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Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities
Nonlinear Analysis 54 (2003) 445 – 456
www.elsevier.com/locate/na
Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities Norimichi Hiranoa;∗ , Wan Se Kimb;1 a Department
of Mathematics, Faculty of Engineering, Yokohama National University, 156 Tokiwadai, Hodogaya-ku, Yokohama, Japan b Department of Mathematics, College of Natural Sciences, Hanyang University, Seoul 133-791, South Korea Received 20 April 2002; accepted 3 December 2002
Abstract In this paper, we establish a multiple existence result of periodic solutions for a nonlinear parabolic equations with singular nonlinearity at zero. To establish our result, We use an approximating process by smooth nonlinear functions to avoid a singularity in nonlinear term. And we adapt topological argument based on the variational structure of functionals corresponding to approximating equations. ? 2003 Elsevier Science Ltd. All rights reserved. MSC: 35k20; 35k55; 35k60 Keywords: Multiplicity; Sublinear grow; Parabolic equation; Stability; Unstability
1. Introduction Let N ¿ 2 and be a bounded domain in RN with a smooth boundary 9 and h : R × 9 → R be a continuous function such that for each x ∈ ; h(·; x) is T -periodic and continuous, and h ¿ 0 on R × . In this paper, we are concerned with multiple
∗
Corresponding author. Tel.: +81-45-339-4212. E-mail addresses: [email protected] (N. Hirano), [email protected] (W.S. Kim).
1
This work was supported by Korea Research Foundation Grant KRF-2000-15-DP0027.
0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0362-546X(03)00101-9
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existence of T -periodic solutions of ut − Ex u = g(u) + h(t; x) u=0 u(0) = u(T )
the following problem: in (0:T ) × ; on (0; T ) × 9;
(P)
9 in ;
where −Ex denotes the Lapalacian on RN and g(t)=|t| −1 t, for t ∈ R, with 0 ¡ ¡ 1. The existence of periodic solutions for semilinear evolution equations has been studied by many authors [5,6,8,10]. But few seems to be known about the case that the nonlinear term g has a singularity at zero in the sense that g(u) lim = ∞: |u|→0 u Hence the usual known scheme, in particular, sub-supersolution method in treating parabolic problems is not applicable directly in proving the existence and multiplicity of solutions due to the singularity. Even in elliptic problems, there have been few results known on the existence and multiplicity results of elliptic problems with above type nonlinearities. We refer [3,4,9] for elliptic problems. In the present paper, we use an approximating process by smooth nonlinear functions to avoid such a singularity, and establish a multiple existence result for (P). To prove the multiple existence of solutions, we need topological argument based on the variational structure of functionals corresponding to problem (P). To avoid unnecessary complexity, we restrict ourselves to the case that g(t) = |t | −1 t, 0 ¡ ¡ 1. One can see that our argument here is valid for any function g ∈ C(R) satisfying that 0 ¡ limt→0 g(t)=| t| −1 t ¡ ∞ and lim| t|→∞ g(t)=t = 0. We put QT = (0; T ) × . A function u ∈ C([0; T ]; H01 ())∩C 1 ([0; T ]; L2 ()) is said to be a solution of (P) if u satisKes (P). We now state our main result. Theorem. There exists m0 ¿ 0 such that for each h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 , there exist at least two solutions of (P). 2. Preliminaries For q ¿ 1, we denote by |·|q and · q the norms of Lq () and W 1; q (), respectively. · stands for the norm of H01 (). We denote by 1 ¡ 2 ¡ · · · the eigenvalues of the problem −Ex u = u;
u ∈ H01 ()
and by ’n the eigenfunction corresponding to the eigenvalue n with ’n = 1 for each n ¿ 1. It is known that the Krst eigenfunction ’1 does not change the sign. Then we may assume that ’1 ¿ 0 on . Let consider an initial boundary value problem ut − Ex u = g(u) + h(t; x) in (0; ∞) × ; u=0 on (0; ∞) × 9; (I) u(0) = u0 in ;
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where u0 ∈ L2 () and h ∈ C 1 (Q9 T ). We denote by t(u0 ) the number such that [0; t(u0 )) is the maximal interval for u(t) to exist. If u is the solution of problem (I) on [0; t(u0 )), then u can be represented by the integral form t u(t) = S(t)u0 + S(t − s)(g(u(s) + h(s; x)) ds 0
for 0 ¡ t ¡ t(u0 ). Here {S(t)} is the semigroup of linear operators generated by −Ex (cf. [12]). It is known that there exists C0 ¿ 0 satisfying S(t)f 6 t −1=2 C0 |f|2
for all f ∈ Lq () and t ¿ 0:
(V)
9 u ¿ 0 on }. Then X+ is a closed cone in (cf. [1,12]). We set X+ = {u ∈ C01 (): 9 We employ the standard order in C 1 (); 9 C01 (). 0 u ¿ v ⇔ u − v ∈ X+ ;
u ¿ v ⇔ u ¿ v; u = v;
uv ⇔ u − v ∈ int X+ :
9 we put For each u; v ∈ C01 (), 9 u 6 w 6 v}: [u; v] = {w ∈ C01 (): 9 is said to be order preserving if Sx ¿ Sy for x; y ∈ [u; v] A mapping S : [u; v] → C01 () with x ¿ y. S is said to be strongly order preserving if SxSy for x; y ∈ [u; v] with 9 ∩ C 0; 1 ((0; T ) × ) 9 is called subsolution for the x ¿ y. A function u ∈ C 1; 2 ((0; T ) × ) initial value problem of (I) if u satisKes ut − Ex u 6 g(u) + h(t; x) in (0; ∞) × ; u=0 on (0; ∞) × 9; u(0) = u0 in : A subsolution is said to be a strict subsolution if it is not a solution of (I). Similarly a supersolution and a strict supersolution are deKned by reversing the inequality sign, correspondingly. Lemma 2.1. There exists an approximating sequence {gn } ⊂ C 1 (R) of g such that gn is Lipschitz continuous and satis:es the following conditions: lim gn (t) = g(t)
n→∞
gn (t) ¿ 1 t→0 t lim
lim
|t|→∞
gn (t) ¡ 1 t
0 6 gn (t) 6
uniformly for n ¿ 1; uniformly for n ¿ 1; uniformly for n ¿ 1;
C on R\{0} | t |1−
for some constant C ¿ 0:
(2.1) (2.2) (2.3) (2.4)
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Proof. We Krst Kx a smooth function $ : (−∞; ∞) → [0; 1] such that $ (t) 6 a, and 0 for t ∈ (−∞; −1] ∪ [1; ∞); $(t) = 1 for t ∈ [ − 12 ; 12 ]: Let n ¿ 1 and tn± be the numbers such that tn− ¡ 0 ¡ tn+ and g(2tn± ) = 2ntn± . We put + for t ¿ 0; n$n (t)t + (1 − $+ n (t))g(t) gn (t) = − − n$n (t)t + (1 − $n (t))g(t) for t 6 0; − + + − − where $+ n (t)=$(t=2tn ) and $n (t)=$(−t=2tn ). Then we Knd that gn (t)=nt on [tn ; tn ] − + ± and gn (t) = g(t) on (−∞; 2tn ) ∪ (2tn ; ∞). Since tn → 0, as n → ∞. Then we Knd that gn (t) → g(t) as n → ∞. (2.2) and (2.3) follows directly from the deKnition of {gn }. We will see that (2.4) holds. Let consider the case that t ¿ 0. From the deKnition, we have gn (t) = nt on [0; tn+ ]. On the other hand, we have that for t ∈ [tn+ ; 2tn+ ]. + + + gn (t) = n($+ n (t) t + $n (t)) + (1 − $n (t))g (t) − ($n (t)) g(t) at at 6n + 1 + 1− + + t + 2tn t 2tn
a a + 1 + + 1− + (2tn+ ) : 6n 2 (tn ) 2
Then we Knd gn (t) 6 C max{n; g (t)} for some C ¿ 0. Moreover, recalling that n(2tn+ )1− ∼ = 1, we Knd that g (t) 6 Cn on [tn+ ; 2tn+ ] for some C ¿ 0, and then each gn is Lipschitz continuous on R. Remark 2.1. From the construction of gn , we have that | gn (t) | 6 | g(t) | for all t ∈ R and n ¿ 1.
3. Proof of theorem We put V = H01 () and H = L2 (). The norm of the dual space V ∗ of V is denoted by · ∗ . ·; · stands for the pairing of V and V ∗ . We also set C([0; T ]; u0 ; H ) = {v ∈ C([0; T ]; H ); v(0) = u0 } for each u0 ∈ H . Let {gn } be the approximating function constructed in Lemma 2.1. For each n ¿ 1, we denote by (Pn ) the problem (P) with g replaced by gn . For each u0 ∈ H and n ¿ 1, we deKne a mapping Ku0 ;n : C([0; T ]; u0 ; H ) → C([0; T ]; u0 ; H ) by t (Ku0 ;n v)(t) = S(t)u0 + S(t − s)(gn (v(s)) + h(s; x)) ds 0
for each v ∈ C([0; T ]; u0 ; H ):
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Then as a direct consequence of the deKnition, we have that Lemma 3.1. For each n ¿ 1 and u0 ∈ H; Ku0 ;n is compact. Moreover, we have that each Ku0 ;n has a unique :xed point vu0 ;n ∈ C([0; T ]; u0 ; H ). Proof. It is known that {S(t)} is a compact semigroup in H . Let u0 ∈ H and n ¿ 1. Then by the Lipschitz continuity of gn , we have that Ku0 ;n is compact (cf. the proof of Theorem 2.1 of [11]). It also follows that Ku0 ;n has a unique Kxed point (cf. also [11]). We deKne a Poincare mapping Sn by (Sn u0 ) = vu0 ;n (T )
for each u0 ∈ H and n ¿ 1:
Then since vu0 ;n is a solution of problem ut − Ex u = gn (u) + h(t; x) in (0; ∞) × ; u=0 on (0; ∞) × 9; u(0) = u0 in ;
(3.1)
(In )
we Knd that each Kxed point un of Sn is a periodic solution of problem (In ). Since (2.2) holds, we can choose ' ¿ 0 and n0 ¿ 1 such that '1 ’1 ¡ gn ('’1 ) for n ¿ n0 . Then we Knd that for n ¿ n0 , −Ex ('’1 ) = '1 ’1 ¡ gn ('’1 ) + h on : That is '’1 is a strict subsolution of problem of (In ) for n ¿ n0 . We next Knd a supersolution of problems (In ). Fix ∈ (0; 1 ). Let t0 be the positive number such that gn (t) ¡ t for all n ¿ 1 and t ¿ t0 . Put c=max{gn (t): 0 6 t 6 t0 ; n ¿ 1}. Since ¡ 1 , Dirichlet boundary value problem −Ex u = u + c + h 9 and v ¿ 0 on has a solution v ∈ H01 (). Noting that c + h ¿ 0, we have that v ∈ C 1 () . Let b ¿ 0 and put u9 = b’1 + v. Then, v(x) + 1 b’1 (x) ¿ (v(x) + b’1 (x)) ¿ gn (v(x) + b’1 (x))
for x ∈ with u(x) 9 ¿ t0
and 9 c ¿ gn (u(x))
for x ∈ with u(x) 9 ¡ t0 :
Then we Knd −Ex u9 = v + 1 b’1 + c + h ¿ gn (u) 9 +h
for all n ¿ 1:
Therefore we have that u9 is a supersolution of (In ) for n ¿ 1. Recall that 9’1 =9n ¿ 0 and 9v=9n ¿ 0 on 9 by the maximal principle (cf. [7]). Then we can choose b ¿ 0
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so large that '’1 u9 on . Then we have, by the parabolic maximal principle (cf. [7]) 9 and that Sn is strongly order preserving on ['’1 ; u] Sn ['’1 ; u] 9 9 ⊂ ['’1 ; u]
for all n ¿ n0 :
9 (cf. Proposition 21.2 of We also have that Sn ['’1 ; u] 9 is relatively compact in C01 () [7]). This implies by Theorem 4.2 of [7] that Snk ('’1 ) and Snk (u) 9 converges to a Kxed (1) (2) (1) point un and un of Sn as k → ∞, respectively and '’1 ¡ un 6 un(2) ¡ u9 holds for each n ¿ n0 . From the deKnition of Sn , we have that problem (Pn ) has a solution un 9 with un (0) = un (T ) = un(i) for n ¿ 1 and i = 1; 2. We have that un ∈ C 1; 2 ([0; T ] × ) (cf. Lemma 20.1 of [7]). Summarizing the argument above, we have 9 of (Pn ) such Lemma 3.2. For each n ¿ n0 , there exists a solution un ∈ C 1; 2 ([0; T ]× ) that '’1 ¡ un (t) ¡ u9 on [0; T ]. It then follows Lemma 3.3. There exists a solution u1 of (P) such that '’1 ¡ u1 (t) ¡ u9 on [0; T ]. Proof. By Lemma 3.2, there exists a sequence {un } such that un is a solution of (Pn ) with '’1 ¡ un ¡ u9 for each n ¿ 1. Then we have t un (t) = S(t)un (0) + S(t − s)fn (s) ds for t ¿ 0; (3.2) 0
where fn (s) = gn (un (s)) + h(s) for all n ¿ 1 and s ¿ 0. Since {un (t): n ¿ 1; t ¿ 0} is bounded in H and {S(t)} is a compact semigroup in H , we Knd by (3.2) that {un (t): n ¿ 1; t ¿ 0} is relatively compact in H . Then to prove the assertion, it is suMcient to show that {un } is equicontinuous in C([0; T ]: H ). In fact, if {un } is equicontinuous, we have that {un } is relatively compact in C([0; T ]: H ). Then there exists a convergent subsequence {uni } of {un } and one can see that the limit point is a solution of (P). Since each un is T -periodic, to prove the assertion, it is suMcient to show that {un } is equicontinuous in C([T; 2T ]: H ). For t; h ¿ 0 with T 6 t 6 2T , t un (t + h) − un (t) = (S(t + h) − S(t))un (0) + (S(h) − I ) S(t − s)fn (s) ds +
0
t+h
t
S(t + h − s)fn ds:
Since | Sv|2 6 | v|2 for v ∈ H , we have t+h S(t + h − s)fn ds 6 M1 h; t 2
(3.3)
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451
where M1 = sup{| fn (s)|2 : s ¿ 0; n ¿ 1}. Let a ¿ 0 be an arbitrary positive number. Then t−a t S(t − s)fn (s) ds = (S(h) − I )S(a) S(t − a − s)fn (s) ds (S(h) − I ) 0
+ (S(h) − I )
and
(S(h) − I )
t
t−a
0
t
t−a
S(t − s)fn (s) ds;
(3.4)
S(t − s)fn (s) ds 6 2aM1 : 2
t−a
S(t − a − s)fn (s) ds. Recalling that h h d S(s)v ds = Ex S(s)v ds; (S(h) − I )v = 0 ds 0
Here we put v =
0
we Knd by Lemma 3.6.2 of [12] that a+h h E S(s + a)v ds 6 C s−3=2 | v |2 | (S(h) − I )S(a)v |2 6 0 x a 2
6 C(a−1=2 − (a + h)−1=2 )| v |2 : Here we put M2 = 2T sup{| fn (s)|: s ¿ 0; n ¿ 1} and a = h1=3 . Then, for h suMciently small, we have that t−a 6 CM2 h1=2 : (S(h) − I )S(a) S(t − a − s)f (s) ds n 0
2
Then we have by (3.4) that t (S(h) − I ) 6 CM2 h1=2 + 2M1 h1=3 : S(t − s)f (s) ds n 0
2
On the other hand, recalling that t ∈ [T; 2T ], we have h Ex S(t + s)un ds | (S(t + h) − S(t))un (0)| 6 0
2
t+h 6 Ex S(s)un (s − t) ds t 6
t
t+h
(3.5)
2
s−3=2 |un (s − t)|2 ds
6 CM3 h; where M3 = sup{|un (s)|2 : n ¿ 1; s ¿ 0}. Then by (3.3), (3.5) and the inequality above, we Knd that |un (t + h) − un (t) |2 6 C1 h + C2 h1=2 + C3 h1=3 ;
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where C1 ; C2 and C3 are constant independent of n, t and h. Then the proof is completed. Next, we prove the existence of the second solution. In the following, {un(i) }; i = 1; 2 are the sequences deKned above. Lemma 3.4. Suppose that lim supn→∞ un(1) − un(2) ¿ 0. Then there exists two solutions v1 ; v2 of (P) such that '’1 ¡ v1 (0) ¡ v2 (0) ¡ u. 9 Proof. Let {un; 1 } and {un; 2 } be the sequence of solutions of (In ) with initial values un; i (0) = un(i) for n ¿ 1 and i = 1; 2. From the proof of Lemma 3.3, we have that {un; 1 } and {un; 2 } are relatively compact in C([0; T ]; H ). Then by subtracting subsequence, we may assume that there exist v1 ; v2 ∈ C([0; T ]; H ) such that un; 1 → v1 and un; 2 → v2 in C([0; T ]; H ) as n → ∞, and that v1 6 v2 on [0; T ]. It is obvious that v1 ; v2 are solutions of (P). By putting un = un; i , n ¿ 1 and i = 1; 2 in (3.2), we Knd by (V) that un; 1 (T ) − un; 2 (T ) 6 C0 T 1=2 | un(1) − un(2) |2 +C0 (2T 3=2 =3) | un; 1 − un; 2 |C([0; T ]; H ) : Suppose that v1 = v2 . Then we have limn→∞ | un; 1 − un; 2 |C([0; T ]; H ) = 0. Then we have by the inequality above that lim un(1) − un(2) = lim un; 1 (T ) − un; 2 (T ) = 0:
n→∞
n→∞
This contradicts the assumption. Therefore, we have that v1 ¡ v2 on [0; T ] and this completes the proof. To completes the proof of Theorem, we assume in the following that lim sup un(1) − un(2) = 0:
(3.6)
n→∞
Let u1 be the solution of (P) obtained in Lemma 3.3. Let In : V → R be a functional deKned by 1 2 In (v) = |∇v | − Gn (v) d x for v ∈ V; 2 t where Gn (t)= 0 gn (s) ds. We denotes by Inc the level set deKned by Inc ={v∈V : In (v)6c}. From the deKnition of gn and In , we can see that limv→∞ In (v) = ∞ for all n ¿ 1. Then we have that −∞ ¡ mn = min{In (v): v ∈ V }
for all n ¿ 1:
One can see that for any v ∈ V and n ¿ 1, we can choose t ¿ 0 suMciently small that In (tv) ¡ 0. That is mn ¡ 0. Recalling that limn→∞ gn (t) = g(t) uniformly on R, we have that there exists m0 ¡ 0 such that mn ¡ m0
for all n ¿ 1:
(3.7)
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Lemma 3.5. For each n ¿ 1 and - ∈ [mn ; 0], there exist m ¿ 1 and a continuous function h : S m → In- such that h(S m ) is not contractible in In- , where S m denotes the unit sphere in Rm . Proof. Let n ¿ 1. We put Vk = span{’1 ; : : : ; ’k } for k ¿ 1. Fix - ∈ [mn ; 0]. Let v ∈ V with v = 1. From conditions (2.2)–(2.4), we Knd that the mapping s → In (sv) decreases on an interval [0; t], where t ¿ 0 satisKes In (tv) = min{In (sv): s ¿ 0}; and increases on [t; ∞). Then from the deKnition of In , we have gn (tv)tv d x for n ¿ 1 t 2 v 2 =
and then recalling that | gn (t) | 6 | g(t) | for t ∈ R, one can see that t v 2 6 t | v | +1
(3.8)
⊥ for some k ¿ 2. Since | v |1+ 6 C1 | v |2 for some C1 ¿ 0 holds. Suppose that v ∈ Vk−1 and that k | v |2 6 v , we have from (3.8) that ( +1) 1 +1 6 C v +1 : t 1− v 2 6 | v | +1 +1 1 k
Therefore we have t6
C1( +1)=(1− ) k(1+ )=(1− )
;
and then t2 t 1+ In (tv) ¿ v 2 − 2 1+
1+
|v|
dx ¿
t 1+ C11+ t2 − 2 (1 + )k1+
:
Since k goes to inKnity as k → ∞, we have that t → 0. Then In (v) goes to 0 as k goes to inKnity. This implies that we can choose k0 ¿ 1 so large that In- ∩ Vk⊥0 = $. Now let v0 ∈ In- . Then since In is an even functional by the deKnition, we have that −v0 ∈ In- . If {v0 ; −v0 } is contractible in In- , we can deKne an odd continuous function h1 : S 1 → In- such that h1 (S 1 ) ⊂ In- due to the Krasnoselski’s result (cf. Lemma 3.2 of [2]). By induction, if hk0 −1 (S k0 −1 ) is contractible, we can construct an odd and continuous function hk0 : S k0 → In- . But since h(S k0 ) ∩ Vk⊥0 = $, this is impossible. Then we have that hk0 −1 (S k0 −1 ) is not contractible in In- . Remark 3.1. The assertion of the lemma above is rewritten as /m (In- ) = {0} for some m ¿ 1, where /m denotes the homotopy group. Let c1 ¿ 0 such that t − c1 ¡ gn (t) for all t 6 0 and n ¿ 1, where is a positive 9 of problem number satisfying ¡ 1 . Then there exists a negative solution v− ∈ C 1 () −Ex v− = v− − c1 :
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N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
Let a ¿ 1. Then u = av− satisKes −Ex u = av− − c1 ¡ gn (av− ) ¡ gn (u) + h: That is u is a subsolution of (Pn ). Lemma 3.6. For any - ¡ 0, there exists -1 ; -2 ¡ 0 such that - ¡ -1 ¡ -2 ¡ 0 and the interval [-1 ; -2 ] contains no critical point of In for n su
for all i ¿ 1:
Then since
- = lim Ini (ui ) = lim i→∞
i→∞
¿ lim
i→∞
1 ui 2 − 2
0
ui (x)
gni (t) dt d x
1 1 2 1+ ui − | ui |1+ ; 2 1+
we have that {ui } is bounded in V . Then we may assume that ui converges to u ∈ V strongly in H and weakly in V . Since gn → g, and | ∇(ui − uj ) |2 6 |gni (ui ) − gnj (uj )|2 | ui − uj |2
for i; j ¿ 1;
we Knd that ui converges to u strongly in V . This implies that ∇In (u)=0 and In (u)=-. Therefore we Knd that the interval (-0 ; 0) is the set of critical value of In . This is impossible. Thus we have the assertion. Lemma 3.7. There exists m0 ¿ 0 such that for each h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 and each n ¿ n0 , there exists a periodic solution wn of (In ) such that wn (0) ∈ int['’1 ; u]. 9 Proof. Let -0 ¡ 0 such that In ('’1 ) ¡ -0 for all n ¿ 1. Let u1 be the solution of (P) 9 We assume that obtained in Lemma 3.3. We put u(1) = u1 (0). Then '’1 ¡ u(1) ¡ u. there is no Kxed point of Sn in V \ ['’1 ; u] 9 for n ¿ n0 . Let -1 ; -2 be the constants such that -0 ¡ -1 ¡ -2 ¡ 0 for all n ¿ 1, and satisfying the assertion of Lemma 3.6. Then we have that m˜ 0 = inf { ∇In (v) ∗ : v ∈ In-2 \ In-1 ; n ¿ n0 } ¿ 0. We put m0 = m˜ 0 =||1=2 . Now let h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 ; n ¿ 1 and vn be the solution of (In ). We claim that if vn (0) ∈ In- for some - ∈ [-1 ; -2 ]; then vn (t) ∈ In-
for t ¿ 0:
(3.9)
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Suppose that - ∈ [-1 ; -2 ], vn (0) ∈ In- and that vn (t) ∈ In-2 on an interval [0; tvn ]. Then from the deKnition of m0 , we have that for t ∈ [0; tvn ], t dvn In (vn (t)) − In (vn (0)) = ∇In (vn (s)) · dt 0 t 6 (− ∇In (vn ) 2∗ + h(t) ∇In (vn ) ∗ ) 0
6
0
t
(− ∇In (vn ) 2∗ + h(t) ∇In (vn ) ∗ ) ¡ 0:
Then we have that In (vn (t)) ¡ In (vn (0)). From the argument above, it follows that vn (t) ∈ In-
for all t ¿ 0:
Then we have that (3.9) holds. Since '’1 ∈ In-0 and un(1) = limk→∞ Snk ('’1 ) for each n ¿ 1, we have that {un(1) } ⊂ In-1 for n ¿ 1. Since un(1) → u(1) strongly in H , and weakly in V , we Knd that u(1) ∈ In-1 . Let ' ¿ 0 such that -1 + 2' ¡ -2 . Then recalling that (3.6) holds, we can choose n1 ¿ n0 such that un(1) ; un(2) ∈ In-1 +'=2
for all n ¿ n1 :
(3.10)
On the other hand, we may assume that n1 is so large that 1 1+ (1) Gn (un ) d x ¡ ' 1 + | v | dx −
(3.11)
for all v ∈ [un(1) ; un(2) ]. Then we have by choosing r ¿ 0 is so small that [v; z] ∩ In-1 +' is contractible in In-2 ;
(3.12)
9 such that v − un(1) ¡ r and z − un(2) ¡ r. In fact, noting that · 2 where v; z ∈ C01 () is a convex function, we have by (3.10) and (3.11) that In ( v + (1 − )un(1) ) 6 -1 + 2'
for all v ∈ [un(1) ; un(2) ] ∩ In-1 +'
and
∈ [0; 1]:
Then we have by choosing r ¿ 0 suMciently small that (3.12) holds. Now Kx n ¿ n1 . Then by Lemma 3.5, we have that there exists m ¿ 0 and a continuous function 9 ∩ In-2 is dense h : S m → In-2 such that h(S m ) is not contractible in In-2 . Since C01 () -2 m 1 9 -2 in In , we may assume that h(S ) ⊂ C0 () ∩ In . We denote by uz the solution of (In ) with u(0) = h(z), z ∈ S m . by choosing b ¿ 0 in the deKnition of u9 suMciently large, we have that v ¡ u9 for all v ∈ h(S m ). Similarly, by choosing a ¿ 0 in the deKnition of u, we have that u ¡ '’1 and u ¡ v for all v ∈ h(S m ). Recall that u and u9 are sub and supersolution of (In ), respectively, and that u ¡ '’1 . Then since we assume that 9 we have that Snk (u) → un(1) and there exist no Kxed point of Sn outside of ['’1 ; u], (2) 9 → un as k → ∞. Then since Snk (u) → un(1) and Snk (u) 9 → un(2) , we have by Snk (u) 9 − un(2) ¡ r. Then since choosing k so large that Snk (u) − un(1) ¡ r and Snk (u) Snk (u) 6 uz (kT ) 6 Snk (u) 9
for all z ∈ S m :
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We have by (3.12) that {uz (kT ): z ∈ S m } is contractible in In-2 . By (3.9), we can deKne a homotopy 3 : [0; kT ] × h(S m ) → C01 () ∩ In-2 by 3(s; h(z)) = uz (s)
for 0 6 s 6 kT and z ∈ S m :
Then since {uz (kT ): z ∈ S m } is contractible in In-2 , we Knd that h(S m ) is contractible in In-2 . This is a contradiction. Thus we have that there exists a Kxed point v of Sn in V \ ['’1 ; u]. 9 Then the assertion follows. Lemma 3.8. Let h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 . Then there exists a solution u2 of (P) such that u2 (0) ∈ int['’1 ; u]. 9 Proof. Let {wn } be the sequence obtained in Lemma 3.7. Then by the same argument as in the proof of Lemma 3.3, we Knd that there exists a convergent subsequence {wni } 9 of {wn } and the limit point w is a solution of (P) satisfying w(0) ∈ int['’1 ; u]. References [1] H. Amann, Periodic solutions of semilinear parabolic equations, in: Nonlinear Analysis: A Collection of Papers in Honor of Erich Roth, Academic Press, New York, 1978, 1–29. [2] A. Bahri, H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981) 1–32. [3] H, Brezis, S. Kamin, Sublinear elliptic equations in Rn , Manuscripta Math. 74 (1992) 87–106. [4] H. Brezis, L. Qsward, Remarks on sublinear elliptic equations, Nonlinear Anal. TMA 10 (1) (1986) 55–64. [5] A. Castro, A.C. Lazer, Results on periodic solutions of parabolic equations suggested by elliptic theory, Boll. Un. Mat. Ital. 6 (I-B) (1982) 1089–1104. [6] E.N. Dancer, N. Hirano, Existence of stable and unstable periodic solutions for semilinear parabolic problems, Discrete Continuous Dyn. System 3 (2) (1997) 207–216. [7] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, in: Pitman Research Note in Mathematics Series, Vol. 247, Longman ScientiKc & Technical, New York, 1991. [8] N. Hirano, Wan Se. Kim, Multiplicity and stability result for semilinear parabolic equations, Discrete Continuous Dyn. System 2 (2) (1996) 271–280. [9] Y. Kajikiya, Non-radials with group invariance for the sublinear Emden–Fowler equations, preprint. [10] W.S. Kim, B. Ko, Multiplicity and stability of solutions for semilinear elliptic equations having not non-negative mass, J. Korean Math. Soc. 37 (1) (2000) 85–109. [11] A. Pazy, Semigroup of linear operators and applications to partial diTerential equations, Appl. Math. Ser. 44 (1983). [12] H. Tanabe, Equations of Evolution, in: Monographs and Studies in Mathematics, Vol. 6, Pitman, Boston, London, Melbourne, 1976.
www.elsevier.com/locate/na
Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities Norimichi Hiranoa;∗ , Wan Se Kimb;1 a Department
of Mathematics, Faculty of Engineering, Yokohama National University, 156 Tokiwadai, Hodogaya-ku, Yokohama, Japan b Department of Mathematics, College of Natural Sciences, Hanyang University, Seoul 133-791, South Korea Received 20 April 2002; accepted 3 December 2002
Abstract In this paper, we establish a multiple existence result of periodic solutions for a nonlinear parabolic equations with singular nonlinearity at zero. To establish our result, We use an approximating process by smooth nonlinear functions to avoid a singularity in nonlinear term. And we adapt topological argument based on the variational structure of functionals corresponding to approximating equations. ? 2003 Elsevier Science Ltd. All rights reserved. MSC: 35k20; 35k55; 35k60 Keywords: Multiplicity; Sublinear grow; Parabolic equation; Stability; Unstability
1. Introduction Let N ¿ 2 and be a bounded domain in RN with a smooth boundary 9 and h : R × 9 → R be a continuous function such that for each x ∈ ; h(·; x) is T -periodic and continuous, and h ¿ 0 on R × . In this paper, we are concerned with multiple
∗
Corresponding author. Tel.: +81-45-339-4212. E-mail addresses: [email protected] (N. Hirano), [email protected] (W.S. Kim).
1
This work was supported by Korea Research Foundation Grant KRF-2000-15-DP0027.
0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0362-546X(03)00101-9
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N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
existence of T -periodic solutions of ut − Ex u = g(u) + h(t; x) u=0 u(0) = u(T )
the following problem: in (0:T ) × ; on (0; T ) × 9;
(P)
9 in ;
where −Ex denotes the Lapalacian on RN and g(t)=|t| −1 t, for t ∈ R, with 0 ¡ ¡ 1. The existence of periodic solutions for semilinear evolution equations has been studied by many authors [5,6,8,10]. But few seems to be known about the case that the nonlinear term g has a singularity at zero in the sense that g(u) lim = ∞: |u|→0 u Hence the usual known scheme, in particular, sub-supersolution method in treating parabolic problems is not applicable directly in proving the existence and multiplicity of solutions due to the singularity. Even in elliptic problems, there have been few results known on the existence and multiplicity results of elliptic problems with above type nonlinearities. We refer [3,4,9] for elliptic problems. In the present paper, we use an approximating process by smooth nonlinear functions to avoid such a singularity, and establish a multiple existence result for (P). To prove the multiple existence of solutions, we need topological argument based on the variational structure of functionals corresponding to problem (P). To avoid unnecessary complexity, we restrict ourselves to the case that g(t) = |t | −1 t, 0 ¡ ¡ 1. One can see that our argument here is valid for any function g ∈ C(R) satisfying that 0 ¡ limt→0 g(t)=| t| −1 t ¡ ∞ and lim| t|→∞ g(t)=t = 0. We put QT = (0; T ) × . A function u ∈ C([0; T ]; H01 ())∩C 1 ([0; T ]; L2 ()) is said to be a solution of (P) if u satisKes (P). We now state our main result. Theorem. There exists m0 ¿ 0 such that for each h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 , there exist at least two solutions of (P). 2. Preliminaries For q ¿ 1, we denote by |·|q and · q the norms of Lq () and W 1; q (), respectively. · stands for the norm of H01 (). We denote by 1 ¡ 2 ¡ · · · the eigenvalues of the problem −Ex u = u;
u ∈ H01 ()
and by ’n the eigenfunction corresponding to the eigenvalue n with ’n = 1 for each n ¿ 1. It is known that the Krst eigenfunction ’1 does not change the sign. Then we may assume that ’1 ¿ 0 on . Let consider an initial boundary value problem ut − Ex u = g(u) + h(t; x) in (0; ∞) × ; u=0 on (0; ∞) × 9; (I) u(0) = u0 in ;
N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
447
where u0 ∈ L2 () and h ∈ C 1 (Q9 T ). We denote by t(u0 ) the number such that [0; t(u0 )) is the maximal interval for u(t) to exist. If u is the solution of problem (I) on [0; t(u0 )), then u can be represented by the integral form t u(t) = S(t)u0 + S(t − s)(g(u(s) + h(s; x)) ds 0
for 0 ¡ t ¡ t(u0 ). Here {S(t)} is the semigroup of linear operators generated by −Ex (cf. [12]). It is known that there exists C0 ¿ 0 satisfying S(t)f 6 t −1=2 C0 |f|2
for all f ∈ Lq () and t ¿ 0:
(V)
9 u ¿ 0 on }. Then X+ is a closed cone in (cf. [1,12]). We set X+ = {u ∈ C01 (): 9 We employ the standard order in C 1 (); 9 C01 (). 0 u ¿ v ⇔ u − v ∈ X+ ;
u ¿ v ⇔ u ¿ v; u = v;
uv ⇔ u − v ∈ int X+ :
9 we put For each u; v ∈ C01 (), 9 u 6 w 6 v}: [u; v] = {w ∈ C01 (): 9 is said to be order preserving if Sx ¿ Sy for x; y ∈ [u; v] A mapping S : [u; v] → C01 () with x ¿ y. S is said to be strongly order preserving if SxSy for x; y ∈ [u; v] with 9 ∩ C 0; 1 ((0; T ) × ) 9 is called subsolution for the x ¿ y. A function u ∈ C 1; 2 ((0; T ) × ) initial value problem of (I) if u satisKes ut − Ex u 6 g(u) + h(t; x) in (0; ∞) × ; u=0 on (0; ∞) × 9; u(0) = u0 in : A subsolution is said to be a strict subsolution if it is not a solution of (I). Similarly a supersolution and a strict supersolution are deKned by reversing the inequality sign, correspondingly. Lemma 2.1. There exists an approximating sequence {gn } ⊂ C 1 (R) of g such that gn is Lipschitz continuous and satis:es the following conditions: lim gn (t) = g(t)
n→∞
gn (t) ¿ 1 t→0 t lim
lim
|t|→∞
gn (t) ¡ 1 t
0 6 gn (t) 6
uniformly for n ¿ 1; uniformly for n ¿ 1; uniformly for n ¿ 1;
C on R\{0} | t |1−
for some constant C ¿ 0:
(2.1) (2.2) (2.3) (2.4)
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Proof. We Krst Kx a smooth function $ : (−∞; ∞) → [0; 1] such that $ (t) 6 a, and 0 for t ∈ (−∞; −1] ∪ [1; ∞); $(t) = 1 for t ∈ [ − 12 ; 12 ]: Let n ¿ 1 and tn± be the numbers such that tn− ¡ 0 ¡ tn+ and g(2tn± ) = 2ntn± . We put + for t ¿ 0; n$n (t)t + (1 − $+ n (t))g(t) gn (t) = − − n$n (t)t + (1 − $n (t))g(t) for t 6 0; − + + − − where $+ n (t)=$(t=2tn ) and $n (t)=$(−t=2tn ). Then we Knd that gn (t)=nt on [tn ; tn ] − + ± and gn (t) = g(t) on (−∞; 2tn ) ∪ (2tn ; ∞). Since tn → 0, as n → ∞. Then we Knd that gn (t) → g(t) as n → ∞. (2.2) and (2.3) follows directly from the deKnition of {gn }. We will see that (2.4) holds. Let consider the case that t ¿ 0. From the deKnition, we have gn (t) = nt on [0; tn+ ]. On the other hand, we have that for t ∈ [tn+ ; 2tn+ ]. + + + gn (t) = n($+ n (t) t + $n (t)) + (1 − $n (t))g (t) − ($n (t)) g(t) at at 6n + 1 + 1− + + t + 2tn t 2tn
a a + 1 + + 1− + (2tn+ ) : 6n 2 (tn ) 2
Then we Knd gn (t) 6 C max{n; g (t)} for some C ¿ 0. Moreover, recalling that n(2tn+ )1− ∼ = 1, we Knd that g (t) 6 Cn on [tn+ ; 2tn+ ] for some C ¿ 0, and then each gn is Lipschitz continuous on R. Remark 2.1. From the construction of gn , we have that | gn (t) | 6 | g(t) | for all t ∈ R and n ¿ 1.
3. Proof of theorem We put V = H01 () and H = L2 (). The norm of the dual space V ∗ of V is denoted by · ∗ . ·; · stands for the pairing of V and V ∗ . We also set C([0; T ]; u0 ; H ) = {v ∈ C([0; T ]; H ); v(0) = u0 } for each u0 ∈ H . Let {gn } be the approximating function constructed in Lemma 2.1. For each n ¿ 1, we denote by (Pn ) the problem (P) with g replaced by gn . For each u0 ∈ H and n ¿ 1, we deKne a mapping Ku0 ;n : C([0; T ]; u0 ; H ) → C([0; T ]; u0 ; H ) by t (Ku0 ;n v)(t) = S(t)u0 + S(t − s)(gn (v(s)) + h(s; x)) ds 0
for each v ∈ C([0; T ]; u0 ; H ):
N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
449
Then as a direct consequence of the deKnition, we have that Lemma 3.1. For each n ¿ 1 and u0 ∈ H; Ku0 ;n is compact. Moreover, we have that each Ku0 ;n has a unique :xed point vu0 ;n ∈ C([0; T ]; u0 ; H ). Proof. It is known that {S(t)} is a compact semigroup in H . Let u0 ∈ H and n ¿ 1. Then by the Lipschitz continuity of gn , we have that Ku0 ;n is compact (cf. the proof of Theorem 2.1 of [11]). It also follows that Ku0 ;n has a unique Kxed point (cf. also [11]). We deKne a Poincare mapping Sn by (Sn u0 ) = vu0 ;n (T )
for each u0 ∈ H and n ¿ 1:
Then since vu0 ;n is a solution of problem ut − Ex u = gn (u) + h(t; x) in (0; ∞) × ; u=0 on (0; ∞) × 9; u(0) = u0 in ;
(3.1)
(In )
we Knd that each Kxed point un of Sn is a periodic solution of problem (In ). Since (2.2) holds, we can choose ' ¿ 0 and n0 ¿ 1 such that '1 ’1 ¡ gn ('’1 ) for n ¿ n0 . Then we Knd that for n ¿ n0 , −Ex ('’1 ) = '1 ’1 ¡ gn ('’1 ) + h on : That is '’1 is a strict subsolution of problem of (In ) for n ¿ n0 . We next Knd a supersolution of problems (In ). Fix ∈ (0; 1 ). Let t0 be the positive number such that gn (t) ¡ t for all n ¿ 1 and t ¿ t0 . Put c=max{gn (t): 0 6 t 6 t0 ; n ¿ 1}. Since ¡ 1 , Dirichlet boundary value problem −Ex u = u + c + h 9 and v ¿ 0 on has a solution v ∈ H01 (). Noting that c + h ¿ 0, we have that v ∈ C 1 () . Let b ¿ 0 and put u9 = b’1 + v. Then, v(x) + 1 b’1 (x) ¿ (v(x) + b’1 (x)) ¿ gn (v(x) + b’1 (x))
for x ∈ with u(x) 9 ¿ t0
and 9 c ¿ gn (u(x))
for x ∈ with u(x) 9 ¡ t0 :
Then we Knd −Ex u9 = v + 1 b’1 + c + h ¿ gn (u) 9 +h
for all n ¿ 1:
Therefore we have that u9 is a supersolution of (In ) for n ¿ 1. Recall that 9’1 =9n ¿ 0 and 9v=9n ¿ 0 on 9 by the maximal principle (cf. [7]). Then we can choose b ¿ 0
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N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
so large that '’1 u9 on . Then we have, by the parabolic maximal principle (cf. [7]) 9 and that Sn is strongly order preserving on ['’1 ; u] Sn ['’1 ; u] 9 9 ⊂ ['’1 ; u]
for all n ¿ n0 :
9 (cf. Proposition 21.2 of We also have that Sn ['’1 ; u] 9 is relatively compact in C01 () [7]). This implies by Theorem 4.2 of [7] that Snk ('’1 ) and Snk (u) 9 converges to a Kxed (1) (2) (1) point un and un of Sn as k → ∞, respectively and '’1 ¡ un 6 un(2) ¡ u9 holds for each n ¿ n0 . From the deKnition of Sn , we have that problem (Pn ) has a solution un 9 with un (0) = un (T ) = un(i) for n ¿ 1 and i = 1; 2. We have that un ∈ C 1; 2 ([0; T ] × ) (cf. Lemma 20.1 of [7]). Summarizing the argument above, we have 9 of (Pn ) such Lemma 3.2. For each n ¿ n0 , there exists a solution un ∈ C 1; 2 ([0; T ]× ) that '’1 ¡ un (t) ¡ u9 on [0; T ]. It then follows Lemma 3.3. There exists a solution u1 of (P) such that '’1 ¡ u1 (t) ¡ u9 on [0; T ]. Proof. By Lemma 3.2, there exists a sequence {un } such that un is a solution of (Pn ) with '’1 ¡ un ¡ u9 for each n ¿ 1. Then we have t un (t) = S(t)un (0) + S(t − s)fn (s) ds for t ¿ 0; (3.2) 0
where fn (s) = gn (un (s)) + h(s) for all n ¿ 1 and s ¿ 0. Since {un (t): n ¿ 1; t ¿ 0} is bounded in H and {S(t)} is a compact semigroup in H , we Knd by (3.2) that {un (t): n ¿ 1; t ¿ 0} is relatively compact in H . Then to prove the assertion, it is suMcient to show that {un } is equicontinuous in C([0; T ]: H ). In fact, if {un } is equicontinuous, we have that {un } is relatively compact in C([0; T ]: H ). Then there exists a convergent subsequence {uni } of {un } and one can see that the limit point is a solution of (P). Since each un is T -periodic, to prove the assertion, it is suMcient to show that {un } is equicontinuous in C([T; 2T ]: H ). For t; h ¿ 0 with T 6 t 6 2T , t un (t + h) − un (t) = (S(t + h) − S(t))un (0) + (S(h) − I ) S(t − s)fn (s) ds +
0
t+h
t
S(t + h − s)fn ds:
Since | Sv|2 6 | v|2 for v ∈ H , we have t+h S(t + h − s)fn ds 6 M1 h; t 2
(3.3)
N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
451
where M1 = sup{| fn (s)|2 : s ¿ 0; n ¿ 1}. Let a ¿ 0 be an arbitrary positive number. Then t−a t S(t − s)fn (s) ds = (S(h) − I )S(a) S(t − a − s)fn (s) ds (S(h) − I ) 0
+ (S(h) − I )
and
(S(h) − I )
t
t−a
0
t
t−a
S(t − s)fn (s) ds;
(3.4)
S(t − s)fn (s) ds 6 2aM1 : 2
t−a
S(t − a − s)fn (s) ds. Recalling that h h d S(s)v ds = Ex S(s)v ds; (S(h) − I )v = 0 ds 0
Here we put v =
0
we Knd by Lemma 3.6.2 of [12] that a+h h E S(s + a)v ds 6 C s−3=2 | v |2 | (S(h) − I )S(a)v |2 6 0 x a 2
6 C(a−1=2 − (a + h)−1=2 )| v |2 : Here we put M2 = 2T sup{| fn (s)|: s ¿ 0; n ¿ 1} and a = h1=3 . Then, for h suMciently small, we have that t−a 6 CM2 h1=2 : (S(h) − I )S(a) S(t − a − s)f (s) ds n 0
2
Then we have by (3.4) that t (S(h) − I ) 6 CM2 h1=2 + 2M1 h1=3 : S(t − s)f (s) ds n 0
2
On the other hand, recalling that t ∈ [T; 2T ], we have h Ex S(t + s)un ds | (S(t + h) − S(t))un (0)| 6 0
2
t+h 6 Ex S(s)un (s − t) ds t 6
t
t+h
(3.5)
2
s−3=2 |un (s − t)|2 ds
6 CM3 h; where M3 = sup{|un (s)|2 : n ¿ 1; s ¿ 0}. Then by (3.3), (3.5) and the inequality above, we Knd that |un (t + h) − un (t) |2 6 C1 h + C2 h1=2 + C3 h1=3 ;
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N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
where C1 ; C2 and C3 are constant independent of n, t and h. Then the proof is completed. Next, we prove the existence of the second solution. In the following, {un(i) }; i = 1; 2 are the sequences deKned above. Lemma 3.4. Suppose that lim supn→∞ un(1) − un(2) ¿ 0. Then there exists two solutions v1 ; v2 of (P) such that '’1 ¡ v1 (0) ¡ v2 (0) ¡ u. 9 Proof. Let {un; 1 } and {un; 2 } be the sequence of solutions of (In ) with initial values un; i (0) = un(i) for n ¿ 1 and i = 1; 2. From the proof of Lemma 3.3, we have that {un; 1 } and {un; 2 } are relatively compact in C([0; T ]; H ). Then by subtracting subsequence, we may assume that there exist v1 ; v2 ∈ C([0; T ]; H ) such that un; 1 → v1 and un; 2 → v2 in C([0; T ]; H ) as n → ∞, and that v1 6 v2 on [0; T ]. It is obvious that v1 ; v2 are solutions of (P). By putting un = un; i , n ¿ 1 and i = 1; 2 in (3.2), we Knd by (V) that un; 1 (T ) − un; 2 (T ) 6 C0 T 1=2 | un(1) − un(2) |2 +C0 (2T 3=2 =3) | un; 1 − un; 2 |C([0; T ]; H ) : Suppose that v1 = v2 . Then we have limn→∞ | un; 1 − un; 2 |C([0; T ]; H ) = 0. Then we have by the inequality above that lim un(1) − un(2) = lim un; 1 (T ) − un; 2 (T ) = 0:
n→∞
n→∞
This contradicts the assumption. Therefore, we have that v1 ¡ v2 on [0; T ] and this completes the proof. To completes the proof of Theorem, we assume in the following that lim sup un(1) − un(2) = 0:
(3.6)
n→∞
Let u1 be the solution of (P) obtained in Lemma 3.3. Let In : V → R be a functional deKned by 1 2 In (v) = |∇v | − Gn (v) d x for v ∈ V; 2 t where Gn (t)= 0 gn (s) ds. We denotes by Inc the level set deKned by Inc ={v∈V : In (v)6c}. From the deKnition of gn and In , we can see that limv→∞ In (v) = ∞ for all n ¿ 1. Then we have that −∞ ¡ mn = min{In (v): v ∈ V }
for all n ¿ 1:
One can see that for any v ∈ V and n ¿ 1, we can choose t ¿ 0 suMciently small that In (tv) ¡ 0. That is mn ¡ 0. Recalling that limn→∞ gn (t) = g(t) uniformly on R, we have that there exists m0 ¡ 0 such that mn ¡ m0
for all n ¿ 1:
(3.7)
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453
Lemma 3.5. For each n ¿ 1 and - ∈ [mn ; 0], there exist m ¿ 1 and a continuous function h : S m → In- such that h(S m ) is not contractible in In- , where S m denotes the unit sphere in Rm . Proof. Let n ¿ 1. We put Vk = span{’1 ; : : : ; ’k } for k ¿ 1. Fix - ∈ [mn ; 0]. Let v ∈ V with v = 1. From conditions (2.2)–(2.4), we Knd that the mapping s → In (sv) decreases on an interval [0; t], where t ¿ 0 satisKes In (tv) = min{In (sv): s ¿ 0}; and increases on [t; ∞). Then from the deKnition of In , we have gn (tv)tv d x for n ¿ 1 t 2 v 2 =
and then recalling that | gn (t) | 6 | g(t) | for t ∈ R, one can see that t v 2 6 t | v | +1
(3.8)
⊥ for some k ¿ 2. Since | v |1+ 6 C1 | v |2 for some C1 ¿ 0 holds. Suppose that v ∈ Vk−1 and that k | v |2 6 v , we have from (3.8) that ( +1) 1 +1 6 C v +1 : t 1− v 2 6 | v | +1 +1 1 k
Therefore we have t6
C1( +1)=(1− ) k(1+ )=(1− )
;
and then t2 t 1+ In (tv) ¿ v 2 − 2 1+
1+
|v|
dx ¿
t 1+ C11+ t2 − 2 (1 + )k1+
:
Since k goes to inKnity as k → ∞, we have that t → 0. Then In (v) goes to 0 as k goes to inKnity. This implies that we can choose k0 ¿ 1 so large that In- ∩ Vk⊥0 = $. Now let v0 ∈ In- . Then since In is an even functional by the deKnition, we have that −v0 ∈ In- . If {v0 ; −v0 } is contractible in In- , we can deKne an odd continuous function h1 : S 1 → In- such that h1 (S 1 ) ⊂ In- due to the Krasnoselski’s result (cf. Lemma 3.2 of [2]). By induction, if hk0 −1 (S k0 −1 ) is contractible, we can construct an odd and continuous function hk0 : S k0 → In- . But since h(S k0 ) ∩ Vk⊥0 = $, this is impossible. Then we have that hk0 −1 (S k0 −1 ) is not contractible in In- . Remark 3.1. The assertion of the lemma above is rewritten as /m (In- ) = {0} for some m ¿ 1, where /m denotes the homotopy group. Let c1 ¿ 0 such that t − c1 ¡ gn (t) for all t 6 0 and n ¿ 1, where is a positive 9 of problem number satisfying ¡ 1 . Then there exists a negative solution v− ∈ C 1 () −Ex v− = v− − c1 :
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N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
Let a ¿ 1. Then u = av− satisKes −Ex u = av− − c1 ¡ gn (av− ) ¡ gn (u) + h: That is u is a subsolution of (Pn ). Lemma 3.6. For any - ¡ 0, there exists -1 ; -2 ¡ 0 such that - ¡ -1 ¡ -2 ¡ 0 and the interval [-1 ; -2 ] contains no critical point of In for n su
for all i ¿ 1:
Then since
- = lim Ini (ui ) = lim i→∞
i→∞
¿ lim
i→∞
1 ui 2 − 2
0
ui (x)
gni (t) dt d x
1 1 2 1+ ui − | ui |1+ ; 2 1+
we have that {ui } is bounded in V . Then we may assume that ui converges to u ∈ V strongly in H and weakly in V . Since gn → g, and | ∇(ui − uj ) |2 6 |gni (ui ) − gnj (uj )|2 | ui − uj |2
for i; j ¿ 1;
we Knd that ui converges to u strongly in V . This implies that ∇In (u)=0 and In (u)=-. Therefore we Knd that the interval (-0 ; 0) is the set of critical value of In . This is impossible. Thus we have the assertion. Lemma 3.7. There exists m0 ¿ 0 such that for each h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 and each n ¿ n0 , there exists a periodic solution wn of (In ) such that wn (0) ∈ int['’1 ; u]. 9 Proof. Let -0 ¡ 0 such that In ('’1 ) ¡ -0 for all n ¿ 1. Let u1 be the solution of (P) 9 We assume that obtained in Lemma 3.3. We put u(1) = u1 (0). Then '’1 ¡ u(1) ¡ u. there is no Kxed point of Sn in V \ ['’1 ; u] 9 for n ¿ n0 . Let -1 ; -2 be the constants such that -0 ¡ -1 ¡ -2 ¡ 0 for all n ¿ 1, and satisfying the assertion of Lemma 3.6. Then we have that m˜ 0 = inf { ∇In (v) ∗ : v ∈ In-2 \ In-1 ; n ¿ n0 } ¿ 0. We put m0 = m˜ 0 =||1=2 . Now let h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 ; n ¿ 1 and vn be the solution of (In ). We claim that if vn (0) ∈ In- for some - ∈ [-1 ; -2 ]; then vn (t) ∈ In-
for t ¿ 0:
(3.9)
N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
455
Suppose that - ∈ [-1 ; -2 ], vn (0) ∈ In- and that vn (t) ∈ In-2 on an interval [0; tvn ]. Then from the deKnition of m0 , we have that for t ∈ [0; tvn ], t dvn In (vn (t)) − In (vn (0)) = ∇In (vn (s)) · dt 0 t 6 (− ∇In (vn ) 2∗ + h(t) ∇In (vn ) ∗ ) 0
6
0
t
(− ∇In (vn ) 2∗ + h(t) ∇In (vn ) ∗ ) ¡ 0:
Then we have that In (vn (t)) ¡ In (vn (0)). From the argument above, it follows that vn (t) ∈ In-
for all t ¿ 0:
Then we have that (3.9) holds. Since '’1 ∈ In-0 and un(1) = limk→∞ Snk ('’1 ) for each n ¿ 1, we have that {un(1) } ⊂ In-1 for n ¿ 1. Since un(1) → u(1) strongly in H , and weakly in V , we Knd that u(1) ∈ In-1 . Let ' ¿ 0 such that -1 + 2' ¡ -2 . Then recalling that (3.6) holds, we can choose n1 ¿ n0 such that un(1) ; un(2) ∈ In-1 +'=2
for all n ¿ n1 :
(3.10)
On the other hand, we may assume that n1 is so large that 1 1+ (1) Gn (un ) d x ¡ ' 1 + | v | dx −
(3.11)
for all v ∈ [un(1) ; un(2) ]. Then we have by choosing r ¿ 0 is so small that [v; z] ∩ In-1 +' is contractible in In-2 ;
(3.12)
9 such that v − un(1) ¡ r and z − un(2) ¡ r. In fact, noting that · 2 where v; z ∈ C01 () is a convex function, we have by (3.10) and (3.11) that In ( v + (1 − )un(1) ) 6 -1 + 2'
for all v ∈ [un(1) ; un(2) ] ∩ In-1 +'
and
∈ [0; 1]:
Then we have by choosing r ¿ 0 suMciently small that (3.12) holds. Now Kx n ¿ n1 . Then by Lemma 3.5, we have that there exists m ¿ 0 and a continuous function 9 ∩ In-2 is dense h : S m → In-2 such that h(S m ) is not contractible in In-2 . Since C01 () -2 m 1 9 -2 in In , we may assume that h(S ) ⊂ C0 () ∩ In . We denote by uz the solution of (In ) with u(0) = h(z), z ∈ S m . by choosing b ¿ 0 in the deKnition of u9 suMciently large, we have that v ¡ u9 for all v ∈ h(S m ). Similarly, by choosing a ¿ 0 in the deKnition of u, we have that u ¡ '’1 and u ¡ v for all v ∈ h(S m ). Recall that u and u9 are sub and supersolution of (In ), respectively, and that u ¡ '’1 . Then since we assume that 9 we have that Snk (u) → un(1) and there exist no Kxed point of Sn outside of ['’1 ; u], (2) 9 → un as k → ∞. Then since Snk (u) → un(1) and Snk (u) 9 → un(2) , we have by Snk (u) 9 − un(2) ¡ r. Then since choosing k so large that Snk (u) − un(1) ¡ r and Snk (u) Snk (u) 6 uz (kT ) 6 Snk (u) 9
for all z ∈ S m :
456
N. Hirano, W.S. Kim / Nonlinear Analysis 54 (2003) 445 – 456
We have by (3.12) that {uz (kT ): z ∈ S m } is contractible in In-2 . By (3.9), we can deKne a homotopy 3 : [0; kT ] × h(S m ) → C01 () ∩ In-2 by 3(s; h(z)) = uz (s)
for 0 6 s 6 kT and z ∈ S m :
Then since {uz (kT ): z ∈ S m } is contractible in In-2 , we Knd that h(S m ) is contractible in In-2 . This is a contradiction. Thus we have that there exists a Kxed point v of Sn in V \ ['’1 ; u]. 9 Then the assertion follows. Lemma 3.8. Let h ∈ C 1 (Q9 T ) with | h |C 1 (Q9 T ) ¡ m0 . Then there exists a solution u2 of (P) such that u2 (0) ∈ int['’1 ; u]. 9 Proof. Let {wn } be the sequence obtained in Lemma 3.7. Then by the same argument as in the proof of Lemma 3.3, we Knd that there exists a convergent subsequence {wni } 9 of {wn } and the limit point w is a solution of (P) satisfying w(0) ∈ int['’1 ; u]. References [1] H. Amann, Periodic solutions of semilinear parabolic equations, in: Nonlinear Analysis: A Collection of Papers in Honor of Erich Roth, Academic Press, New York, 1978, 1–29. [2] A. Bahri, H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981) 1–32. [3] H, Brezis, S. Kamin, Sublinear elliptic equations in Rn , Manuscripta Math. 74 (1992) 87–106. [4] H. Brezis, L. Qsward, Remarks on sublinear elliptic equations, Nonlinear Anal. TMA 10 (1) (1986) 55–64. [5] A. Castro, A.C. Lazer, Results on periodic solutions of parabolic equations suggested by elliptic theory, Boll. Un. Mat. Ital. 6 (I-B) (1982) 1089–1104. [6] E.N. Dancer, N. Hirano, Existence of stable and unstable periodic solutions for semilinear parabolic problems, Discrete Continuous Dyn. System 3 (2) (1997) 207–216. [7] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, in: Pitman Research Note in Mathematics Series, Vol. 247, Longman ScientiKc & Technical, New York, 1991. [8] N. Hirano, Wan Se. Kim, Multiplicity and stability result for semilinear parabolic equations, Discrete Continuous Dyn. System 2 (2) (1996) 271–280. [9] Y. Kajikiya, Non-radials with group invariance for the sublinear Emden–Fowler equations, preprint. [10] W.S. Kim, B. Ko, Multiplicity and stability of solutions for semilinear elliptic equations having not non-negative mass, J. Korean Math. Soc. 37 (1) (2000) 85–109. [11] A. Pazy, Semigroup of linear operators and applications to partial diTerential equations, Appl. Math. Ser. 44 (1983). [12] H. Tanabe, Equations of Evolution, in: Monographs and Studies in Mathematics, Vol. 6, Pitman, Boston, London, Melbourne, 1976.