim operator o f index zero.

(b) There exists c > 0 , e , > 0 such that the spectrum a ( A ) o f A in the strip SA = ( - ~ , c ) X ( - i e , ie,) consists o f finitely many eigenvalues o f finite (algebraic) multiplicity.

Definition 3.2. Let fl C D be a bounded domain. A map G: f~ --* E is called admissible if G E C2(fL E). Its Frechet-derivative D G ( u ) = G'(u) is admissible in the sense o f Definition 3.1 for all u E ~ . G is proper, i.e. the inverse image in ~ o f any compact set in E is compact in D.

Note. The definition o f "crossing number" is more complicated, so for simplicity, we omit it here. For details, see KielhSfer [6].

THEOREM 3.3 (KielhSfer). Let G: R ÷ x D - ~ E be a C2-map satisfying the following conditions:

(a) G is proper on any bounded and closed domain in R" x D. (b) G0;t, -) is admissible for any ), E R ~ .

Assume that G(2, 0) = 0 for all ,;t • R +, that at some 2o E R *, the operator A(2) = G~(2, 0) has an eigenvalue o f zero, and that A(),) for 0 < 12 - 2O1 < 6 has no eigenvalue zero. If A(2) has an odd crossing number Z(20) at ,;. = 20 (i.e. A(,() has an odd crossing number X(0) at ,~ = 0 for ~. = ,;. - 2o; see KielhSfer [6]), then (20, 0) is an (isolated) bifurcation point for G(;t, u) = 0. Call

N S = cl{(/~,0) E R + × D, G(2, u) = 0, u :~ 0}

the closure o f the nontrivial solution set. Then the component NS~a,,.o~ o f N S connected to the bifurcation point (2O) is either unbounded in R* x D, or NS~z,.o~ meets a different bifurcation point (~,, 0).

LEMMA 3.4. For each (2, o) E R + x D, the linear operator G,,(2, u): D ~ E is a Fredholm operator o f index zero, and the spectrum o f -[G,,(2, o)] in the strip {z E C: R e z <- 1,

472

Proof

YI-WEN SHIH

1

02h

32h

G,,(2, v)h = ~. Ot-- 7 + --3y 2 + f'(uo + o)h for all h E D.

So Gd2, v) is a uniformly elliptic operator for each (2, o) E R + x D. By periodicity we then

have the following Schaulder estimate

11hl[2.,~ -< C(][hl[0,,~ + 11G,,(2, o)hi[0.<,) for all (2, o) E R + × D, h 67 D, (3.1) where C is independent o f h (see Gilbarg-Trudinger [8]). Since the embedding o f D into E is compact, the estimate (3.1) implies that Go(2, v) is a semi-Fredholm operator for each (2, o) (see Grisvard [9, L e m m a 4.4.1.1]). In a Hilbert-space setting (L](f~) = {u E L2(f~): u has period 4 in t} and H ~ ( f ~ ) = {u ~ H 2 ( f ~ ) : u has period 4 in t}) the symmetry o f G,,O., o) implies that the co-dimension o f R(G,~(2, ~))) is equal to the dimension o f N(G,,(2, o)) i.e.

G,(2, o) is a Fredholm operator o f index zero. By standard regularity theory, G,,(2, tJ) is also a Fredholm operator o f index zero in our H61der-space setting.

Establishing the second assertion duplicates L e m m a 2.1 in Healey-Kielh6fer [4], so we omit it. •

Remark 3.5. By periodicity, the norms with respect to fl, and ~ are identical, so we have the

same Schoulder estimate. Therefore, the dimension o f N(G,(~, 0)) is equal to the co-dimension

o f R(G,,(~, 0)).

LEMMA 3.6. The mapping G: R+ × D ~ E is proper on any bounded closed domain; i.e. G - ' ( ( K ) f'l B is compact in R ÷ × D whenever K C E is compact and B C R+ × D is bounded and closed.

Proof Decompose G as the sum

G0., ,)) = A(2)~) + F(o),

where

] c)20

020

F(v) = f(uo + o) - f(uo).

Let G().., o.) = f . , w h e r e f . --+fin E and {(2., o.)} C B is bounded and closed in R+ × D. Without loss o f generality, we note that {2.} converges to 2 in R + and, by compact embedding, {o.} converges to o in the Banach space C4""(fA). This implies that

F(u.) --* F(u) in E

I/[A(2.) - A()-)]v.llo.~ - el[v.l/z.. <- eM for all n -> N(e). Hence, the estimate means that

A(2,,)v. - A(2)t). --* 0 in E.

Let/x ~ R be such that A(2) - / . d : D --" E is bijective [i.e./t is in the resolvent set o f A(2)]. Then,

Symmetry breaking 473

which converges to - F ( o ) - / a t ) + f i n E. Finally, the convergence o f {[A(2) - / d ] o , } in E is equivalent to the convergence o f {v,} in D.

Now define

{ 4 1 ! }

D~ = u E C2'~(f~): u i s - p e r i o d i c , odd in t = - 1 + - a n d e v e n in t = - I +

n n

E~ = u E C ° ' ~ ( ~ ) : u is 4 periodic, odd in t 1 + - and even in t 1 +

n n

Consider G: R ~ x D, --* E,, the restriction o f G(;t, .) to D, (see Healey--Kielh6fer [4]). By Theorem 1.3, we have G,,(2, 0) has an odd crossing number Z(,~), where 2 = 2t, 22 . . . . and all properties o f G remain valid for its restriction, so all conditions o f Theorem 3.3 are fulfilled. •

Therefore, we may summarize the following.

THEOREM 3.7. Assume that (H-0)-(H-2) are satisfied. (2,, 0) is then a bit'urcation point o f the global branch E " C R ' x D, o f nontrivial solutions (subject to Rabinowitz alternative; i.e. either unbounded in R ÷ × D, or meeting a different bifurcation point (2,,, 0)) o f G(2, 0).

Now each eigenfunction I~n is positive or negative on the open rectangle 12,, =

( - I - l/n, - 1 + 1/n) × (0, 1). By the proof o f Theorem 3.1 in Healey-Kielh6fer [4], it will preserve the sign in fin along the branch, so we have the following theorem.

THEOREM 3.8. Assume that (H-0)-(H-2) are satisfied. ().,, 0) is then a bifurcation point o f a global, unbounded branch o f solutions o f G(,:., o) = 0 having precisely the nodal configuration o f v, along the entire continuum. Therefore, continuua emanating from different nodal con- figurations are globally separated.

Theorem 1.4 has thus been proven. To prove Theorem 1.5, as well as Theorem 3.8, we need the following result from Lin [1].

THEOREM 3.9 (Lin). Assume condition (H-3) is satisfied, there then must exist a constant C > 0 such that for any a > 0 and any positive solution u, o f (!.2), we have

Ilu, ll~ = max{lu,,(x)l: x ~ ~:~,} ~ C.

P r o o f o f Theorem 1.5. Recall the scaling x = x/,;d, ). = a z, r)(t,y) = u ( x , y ) - uo(x,y). So by Theorems 3.8 and 3.9, Fig. 1 represents the bifurcation diagram for (1.2). Therefore, we have also proven Theorem 1.5 here. •

R e m a r k 3.10. It is o f interest to obtain results similar to Theorems 1.3-1.5 in the case in which n > _ 2 .

47,1 YI-WEN SHIH Ilull. u ~ C . . . i i a l ~ 2 a ~ D 0 Fig. 1.

Acknowledgement--The author wishes to thank his adviser, Professor S. S. Lin, for his guidance. R E F E R E N C E S

I. Lin, S. S., Symmetry-breakings for scmilinear elliptic equations on finite cylindrical domains. Proc. A.M.S., 1993, 117(3), 803-811.

2. Lin, S. S., On the existence of positive radial solutions for nonlinear elliptic equations in annular domains, d~ Differential Equations, 1989, 81,221-233.

3. Ni, W. M. and Nussbaum, R., Uniqueness and nonuniqueness for positive radial solution of Au +f(u,r) = O. Comm Pure Appl. Math., 1985, 38, 67.-108.

4. Healey, 1". J. and Kielh6fer, H., Symmetry and nodel properties in the global bifurcation analysis of quasi-linear elliptic equations. Arch. Rational. Mech. Anal., 1991, 113, 299-311.

5. Crandall, M. G. and Rabinowitz, P. H., Bifurcation from simple eigenvalues, d Functional Analysis, 1971, 8, 321-340.

6. Kielh6fer, H., Multiple eigenvalue bifurcation for Fredholm operators, d. Reine Ang Math., 1985, 358, 104-124. 7. Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems..1 Functional Anal., 1971, 7, 487-513. 8. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order. Springer Verlag, New York,

1977.

9. Grisvard, P., Elliptic Problems in Nonsmooth Domain. Monographs and Studies in Mathematics 24. Pitman, Boston, 1985.