Multiplicity of solutions of semilinear elliptic boundary value problems with jumping nonlinearities at zero

Nonlinear Analysis 48 (2002) 1051 – 1063

www.elsevier.com/locate/na

Multiplicity of solutions of semilinear elliptic boundary value problems with jumping nonlinearities at zero
Zhaoli Liu ∗ Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China Received 30 March 2000; accepted 9 June 2000

Keywords: Variational method; Elliptic problem; Multiplicity of solutions; Sub- and supersolutions; Jumping nonlinearity

1. Introduction In this paper, we consider the elliptic boundary value problem −4u = f(u); x ∈ 6; u = 0; x ∈ @6;

(1)

where 6 is a bounded domain in RN with smooth boundary @6 and f : R → R a Lipschitz continuous function. We will assume that f(0) = 0, therefore 0 is a trivial solution of (1) and the purpose is to seek for nontrivial solutions. It is well known that nontrivial solutions of (1) correspond to nontrivial critical points of the functional

1 J (u) = |∇u|2 − F(u); u ∈ H01 (6); 2 6 6 t where F(t) = 0 f(s) ds. We will use the following two conditions in this paper.



(H1 ) f+ (0)=a; f− (0)=d, where f+ (0) and f− (0) are the right and left derivatives of f at 0, respectively.


This work is supported by NNSF and RFDP of China.

∗

Corresponding author. E-mail address: [email protected] (Z. Liu).

0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 2 3 4 - 0

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Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

(H2 ) Problem (1) has a negative strict subsolution  and a positive strict supersolution , that is,  and satisfy −4 ¡ f() in 6;

 ¡ 0 in 6;  = 0 on @6;

−4 ¿ f( ) in 6;

¿ 0 in 6;

= 0 on @6:

Under conditions (H1 ) and (H2 ), we will prove that if (a; d) is above a curve E in R2 then (1) has a minimal positive solution u1 , a maximal negative solution u2 , and a sign-changing solution u3 , and the three solutions have the properties u2 ¡ u3 ¡ u1 ; J (u1 ) ¡ 0; J (u2 ) ¡ 0; and J (u3 ) ¡ 0. Under assumption (H1 ), properties of problem (1) in a neighborhood of the origin of H01 (6) is closely related to the following homogeneous jumping nonlinear problem: −4u = au+ + du− ; u = 0;

x ∈ 6;

(2)

x ∈ @6;

where u+ = max{u; 0} and u− = min{u; 0}. If a = d, problem (1) has been well studied (see [4,9]). In the case of a = d, especially j ∈ (a; d)∪(d; a) for some j , problem (1) involves much more complexity, and f is called to have jumping nonlinearities at zero. Here 1 ¡ 2 ¡ · · · ¡ j ¡ · · · are the eigenvalues of −4 with homogeneous Dirichlet boundary condition. It is well known that 1 ¿ 0 is simple and any eigenfunction corresponding to 1 does not change sign in 6. In a recent paper [12], Li and Zhang proved that (1) has at least three nontrivial solutions, one positive, one negative and one sign-changing, under the following conditions [12, Theorem 1]: lim

t→0+

|f(t) − at| = m1 ; |t|1+1

lim sup |t|→∞

lim

t→0−

f(t) ¡ 1 t

|f(t) − dt| = m2 ; |t|1+2

(3) (4)

˜ and 1 ; 2 ; m1 ; m2 are positive in which a and d were assumed to satisfy (a; d) ∈ S, numbers. Results of Li and Zhang [12] improved former results of Dancer and Du [6] in a special case, that is, the case of (3). Dancer and Du [6] proved the same ˜ and that (2) has only conclusion as [12, Theorem 1] assuming (H1 ), (4), (a; d) ∈ S, the zero solution. It is known that (4) implies (H2 ). Also, it is known that (H2 ) is satisJed if there are two numbers t1 ¡ 0 ¡ t2 such that f(t1 ) = f(t2 ) = 0 or if there is a number k ¿ 0 such that |f(t)| ¡ k for any t ∈ [ − c; c] where c = max6 e and e satisJes −4e = k in 6;

e = 0 on @6:

˜ which is a subset of R2 , as in [5]. It is known Let us give a description of the set S, that (see [7] or [5, p. 1166]), there exists a continuous function (t) deJned on (1 ; 2 ] with the property that: (a)  is strictly decreasing, (2 ) = 2 ; lim→1 +0 () = +∞;

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(b) Eq. (2) has a nontrivial solution for (a; d) = (a; (a)); a ∈ (1 ; 2 ] or (a; d) = ((d); d); d ∈ (1 ; 2 ]; (c) Eq. (2) has no nontrivial solution for 1 ¡ d ¡ (a); a ∈ (1 ; 2 ] or 1 ¡ a ¡ (d); d ∈ (1 ; 2 ]. If we denote by E the curve {(a; (a)): 1 ¡ a ≤ 2 } ∪ {((d); d): 1 ¡ d ≤ 2 }, then S˜ is the set of points (a; d) which are above E in the ad-plane. That is, S˜ = {(a; d) | d ¿ (a) and 1 ¡ a ≤ 2 ; or d ¿ −1 (a) and 2 ≤ a ¡ + ∞}: In this paper, we will mainly prove the following two Theorems. ˜ then (1) has Theorem 1. Assume that (H1 ) and (H2 ) are satis5ed and (a; d) ∈ S; a minimal positive solution u1 ; a maximal negative solution u2 ; and a sign-changing solution u3 ; u2 ¡ u3 ¡ u1 in 6; J (u1 ) ¡ 0; J (u2 ) ¡ 0; J (u3 ) ¡ 0: Theorem 2. Under the assumptions of Theorem 1; if we assume in addition that ([a; d]∪[d; a])∩{j }∞ 1 =∅; then (1) has at least four nontrivial solutions u1 ; u2 ; u3 , and u4 ; u1 is the minimal positive solution; u2 is the maximal negative solution; both u3 and u4 are sign-changing solutions; u2 ¡ u3 ¡ u1 ; u2 ¡ u4 ¡ u1 ; J (u1 ) ¡ 0; J (u2 ) ¡ 0; J (u3 ) ¡ 0: Note that the condition that (2) has only the zero solution assumed in [6] is not needed in [12], the condition (H1 ) in [6], however, was strengthened in [12]. In the Jrst theorem here, condition (3) used by Li and Zhang [12] is replaced by (H1 ) and condition (4) used in [6,12] is replaced by (H2 ), and as in [12] we do not need the assumption that (2) has only the zero solution. In fact (H1 ) in Theorem 1 can still be weakened considerably. See Remark 2 at the end of the proof of Theorem 1. Apart from the result of getting one positive solution, one negative solution, and one sign-changing solution, which was proved by Dancer and Du [6] and Li and Zhang [12], we prove that (1) has a minimal positive solution u1 , a maximal negative solution u2 , and a sign-changing solution u3 which is greater than u2 and less than u1 , and that J (u1 ) ¡ 0; J (u2 ) ¡ 0; J (u3 ) ¡ 0: These properties about u1 ; u2 ; and u3 will be proved useful by a forthcoming paper [14] in getting one more sign-changing solution in some cases, for example, in the case that there are at most two eigenvalues k ; k+1 between a and d. Under the assumptions of the second theorem here and assuming that a=d, Hofer [9] Jrst proved that (1) has at least four nontrivial solutions, and after this paper has Jnished we learned that Li and Wang [11] proved that (1) has at least two sign-changing solutions by a local mountain pass argument in order intervals. In our Theorem 2, however, the assertions of the minimality of the positive solution u1 , the maximality of the negative solution u2 , the position of two sign-changing solutions u3 and u4 relative to the order interval [u1 ; u2 ], J (u1 ) ¡ 0; J (u2 ) ¡ 0, and J (u3 ) ¡ 0 seem new even in this special case a = d. The methods of getting two sign-changing solutions in this paper is diMerent from those in [11]. If (H2 ) is satisJed and some additional conditions on the behavior of f at ∞ are assumed, for example, if we assume that f is asymptotically linear at inJnity or f is

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superlinear and subcritical at inJnity, then three solutions can be obtained outside the order interval [; ], one in positive, one is negative, and one is sign-changing (see [13]). Let X be a Banach space and J a C 1 functional deJned on X . Recall that J is said to satisfy the P.S. condition if any sequence {un }∞ 1 ⊂ X such that |J (un )| ≤ c for some constant c and J
(un ) →  as n → +∞ has a convergent subsequence. 2. Proof of Theorem 1 Let  and be as in assumption (H2 ). Since all the discussions made in the sequel are conJned to the order interval [; ], we can assume that f(t) is bounded on R1 and satisJes uniformly the Lipschitz condition without loss of generality. Choose a number m ¿ 0 such that f(t) + mt is strictly increasing in t ∈ R. The inner product of the Hilbert space H01 (6) is then taken to be
(u; v) = (∇u · ∇v + muv) d x; u; v ∈ H01 (6) 6

and the associated norm is denoted by || · ||H01 (6) . With respect to this norm, J is a C 2−0 functional, which satisJes the P.S. condition, and the gradient of J at a point u has the following expression: J
(u) = u − (−4 + m)−1 (f(u) + mu): By the Lp theory of elliptic operators and Sobolev imbedding theorems, it is easy to see that the operator A deJned by (−4 + m)−1 (f(·) + m·) and considered as an O to C 1 (6) O satisJes uniformly the Lipschitz condition. operator from C01 (6) 0 O and consider the initial value problem in C 1 (6) O Let u0 ∈ C01 (6) 0 du = −u + Au; dt u(0) = u0 :

(5)

Assume that u(t; u0 ) is the unique solution of (5) with maximal right existence interval O That is, [0; (u0 )). Now we deJne D to be the order interval [; ] in C01 (6). O | (x) ≤ u(x) ≤ (x) in 6}: D = {u ∈ C01 (6) Lemma 1. For any u0 ∈ D, {u(t; u0 ) | 0 ≤ t ¡ (u0 )} ⊂ D. Proof. Since  and are subsolution and supersolution of problem (1) respectively, A(D) ⊂ D by the maximum principle. Then Lemma 1 is a direct consequence of a result in [15] (see also [4, p. 64]). Remark 1. In fact, if ˜ and ˜ are, respectively, any subsolution and supersolution of (1) satisfying  ≤ ˜ ≤ ˜ ≤ , and if we let ˜ O | (x) D˜ = {u ∈ C01 (6) ≤ u(x) ≤ ˜ (x) in 6};

Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

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˜ {u(t; u0 ) | 0 ≤ t ¡ (u0 )} ⊂ D. ˜ This observation will be used frethen for any u0 ∈ D, quently in the sequel. Lemma 2. For any u0 ∈ D; (u0 ) = +∞: O norm, Proof. Since {u(t; u0 ) | 0 ≤ t ¡ (u0 )} ⊂ D and D is bounded in the C(6) the Lp theory of elliptic operators and the Sobolev imbedding theorem imply that O norm. Since, for 0 ¡ t1 ¡ t2 ¡ {Au(t; u0 ) | 0 ≤ t ¡ (u0 )} is bounded in the C01 (6) (u0 ),
t2 t2 t1 ||et2 u(t2 ; u0 ) − et1 u(t1 ; u0 )||C 1 (6) ≤ es ||Au(s; u0 )||C 1 (6) O ds ≤ C(e − e ); O 0

0

t1

in which C ¿ 0 is a constant, if (u0 ) ¡ + ∞ then limt→(u0 )− u(t; u0 ) exists in the O norm and the solution u(t; u0 ) may be continued to cross the time (u0 ). This C01 (6) is a contradiction. Therefore, (u0 ) = +∞: O we say v ≤ w if and only if v(x) ≤ w(x) for all x ∈ 6. For v; w ∈ C01 (6), Lemma 3. For any v; w ∈ D; if v ≤ w then u(t; v) ≤ u(t; w);

∀ 0 ≤ t ¡ + ∞:

Proof. If v = w, the result is obvious. Suppose that v ≤ w and v = w; the Hopf ’s strong maximum principle and the fact that f(t) + mt is strictly increasing imply that @ @ Av ¿ Aw on @6; @n @n where n is the outer unit normal at a point on @6. Therefore (5) implies that Av ¡ Aw

in 6;

d t d (e u(t; v))|t=0 ¡ (et u(t; w))|t=0 in 6; dt dt       d @ @ d et u(t; w)  et u(t; v)  ¿ @n dt @n dt t=0 t=0

on @6:

It follows that there exists a number t0 ¿ 0 such that, for 0 ¡ t ≤ t0 , u(t; v) ¡ u(t; w) in 6: From these observations, we easily get the result by an argument of contradiction. DeJne a functional on H01 (6) as


1 a d J(a; d) (u) = |∇u|2 − (u+ )2 − (u− )2 : 2 6 2 6 2 6 Then critical points of J(a; d) correspond to solutions of (2). Let −1 1 4(a; d) = J(a; d) (−∞; 0) = {u ∈ H0 (6) | J(a; d) (u) ¡ 0}

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Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

and S = {(a; d) | a; d ¿ 1 ; 4(a; d) is path-connected}: The next lemma is a result of [5]. ˜ Lemma 4. S = S. Proof of Theorem 1. Let 1 be an eigenfunction of −4 corresponding to 1 such that 1 ¿ 0 for x ∈ 6. Since limt→0+ f(t)t −1 = a ¿ 1 , there exists a &0 ¿ 0 such that if 0 ¡ & ≤ &0 then −4(&1 ) ¡ f(&1 ); &1 = 0; x ∈ @6 and

1 J (&1 ) = &2 1 2


6

x ∈ 6;

2

(1 ) −

(6)


6

F(&1 ) ¡ 0:

(7)

Assuming that 0 ¡ & ≤ &0 , from (6) we know that &1 is a strict subsolution of (1). DeJne O | &1 (x) ≤ u(x) ≤ (x) in 6}: D&+ = {u ∈ C01 (6) By Remark 1, {u(t; &1 ) | 0 ≤ t ¡ + ∞} ⊂ D&+ : Since J (u(t; u0 )) is decreasing in t ∈ [0; +∞) for any u0 ∈ D, we see that − ∞ ¡ inf J (u) ≤ J (u(t; &1 )) ≤ J (&1 ) ¡ 0; u∈D

0 ≤ t ¡ + ∞:

(8)

It follows that there exists an increasing sequence {tn }∞ 1 of positive numbers with tn → +∞ as n → +∞ such that d lim (9) J (u(t; &1 ))|t=tn = 0: n→+∞ dt By (5) and (9) we get that lim ||J
(u(tn ; &1 ))||H01 (6) = 0:

n→+∞

(10)

Combining (8), (10), and the P.S. condition, we see that there is a subsequence of {u(tn ; &1 )}, which we denoted by {u(tn ; &1 )} also, and an element u& ∈ H01 (6) such that ||u(tn ; &1 ) − u& ||H01 (6) → 0 as n → +∞:

(11)

It is easy to see that u& ∈ D&+ ; u& is a critical point of J , and u& is a weak solution of (1). Since f is Lipschitz continuous and bounded, any weak solution of (1) is classical. Therefore, u& is a positive classical solution. For any t0 ¿ 0, u(t0 ; &1 ) ∈ D&+ , hence u(t0 ; &1 ) ≥ &1 . Lemma 3 implies that u(t + t0 ; &1 ) ≥ u(t; &1 );

0 ≤ t ¡ + ∞:

Therefore, u(t; &1 ) is increasing in t. In view of (11), we have lim u(t; &1 )(x) = u& (x)

t→+∞

for all x ∈ 6:

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Lemma 3 also implies that if 0 ¡ &1 ¡ &2 ≤ &0 then u(t; &1 1 ) ≤ u(t; &2 1 );

0 ≤ t ¡ + ∞:

It follows that if 0 ¡ &1 ¡ &2 ≤ &0 then u &1 ≤ u & 2 : O as DeJne a function u1 on 6 u1 (x) = lim u& (x); &→0+

O x ∈ 6:

By the regular theory of elliptic boundary value problems, {u& | 0 ¡ & ≤ &0 } is bounded O norm for some ' ∈ (0; 1). Then the Arzela–Ascoli theorem shows that in the C 2+' (6) lim ||u1 − u& ||C 2 (6) O = 0;

&→0+

therefore, u1 is a solution of (1). It is obvious that u1 ≥ 0. If u1 =0 then ||u& ||C 2 (6) O →0 as & → 0+, but this is impossible since u& is a positive solution and a ¿ 1 . In fact, there exists two numbers t ∗ ¿ 0 and ( ¿ 0 such that, for 0 ¡ t ¡ t ∗ , f(t) ¿ (1 + ()t: Choose &∗ ¿ 0 such that, for 0 ¡ & ¡ &∗ ; 0 ¡ u& ¡ t ∗ in 6. It follows that, for 0 ¡ & ¡ &∗ , −4u& = f(u& ) ¿ (1 + ()u& ; u& = 0; @6:

6;

Multiplying it with 1 and integrating, we have, for 0 ¡ & ¡ &∗ ,

& 1 u 1 ¿ (1 + () u & 1 ; 6

6

a contradiction. Therefore, u1 ¿ 0 and u1 is a positive solution of (1). Let u be any positive solution of (1). The Hopf’s strong maximum principle shows that u ¿ 0 in 6 and @u=@n ¡ 0 on @6. It follows that if & ¿ 0 is small enough then &1 ¡ u and Lemma 3 implies that u(t; &1 ) ≤ u;

0 ≤ t ¡ + ∞:

Taking limit as t → +∞, we have u& ≤ u for & ¿ 0 suQciently small. Hence, for & ¿ 0 small enough, u1 ≤ u& ≤ u:

(12)

This means that u1 is the minimal positive solution of (1). Taking u to be u1 in (12), we see that u& ≡ u1 for all & ¿ 0 suQciently small. Eq. (7) and the fact that J (u(t; &1 )) is decreasing imply that, when & ¿ 0 is suQciently small and 0 ¡ t ¡ + ∞, J (u1 ) = J (u& ) ¡ J (u(t; &1 )) ¡ J (&1 ) ¡ 0: A similar argument shows that (1) has a maximal negative solution u2 with J (u2 ) ¡ 0:

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Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

DeJne



+

D = u∈

 @u ≤ (x) in 6 and ¡ 0 on @6 : @n

O | 0 ¡ u(x) C01 (6)

 Then D+ = 0¡&≤&0 D&+ . Since {u(t; u0 ) | 0 ≤ t ¡+∞} ⊂ D&+ for any u0 ∈D&+ , {u(t; u0 ) | 0 ≤ t ¡ + ∞} ⊂ D+ for any u0 ∈ D+ . Similarly, if we let   @u − 1 O D = u ∈ C0 (6) | (x) ≤ u(x) ¡ 0 in 6 and ¿ 0 on @6 ; @n then {u(t; u0 ) | 0 ≤ t ¡ + ∞} ⊂ D− for any u0 ∈ D− . Note that D+ and D− are open O topology. In view of the fact that (a; d) ∈ S, ˜ subset of D with respect to the C01 (6) we have

1 a (1 )2 − (1 )2 ¡ 0; J(a; d) (1 ) = 2 6 2 6 J(a; d) (−1 ) =

1 2


6

(1 )2 −

d 2


6

(1 )2 ¡ 0:

By Lemma 4, there is a path h ∈ C([0; 1]; H01 (6)) such that h(0) = −1 ;

h(1) = 1 ;

J(a; d) (h(s)) ¡ 0;

0 ≤ s ≤ 1:

O is densely imbedded into H 1 (6), there exists a path Since [0; 1] is compact and C02 (6) 0 2 O ∗ h ∈ C([0; 1]; C0 (6)) such that h∗ (0) = −1 ;

h∗ (1) = 1 ;

J(a; d) (h∗ (s)) ¡ 0;

0 ≤ s ≤ 1:

Take a number k ¿ 0 such that J(a; d) (h∗ (s)) ≤ −k;

0 ≤ s ≤ 1;

and let f1 (t) be deJned as  f(t) − at; t ≥ 0; f1 (t) = f(t) − dt; t ¡ 0: Then, for & ¿ 0 and 0 ≤ s ≤ 1, ∗

∗




J (&h (s)) = J(a; d) (&h (s)) − F1 (&h∗ (s)) d x 6
2 ≤ −k& + |F1 (&h∗ (s))| d x; t

6

 where F1 (t) = 0 f1 (s) ds: DeJne M = max0≤s≤1 6 (h∗ (s))2 d x. The condition (H1 ) implies the existence of a number t ∗ ¿ 0 such that if |t| ≤ t ∗ then |f1 (t)| ≤ M −1 k|t|:

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It follows that, for |t| ≤ t ∗ , |F1 (t)| ≤ 12 M −1 kt 2 : ∗ If & ¿ 0 is suQciently small such that & max0≤s≤1 ||h∗ (s)||C(6) O ≤ t , then we have
1 1 J (&h∗ (s)) ≤ −k&2 + M −1 k&2 (h∗ (s))2 ≤ − k&2 ; 0 ≤ s ≤ 1: 2 2 6

Therefore, for & ¿ 0 suQciently small, J (&h∗ (s)) ¡ 0; ∗

In view of h ∈

0 ≤ s ≤ 1:

O C([0; 1]; C02 (6)),

u2 ≤ &h∗ (s) ≤ u1 ;

(13) we have, for & ¿ 0 suQciently small,

0 ≤ s ≤ 1:

(14) ∗

Fix a & ¿ 0 such that (13) and (14) are satisJed. Since &h (0) = −&1 ∈ D− ; O &h∗ (1) = &1 ∈ D+ ; and D− and D+ are open subsets of D with respect to the C01 (6) + − topology and since {u(t; u0 ) | 0 ≤ t ¡ + ∞} ⊂ D (D respectively) for any u0 ∈ D+ (D− respectively), if we let s∗ = sup{s | 0 ≤ s ≤ 1 and for any 0 ≤ s1 ≤ s there exists a number 0 ¡ t1 ¡ + ∞ such that u(t1 ; &h∗ (s1 )) ∈ D− }; then 0 ¡ s∗ ¡ 1 and the general theory of diMerential equations implies that {u(t; &h∗ (s∗ )) | 0 ≤ t ¡ + ∞} ∩ D− = ∅;

(15)

{u(t; &h∗ (s∗ )) | 0 ≤ t ¡ + ∞} ∩ D+ = ∅:

(16)

Just as the discussions made for u1 as above, there exist an increasing sequence {tn }∞ 1 with tn → +∞ and a solution u3 of (1) such that ||u(tn ; &h∗ (s∗ )) − u3 ||H01 (6) → 0 as n → +∞:

(17)

From (13) we obtain J (u3 ) ¡ J (u(tn ; &h∗ (s∗ ))) ¡ J (&h∗ (s∗ )) ¡ 0: By Lemma 3 and (14), we deduce that u2 ≤ u(tn ; &h∗ (s∗ )) ≤ u1 ;

n = 1; 2; : : : :

Taking limit as n → +∞, we have u2 ≤ u3 ≤ u1 : At last, we prove that u3 is sign-changing. Note that {u(t; &h∗ (s∗ )) | 0 ≤ t ¡ + ∞} ⊂ D. Using the formula
t u(t; &h∗ (s∗ )) = e−t &h∗ (s∗ ) + e−t+s Au(s; &h∗ (s∗ )) ds; 0 ≤ t ¡ + ∞; 0

∗

∗

we see that {u(t; &h (s )) | 0 ≤ t ¡ + ∞} is bounded in the W 2; p (6) norm for any p ¿ 1. The Sobolev imbedding theorem shows that {u(t; &h∗ (s∗ ))|0 ≤ t ¡ + ∞} is

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Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

O norm for any ' ∈ (0; 1). Then (17) and a compact argument bounded in the C01; ' (6) shows that ||u(tn ; &h∗ (s∗ )) − u3 ||C 1 (6) O →0 0

as n → +∞:

(18)

Combining (15), (16), and (18) together, we see that u3 ∈ D− ; u3 ∈ D+ . Since J (u3 ) ¡ 0 and J () = 0, u3 = . Therefore, u3 is sign-changing and u2 ¡ u3 ¡ u1 . The proof is complete. Remark 2. From the proof above we see that, in Theorem 1, the assumption (H1 ) can be weakened to be lim inf t→0+ f(t)t −1 = a and lim inf t→0− f(t)t −1 = d. The only step in the proof of theorem 1 which should be changed is what induces (13) from J(a; d) (h∗ (s)) ¡ 0 for 0 ≤ s ≤ 1. Indeed, by taking smaller a and d we can assume that lim inf t→0+ f(t)t −1 ¿ a and lim inf t→0− f(t)t −1 ¿ d and that (a; d) ∈ S˜ is still valid. Then for & ¿ 0 small and 0 ≤ s ≤ 1, we have J (&h∗ (s)) ¡ ∗ J(a; d) (&h (s)) ¡ 0. Note that we have obtained a minimal positive solution u1 and a maximal negative solution u2 . In [2], a minimal positive solution was obtained for a concrete nonlinear function f by an argument suitable to that special case. The method in [2] seems to have no eMect in the present case. In the case that f
(0) exists and k ¡ f
(0) ¡ k+1 for some k ≥ 2, the existence of the third nontrivial solution was proved by various methods in literature, for example, by an argument of the mountain pass lemma or by an argument of the topological degree theory. Nevertheless, with any of these methods we might not give a signchanging solution u3 with u2 ¡ u3 ¡ u1 and J (u3 ) ¡ 0. If u1 and u2 are strict local minimizers of J and all the solutions of (1) are discrete, we can select a sign-changing solution u3 which is of the mountain pass type with Jxed point index −1. This observation can be used to get another sign-changing solution in some cases. Theorem 2 deals with one of these cases. Recall that a critical point u of J is called of mountain pass type if for every open neighborhood U of u the set J −1 (−∞; J (u)) ∩ U = ∅ and J −1 (−∞; J (u)) ∩ U is not path-connected (see [10]).

3. Proof of Theorem 2 Under the conditions of Theorem 2, it is a standard argument to prove the existence of at least four nontrivial solutions by using the mountain pass lemma and the Jxed point index. But the standard argument can neither guarantee the existence of at least two sign-changing solutions nor give the relationships between u1 ; u2 and the two sign-changing solutions. Since u1 and u2 may not be strict local minimizers of J , we do not know at this stage if u3 is of the mountain pass type. Since u1 is the minimal positive solution and u2 is the maximal negative solution, it is allowed to modify the function f such that u1 and u2 are the unique positive solution and the unique negative solution of the modiJed problem, respectively. Then, the standard argument has its eMect.

Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

Proof of Theorem 2. Let u1   f(u2 (x)); ˜ t) = f(t); f(x;  f(u1 (x)); and let ˜ t) = F(x;


0

t

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O × R → R as and u2 be as in Theorem 1. DeJne f˜ : 6 if t ¡ u2 (x); if u2 (x) ≤ t ≤ u1 (x); if t ¿ u1 (x)

˜ s) ds: f(x;

O × R; R), f˜ is bounded and Lipschitz continuous. Consider Then f˜ ∈ C(6 ˜ u); x ∈ 6; −4u = f(x; u = 0; x ∈ @6:

(19)

By the maximum principle, any solution u of (19) must satisfy u2 (x) ≤ u(x) ≤ u1 (x) for x ∈ 6, and therefore is a solution of (1). Solutions of (19) correspond to critical points of the functional

1 2 ˜ ˜ u); u ∈ H01 (6): J (u) = |∇u| − F(x; 2 6 6 It is obvious that J˜ ∈ C 2−0 (H01 (6); R) and the gradient of J˜ at a point u ∈ H01 (6) is ˜ u) + mu): J˜
(u) = u − (−4 + m)−1 (f(x; Since u1 is the unique positive solution of (19), it is easy to see that u1 is a strict local minimizer of J˜|C 1 (6) O . By a result of [3,8], u1 is also a strict local minimizer of 0 ˜ J . Similarly, u2 is a strict local minimizer of J˜. Note that &h∗ is a path connecting −&1 with &1 and satisfying (13) and (14). Since J (&1 ) ¡ 0; since u(t; &1 ) ¡ u1 for t ¿ 0, and since limt→+∞ ||u(t; &1 ) − u1 ||H01 (6) = 0 for & small enough, there is a path h1 (s) connecting &1 with u1 such that J (h1 (s)) ¡ 0;

0 ≤ s ≤ 1;

u2 ≤ h1 (s) ≤ u1 ;

0 ≤ s ≤ 1:

Similarly, there is a path h2 (s) connecting u2 with −&1 such that J (h2 (s)) ¡ 0;

0 ≤ s ≤ 1;

u2 ≤ h2 (s) ≤ u1 ;

0 ≤ s ≤ 1:

DeJne a path h∗∗ as  1   0 ≤ s¡ ; h (3s);   2 3      2 1 h∗∗ (s) = &h∗ (3s − 1); ≤ s¡ ;  3 3       2   h1 (3s − 2); ≤ s ≤ 1: 3

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Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

Then h∗∗ connects u1 with u2 and satisJes J˜(h∗∗ (s)) ¡ 0;

0 ≤ s ≤ 1:

By the mountain pass lemma, J˜ has a third critical point u3 of the mountain pass type with J˜(u3 ) ¡ 0. It is obvious that u2 ¡ u3 ¡ u1 ; u3 is a sign-changing solution of (1), and J (u3 ) ¡ 0. Now, we assume that (19) has only a Jnite number of solutions. It is known that (see [1,4,8]), ˜ u1 ) = 1; i(A;

˜ u2 ) = 1; i(A;

˜ u3 ) = −1; i(A;

˜ u) + ˜ = (−4 + m)−1 (f(x; where A˜ is the operator from H01 (6) to H01 (6) deJned by Au ˜ ˜ mu) and i(A; u) is the Jxed point index of A at an isolated Jxed point u. ˜ Since ([a; d] ∪ [d; a]) ∩ {j }∞ 1 = ∅ and (a; d) ∈ S, there is an integer k ≥ 2 such 1 O that k ¡ a ¡ k+1 and k ¡ d ¡ k+1 . Let k = 2 (k + k+1 ), and deJne A˜s : H01 (6) → ˜ u) + sOk u + mu) for 0 ≤ s ≤ 1. If u = A˜s u H01 (6) by A˜s u = (−4 + m)−1 ((1 − s)f(x; for some u ∈ H01 (6) and 0 ≤ s ≤ 1, then u is a solution of the problem ˜ u) + sOk u; −4u = (1 − s)f(x; u = 0; x ∈ @6:

x ∈ 6;

(20)

A bootstrap argument shows that for any solution u of (20) if ||u||H01 (6) is suQciently small then so is ||u||C 1 (6) O . Therefore, for any solution u of (20) with ||u||H 1 (6) suQ0 0 ciently small, we have u1 ≤ u ≤ u2 and ˜ u) − Ok u)||L2 (6) ||u||L2 (6) = (1 − s)||(−4 − Ok )−1 (f(x; 2 ˜ u) − Ok u||L2 (6) ||f(x; k+1 − k 2 = ||f(u) − Ok u||L2 (6) : k+1 − k

≤

Here we use the fact that the norm of (−4− Ok )−1 as an operator from L2 (6) to L2 (6) is 2=(k+1 − k ). Take a number ( ¿ 0 such that |f(t) − Ok t| ¡ 12 (k+1 − k − ()|t| for t = 0 and |t| suQciently small, then for any solution u of (20) with ||u||H01 (6) suQciently small, ||u||L2 (6) ≤

k+1 − k − ( ||u||L2 (6) : k+1 − k

It follows that if ||u||H01 (6) is suQciently small and u =  then u = A˜s u for all 0 ≤ s ≤ 1. By the homotopy invariance of Jxed point index and the Leray–Schauder index formula, we have ˜ ) = i(A˜0 ; ) = i(A˜1 ; ) = (−1)/ ; i(A; where / is the dimension of the space ⊕1≤i≤k ker(−4 − i I ). On the other hand, O × R1 , it is easy to see that if R is suQciently large then the since f˜ is bounded on 6 Leray–Schauder degree ˜ B(; R); ) = 1; deg(I − A;

Z. Liu / Nonlinear Analysis 48 (2002) 1051 – 1063

1063

where B(; R) = {u ∈ H01 (6)|||u||H01 (6) ¡ R}. If ; u1 ; u2 ; u3 are the only solutions of (19), then the additivity property of the Leray–Schauder degree shows that ˜ B(; R); ) 1 = deg(I − A; ˜ ) + i(A; ˜ u1 ) + i(A; ˜ u2 ) + i(A; ˜ u3 ) = (−1)/ + 1 + 1 + (−1); = i(A; a contradiction. Therefore, (19) has a fourth nontrivial solution u4 . Since u1 and u2 are the only two nontrivial solutions of (19) with deJnite sign and any solution u of (19) satisJes u2 ≤ u ≤ u1 , u4 is sign-changing and u2 ¡ u4 ¡ u1 . The proof is complete. In Theorem 2, if ([a; d] ∪ [d; a]) ∩ {j }∞ 1 = ∅ then the problem involves more diQculties and we will consider this case in a forthcoming paper [14]. When a = d, the existence part of Theorem 2 is a known result, only the conclusions concerning the properties of the solutions and the relationships between them seem to be new. Acknowledgements The author would like to thank Prof. Shujie Li for valuable conversations. References [1] H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85 (1982) 591–595. [2] A. Ambrosetti, H. Brezis, G. Cerami, Combined eMects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519–543. [3] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris 317 (1993) 465– 475. [4] K.C. Chang, InJnite Dimensional Morse Theory and Multiple Solution Problems, BirkhUauser, Boston, 1993. [5] E.N. Dancer, Y. Du, Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero, Proc. Royal Soc. Edinburgh 124A (1994) 1165–1176. [6] E.N. Dancer, Y. Du, Multiple solutions of some semilinear elliptic boundary value problems via the generalized Conley index, J. Math. Anal. Appl. 189 (1995) 848–871. [7] D.G. de Figueiredo, J.-P. Gossez, On the Jrst curve of the Fucik spectrum of an elliptic operator, DiMerential Integral Equations 7 (1994) 1284 –1302. [8] D.G. de Figueiredo, S. Solimini, A variational approach to superlinear elliptic problems, Comm. Partial DiMerential Equations 9 (1984) 699–717. [9] H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982) 493–514. [10] H. Hofer, A note on the topological degree of a critical point of mountain pass type, Proc. Amer. Math. Soc. 90 (1984) 309–315. [11] S.J. Li, Z.-Q. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math. to appear. [12] S.J. Li, Z.T. Zhang, Sign-changing and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities, Acta Math. Sinica, New Series 16 (2000) 113–122. [13] Z.L. Liu, Multiple solutions of diMerential equations, Ph.D. Thesis, Shandong University, Jinan, 1992. [14] Z.L. Liu, S.J. Li, Contractability of the level sets of the functionals associated with some elliptic boundary value problems and applications, to appear. [15] J.X. Sun, The Schauder condition in the critical point theory, Chinese Sci. Bull. 31 (1986) 1157–1162.