Volume 117, Number 3, March 1993
SYMMETRY-BREAKINGS
FOR SEMILINEAR ELLIPTIC EQUATIONS
ON FINITE CYLINDRICAL DOMAINS
SONG-SUN LIN
(Communicated by Barbara L. Keyfitz)
Abstract. We study the existence and multiplicity of asymmetric positive solu-tions of a semilinear elliptic equation on finite cylinders with mixed type bound-ary conditions. By using a Nehari-type variational method, we prove that the numbers of asymmetric positive solutions are increasing without bound when the lengths of cylinders are increasing. On the contrary, by using the blow up technique, we obtain an a priori bound for positive solutions and then prove that all positive solutions must be symmetric when the cylinders are short enough.
1. INTRODUCTION
Let co be a bounded smooth domain in W, n > 1, a > 0, and Qa = (-a, a) x co be a finite cylindrical domain in K"+1 = {(x, y) : x € E1 and y e W}. In this paper, we shall study the existence and multiplicity of positive solutions of the following semilinear elliptic equation with mixed type boundary
conditions
(1.1) Au + f{u) = 0 inQa,
(1.2) u = 0 on[-a, a]xdco,
(1.3) — = 0 on {-a, a} x co,
where / satisfies the following conditions:
(H-0) feCl(Rl),
f{u)>0 for w>0,
(H-l) /(0) = 0 and /'(0) = 0,(H-2) there exists a > 0 such that uf'(u) > (1 + a)f(u) for all u > 0,
(H-3) f(u) < C(up + 1) for some C > 0 and all u > 0, where 1 < p < (n + 3)/(« - 1) if n > 2 and 1 < p < oo if n = 1 .
A solution u € C2(£la) of (1.1)-(1.3) is said to be symmetric (with respect to the x-axis) if u is constant along the x-axis, i.e., u = u(y) and satisfies
(1.4)
Au + f(u) = 0 in co,
(1.5) u = 0 on dco.
Received by the editors January 30, 1991 and, in revised form, July 5, 1991.
1991 Mathematics Subject Classification. Primary 35B32, 35J65, 35P30.
Key words and phrases. Symmetry breaking, cylinders.
Work partially supported by the National Science Council of the Republic of China.
© 1993 American Mathematical Society
0002-9939/93 $1.00+ $.25 per page
Otherwise, u is called an asymmetric solution. The following problem was
posed by Berestycki:
,p. For /(«) = up , taking the length a as a bifurcation parameter, ^ ' is there symmetry-breaking for (1.1)-(1.3)?
Previously, it has been known that there is an asymmetric solution when a is large enough. In this paper, we answer the problem (P) partially, and hope it will lead to more complete results of the problem. The main results are as
follows.
Theorem 1.1. Assume conditions (H-0)-(H-3) are satisfied. Then there exists an increasing sequence ak — ak(co, a) —► oo as k —> oo, such that for each k > 1 and a e (ak , oo), (1.1)—(1.3) have at least k-many asymmetric positive
solutions.
On the contrary, for short cylinders we have
Theorem 1.2. Assume f satisfies (H-0)-(H-2) and
(H-3)' lim^oo f(u)/uP = C > 0 for 1 < p < (n + 3)/(« - 1) if n>2 and
1 < p < oo if n = 1. Then there exists <z» = a*(co, p) > 0 such that for any ae(0,fl,), every positive solution o/(l.l)-(1.3) is symmetric.
The last two theorems indicate that there are asymmetric bifurcations at certain critical numbers a*., aj-»oo as k —> oc, and the bifurcation branches will go to the direction of increasing a . However, we still need more rigorous justifications.
To prove Theorem 1.1, we need to study the problems of asymmetric insta-bility of symmetric positive solutions and then use a Nehari-type variational method to prove that there exist asymmetric positive solutions when all
sym-metric positive solutions are unstable with respect to certain asymsym-metric mode.
To prove Theorem 1.2, we need to obtain some a priori bounds for posi-tive solutions of (1.1)—(1.3) and then prove that asymmetric posiposi-tive solutions cannot exist when the cylinders are short enough.
As for the related problems, the problems of symmetry-breaking of positive radial solutions on a ball were studied in [4, 6, 15-17], on annular domains in
[3, 5, 8-11, 18, 19], and on sectorial domains in [2, 12].
The paper is organized as follows. In §2 we study the linear eigenvalue prob-lems and obtain some asymmetric instability results for symmetric positive so-lutions for long cylinders. In §3 we prove the existence of asymmetric positive solutions for long cylinders. In §4 we prove the nonexistence of asymmetric positive solutions for short cylinders.
2. Linearized eigenvalue problems
Let g e C(co). Consider the following linear eigenvalue problem:
(2.1) Av + g(y)v = -vv ina;,
(2.2) v = 0 on dco.
Then there is the sequence {v\,v2,...} = {v\ {g), v2{g), ...} of eigenvalues of (2.1) and (2.2) satisfying v\ < v2 < 1/3 < •• • , where each distinct eigenvalue
is repeated according to its multiplicity. By separation of variables, the linear eigenvalue problem
(2.3) Aw + g(y)w = -Xw in ila,
(2.4) w = 0 on [-a, a] x dco,
(2.5) -£— = 0 on {-a, a} x co,
is equivalent to the problems
(2.6) Av + g(y)v = - (A + p)v m co,
(2.7) v = 0 on dco,
and
(2.8)
<p"{x)
= n<p{x), x£(-a,a),
(2.9)
<p'(-a) = 0 = <p'(a),
with w(x, y) = cp(x)v{y). It is clear that the eigenvalues of (2.8) and (2.9) are given by
(2.10) fik(a) = -(kn/2a)2,
k = 0, 1,2, ... , and the associated eigenfunctions are
sin —x if k is odd,
(2.11)
<Pk(x)={
H
cos -r— x if k is even.
*> 2a
Therefore, X is an eigenvalue of (2.3)—(2.5) if and only if X-\-fik = v\ for some k and /. Hence, we denote
(2.12)
h,i=h,i{g,a)
= vt-nk = vt{g)+ l-^J
where k = 0, 1, 2, ... and / = 1,2,.... Note that when k = 0 it corre-sponds to symmetric modes and k > 1 to asymmetric modes.
Let Mo be any symmetric positive solution and v\ = u\(uo) be the least eigenvalue of (2.1) and (2.2) with g(y) = f'(uo(y)). Then we have the following estimate for ^i(wn).
Lemma 2.1. Assume condition (H-2) is satisfied. Then for any symmetric positive solution uq we have
(2.13) "i(«o) < -ov\ ,
where V\ = u\{co) > 0 is the least eigenvalue of the Laplacian -A on co with Dirichlet boundary conditions.
Proof. It is well known that v\(uo) can be characterized by
vx = inf \Q{v)l j v2 : v e 7/0V)\{0}
J ,
where
Q(v)= f \Vv\2-f'(u0)v2.
Since Uq is a solution of (1.4) and (1.5), we have / |Vwo|2 = / u0f{u0).
Jw Jw
Hence, by (H-2), we have
Q{u0) = / |Vw0|2
- f'(u0)ul = / u0f{u0) - f'(u0)ul
Jw J(0
< - a / u0f(u0) = -o /
|V«o|2-Jco Jw
Therefore, by using the Poincare inequality, (2.13) follows. The proof is
com-plete.
Note that the upper bound -avx of vx(uq) is independent of symmetric positive solutions «o ■
A consequence of Lemma 2.1 is the following asymmetric instability results
for long cylinders.
Theorem 2.2. Assume condition (H-2) is satisfied. For each k > 1, let
(2.14)
a* = ^W"1/2.
Then for any a e (ak , oo) and any symmetric positive solution uq we have
4,i("o, a) < 0.
Proof. By(2.12), Xk \{u$, a) = v\{u0) + fik(a); the result follows from Lemma
2.1.
Remark 2.3. Assume conditions (H-0)-(H-3) are satisfied.
(i) If u e C2(fta)nC°(fia) is a solution of (1.1)-(1.3), then u is positive in Q. Indeed, by (H-0) and the maximum principle, the minimum of u in Q.a is achieved on dQa ■ On the other hand, by (1.3) and the strong maximum prin-ciple, the minimum cannot be achieved on {-a, a) xco. Hence the minimum will be achieved on [-a, a] x dco, which implies u is positive in Qa.
(ii) There exist positive solutions of (1.1)—(1.3) and (1.4) and (1.5),
respec-tively, see, e.g., [1, 14].
3. Long cylinders
In this section, we shall prove that if the cylinders are long enough, a > ak , then there exist at least /c-many asymmetric positive solutions. We shall use a Nehari-type variational method.
For any function u defined on Q.a , it can be extended to u on £1^ = (-oo, oo) x co. In fact, we first define
u(x, y) — u(2a - x, y) for x e (a, 3a) and y £ co,
a reflection with respect to x - a. Then « is extended to the whole of Q^o by 4a periodically along x-axis. Due to the Neumann boundary conditions on {-a, a] x co, the eigenfunctions wkj and solutions of (1.1)—(1.3) can be extended smoothly on ^lx in this way.
We need some notation. Let V — {u e Hl(Cla) : u = 0 on [-a, a] x dco}.
Define the functional
J(u)= f Uvu\2-F{u) and M(u) = f \Vu\2-uf(u)
on V , where F(u) = /„" /(/) dt.
For each k > 1, the submanifolds Vk and Foo of V are defined by
^ = ^t,« = <"€K:fifjc + -^,yJ =u{x,y)
for (x, y) € Qoo and M(u) = 0 >
and
Foo = {W€/f0'(w):M(M) = 0}.
Let
Ik = Ik{a) = inf{/(w) : u £ Vk}, /<*, = /^(a) = inf{/(w) : u 6 K^}. Let «o be a symmetric positive solution, and wq = i>i(y) > 0 and wk = <pk{x)v\(y) be eigenfunctions with respect to eigenvalues Ao,i = Xoy\{uo,a)
and Afc>i = Afc)1(«o, a) with /fi ty^ = fa w\ = 1. Note that for k > 1, we have
(3.1)
f wk=0.
Then we have the following results concerning the change of J(u) along the direction of asymmetric mode wk .
Lemma 3.1. Assume conditions (H-0)-(H-2) are satisfied. Then there exist e > 0 and a smooth function 6: (-e, e) -» R1 w/?/z r5(0) = S'{0) = 0 5mc/z that
(3.2)
M(M0
+ <5(0^o + ^t) = 0
/or a// t e (—e , e). Furthermore, we have
(3.3) /(ao + <$(Ot"o + ftofc) = J(u0) + \X0JS(t)2 + {XkAt2 + 0(t4),
as t ~ 0.
Proof. The proofs are similar to Lemmas 6.1 and 6.2 in [11]. (3.2) is proved by using the Implicit Function Theorem and (3.3) is proved by straightforward computations; the details are omitted. Note that (3.1) is used repeatedly in the
proofs.
To prove Theorem 1.1, we need the following result, which indicates that asymmetric minimizers are different for different modes.
Lemma 3.2. Assume conditions (H-0)-(H-3) are satisfied. If Ikm < /^ then
h < hm for k = 1 > 2, ... and m = 2, 3, ... .
Proof. Under the assumptions of (H-0)-(H-3), it is rather standard to verify that the Palais-Smale condition holds, see, e.g., [1, 14]. Therefore, the mini-mizers of Ik and /oo are achieved by some functions uk e Vk and «o € K»
Let m(x, y) be a minimizer of J(u) on Vkm . Since Ikm < 1^ ,
(3.4) du/dx^O onQfl.
Let
u (— , y J for x e (-a, a) when km is odd,
f(*»)0 = < (x + a \
u -, y ] for x e (-a, a) when /cm is even.
( \ m J
Then for km odd we have
(3.5)
v(x + f,y^=u^
+ ^,y^=u(^,y)=v(x,y).
Furthermore, we have
(36) L}^--LLW)+{^2}dxdy
--mLlZ{X)1+(v'u]1)dxdy<Lml-Similarly, if km is even then (3.5) and (3.6) can also be proved. Since u andv are periodic in x of periods km and k, respectively, it can be proved that
(3.7)
/ vf(v) = f uf(u) and / F(v) = I F(u).
Jo.a Jna Jiia Jaa
Therefore, (3.6) and (3.7) imply M(v) < 0. Since f'(0) = 0, we have M(tv) > 0 for t > 0 and sufficiently small. This implies that there exists a t\ e (0, 1) such that M(tiv) = 0.
Now, by (H-2), we have
j-t ^tuf(tu)
- F(tu)} = \{tuf'{tu) - /(/«)} > 0
for m > 0. Therefore, we have
Ik<J{t\v)= f l-t2\Vv\2-F(tiv)
jQa l
= / \txvf(txv)-F(txv)<
f Lf(v)-F(v)
= / ^uf(u)-F(u) = Ikm.
The proof is complete.
Proof of Theorem 1.1. For each k > 1 and a e (ak, oo), ak given by (2.14), the minimizers of Ij(a) and /oo(a) are achieved by some functions Uj e Vj
and Mo e Foo , ;' = 1, ... , k . By (3.3), /,-(a) < /^(a), ;'= 1,..., k.
Now let 1 < i < j < k and / be the least common multiple of i and j. If
j = I then, by Lemma 3.2, m, and Uj are different. If j < I then VjDVj — Vj. If m, = Uj■■ = u € ^/ then
By (3.8) and Lemma 3.2, we obtain /,(a) < //(a), a contradiction to (3.8).
Hence, m, and Uj are different. The proof is complete.
4. Short cylinders
In this section, we shall prove that there is no asymmetric positive solution of (1.1)—(1.3) when the cylinders are short enough.
We need the following a priori estimates.
Lemma 4.1. Assume conditions (H-0)-(H-3)' are satisfied. Then there exists a constant C = C(co) > 0 such that for any a > 0 and any positive solution ua
o/(l.l)-(1.3), we have
(4.1) II"alloc = max{|Ma(x)| : x e fia} < C.
Proof. Since the extension ua of ua is smooth in Qoo , i.e., ua e C2(Qoo), the
boundary {-a, a} x co is now an interior part of Qoo • Since ua > 0 in Q^o
and satisfies
Aua + f{ua) = 0 in Qoo,
ua = 0 on <9Qoo ,
by using the technique of "blow up" as in [7], we can prove (4.1). The details
are omitted.
Proof of Theorem 1.2. For a > 0, if ua is an asymmetric positive solution of
(l.l)-( 1.3), then
(4.2) wa = dua/dx £ 0
and wa satisfies
(4.3) Awa + f'(ua)wa = 0 inQa,
(4.4) wa = 0 on <9Qa.
Let Ai(a) > 0 be the least eigenvalue of the Laplacian -A on Qa with Dirichlet boundary condition. Then it is easy to verify that
(4.5) lim Ai(a) = oo.
a—>0+
On the other hand, the least eigenvalue t]{ (a) of the linearized eigenvalue problem Aw + f'{ua)w — -nw inQa,
w = 0
on <9Qj
can be characterized bynx{a) = inf [qM/J
v2 : v e //0'(Qfl)\{0}} ,
whereQ(v)= f \Vv\2-f'(ua)v2.
By Lemma 4.1, there exists a constant Cx > 0 such that
By the Poincare inequality, we have
(4.7) / |Vu|2 >A,(a) / v2
for all v e H^(£la). Hence, by (4.5), (4.6), and (4.7), there exists a* > 0 such
that
(4.8)
m(a)>0
for all a e (0, a*). Therefore, (4.2), (4.3), (4.4), and (4.8) imply that there is
no asymmetric positive solution for a 6 (0, a*). The proof is complete. Remark 4.2. The results are still valid if f(u) is replaced by f(y, u), which satisfies similar conditions as (H-0)-(H-3) and (H-3)'.
ACKNOWLEDGMENT
The author is grateful for several useful conversations with Professor H. Berestycki during his stay in Taiwan in April 1988.
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Department of Applied Mathematics, National Chiao Tung University, Hsin-chu, Taiwan, Republic of China
