Calculating critical groups of solutions for elliptic problem with jumping nonlinearity

Nonlinear Analysis 49 (2002) 779 – 797

www.elsevier.com/locate/na

Calculating critical groups of solutions for elliptic problem with jumping nonlinearity Liu Jiaquana; ∗ , Wu Shaopingb a Department

b Department

of Mathematics, Peking University, Beijing 100871, People’s Republic of China of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China Received 13 December 1998; accepted 11 August 1999

Keywords: Critical groups; Elliptic problem; Jumping nonlinearity; Morse inequality

1. Introduction Consider the semilinear elliptic equation 2u + g(x; u) = 0 in ; u=0

on @;

(P)

where  is a bounded domain in Rn with Lipschitz boundary, g is a Caratheodory function in 7 × R1 satisfying n+2 |g(x; t)| 6 C(1 + |t|p ); 1 ¡ p ¡ : (1) n−2 The corresponding functional is 
 1 2 I (u) = |∇u| − G(x; u)  2 de9ned on E = H01 , where G is the primitive of g:
t G(x; t) = g(x; s) ds: 0

(2)

(3)

 This work was completed as the authors were vesting ICTP. This research is supported by CNSF and ZNSF. ∗ Corresponding author. Fax: +86-24-685-2421. E-mail address: [email protected] (L. Jiaquan).

c 2002 Elsevier Science Ltd. All rights reserved. 0362-546X/02/$ - see front matter  PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 1 3 9 - 0

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Any critical point u of I is a weak solution for (P). Suppose u is an isolated critical point of I , then we can de9ne its critical groups Cq (u) = Cq (u; I ) = Hq (Ic ∩ U; Ic ∩ U \ {u});

q = 0; 1; 2; : : : ;

(4)

where c = I (u); Ic is the sublevel set {u|I (u 6 c)} and U any neighborhood of u, provided u is the unique critical point in U . By the excision property, Cq is independent of the neighborhood U . On the other hand, it is convenient to de9ne the critical groups at the in9nity as Cq (∞) = Cq (∞; I ) = Hq (E; I−M );

q = 0; 1; 2; : : : :

(5)

We assume that I has no critical values less than −M0 and M ¿ M0 . Then Cq (∞; I ) is independent of M . Now we have the Morse relation  M (t; u) = P(t; ∞) + (1 + t)Q(t); (6) u∈K

where K is the critical set of I : K = {u ∈ E|I  (u) = 0};  M (t; u) = dim Cq (u)t q ; q

P(t; ∞) =



dim Cq (∞)t q

(7)

q

and Q(t) = q aq t q is a series with nonnegative integer coeHcients. So an important problem is to calculate these critical groups. If the critical point u happens to be nondegenerate and its Morse index is m, then Cq (u) = qm G. For a degenerate critical point the calculation of the critical groups is much complicated. There are some papers related to this problem, see [6,8,11,12]. In this paper we present one more example, in which a complete description of the critical groups is possible. We assume g(x; t) lim (8) = +∞; lim = ∈ R1 : t→+∞ t→−∞ t This is the so-called jumping nonlinearity. And it goes back to Ambrosetti–Prodi problem [1], see also [5,9,10]. As a model we 9rst consider the equation: −2u = u + (u+ )p u=0

in ;

on @:

(9)

For 1 ¡ p ¡ (n + 2)=(n − 2); g = t + (t + )q has a jumping at the in9nity, and we calculate the critical groups at the in9nity. We prove that Cq (∞; I ) = 0;

q = 0; 1; 2; : : : :

For 0 ¡ p ¡ 1; g has a jumping at zero: g(x; t) lim = +∞; lim = ∈ R1 : t→+∞ t→−∞ t

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In this case we calculate the critical groups for the trivial critical point " and prove that Cq ("; I ) = 0;

q = 0; 1; 2; : : : :

(12)

Generalizations are considered, again a complete description is present. Existence of (nontrivial) solutions is obtained as a consequence through the Morse relation. In the appendix we discuss the relation of H01 versus C 1 isolated critical points, which is used to calculate the critical groups under some weaker assumptions. 2. Calculating the critical groups, g = t + (t + )p We 9rst study the model case g(x; t) = t + (t + )p , where ∈ R1 is a real parameter. If = 1 , the 9rst eigenvalue of the Laplacian, every function u = −c’ is a solution of (P), where c ¿ 0 and ’ is the positive eigenfunction corresponding to 1 . In the following we always assume = 1 . Let E be the Sobolev space H01 (). The corresponding functional for problem (P) is

1 1 [|∇u|2 − u2 ] − (u+ )p+1 ; (13) I (u) = 2  p+1  which is diLerentiable on E, and its Frechet’s derivative is

[∇u∇v − uv] − (u+ )p v: I  (u); v = 



(14)

We distinguish two cases, namely 0 ¡ p ¡ 1 and 1 ¡ p ¡ (n + 2)=(n − 2). 2.1. The case 1 ¡ p ¡ (n + 2)=(n − 2) As p ¿ 1; g has a jumping at the in9nity. We calculate the critical groups at the in9nity. It is easy to show that the functional I has no negative critical value and zero is the only critical point at the zero level. In fact, suppose u is a solution of (P). Then the functional

1 1 [|∇u|2 − u2 ] − (u+ )p+1 I (u) = 2  p+1  
 1 1 − = (u+ )p+1 ¿ 0: (15) 2 p+1  Moreover, if I (u) = 0, from the above formula we get u+ = 0. So is 2u + u = 0. And = 1 implies u = 0. We have the following theorem: Theorem 2.1. For a given M ¿ 0; Hq (E; I−M ) = 0

for q = 0; 1; : : : :

To prove Theorem 2.1 we need the following proposition:

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L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

Proposition 2.1. Given a constant M ¿ 0; any compact subset of the level set I−M = {u ∈ H01 ()|I (u) 6 −M } is contractible in I−M . Proof. Let M ¿ 0. Take a compact subset A ⊂ I−M . We deform it to a singleton by several steps: Step 1: Deform A to a subset of I−2M . We have

d 2 2 [|∇u| − u ] − (u+ )p+1 I (tu)|t=1 = dt  
p−1 (u+ )p+1 = 2I (u) − p+1  6 −2M;

(17)

as u ∈ I−M . Suppose −2M 6 I (u) 6 −M . There is a unique t ∗ ¿ 0 such that I (t ∗ u) = −2M . For I (u) 6 −2M , set t ∗ = 1. By (17) and the implicit function theorem t ∗ is a continuous function of u. We de9ne a Now )1 : A × [0; 1] → I−M by )1 (u; t) = ((1 − t) + tt ∗ )u:

(18)

Let A1 = )1 (A; 1). We have A1 ⊂ I−2M . Step 2: Deform to a subset of smooth functions. We replace each element u of the set A1 by a smooth function u* ∈ H01 () with u − u*  ¡ *. Since the set A1 is still a compact set, there is a 9nite family of smooth functions {ui* } and a corresponding decomposition of unit {+i } such that       * (19)  +i (u)ui − u 6 * for all u ∈ A1 ;   i

where {+i } are Lipschitz continuous functions of u in H01 . Let u∗ =i +i (u)ui* . It is easy to see that u∗ is smooth. For * small enough, we can de9ne a Now )2 : A1 × [0; 1] → I− 3 M by 2

)2 (u; t) = (1 − t)u + tu∗ : Let A2 = )2 (A1 ; 1). We have A2 ⊂ I− 3 M ∩ C01 . 2 Step 3: Deform to a subset of functions with nonzero positive part. Let d(x) = dist(x; @). De9ne a function ’* on 7 by  M7 d(x); d(x) 6 *; ’* (x) = M7 *; d(x) ¿ *:

(20)

(21)

7 Take M7 large For u in A2 we have a uniform bound for |∇u| on the domain . enough, near the boundary |∇’* | is rather large. Thus for each element u ∈ A2 , the element v = u + ’* always has a nonzero positive part, namely v+ = max{v; 0} = 0. On the other hand, as the constant * is small enough, ’*  1. We de9ne a Now )3 : A2 × [0; 1] → I−M by )3 (u; t) = u + t’* . And let A3 = )3 (A2 ; 1).

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Step 4: Make I (u+ ) negative. For each element u ∈ A3 , we have

1 [|∇u+ |2 − (u+ )2 ] − (u+ )p+1 I (u) = I (u+ ) + I (u− ) = 2  
1 [|∇u− |2 − (u− )2 ] 6 −M: + 2 

(22)

From (17) I (tu) is decreasing as t ¿ 1. Since the positive part u+ is nonzero for u ∈ A3 , we can 9nd T4 ¿ 0 such that I (tu+ ) ¡ − M as t ¿ T4 for u ∈ A3 by the compactness of the set A3 . Now de9ne a Now )4 : A3 × [0; 1] → I−M by )4 (u; t) = (1 − t + tT4 )u. Let A4 = )4 (A3 ; 1). We have A4 ⊂ I−M and I (u+ ) 6 −M for u ∈ A4 . Step 5: Make I (u+ ) negatively large enough. There is a constant C5 such that I (su− ) 6 C5

for all u ∈ A4 ; 0 6 s 6 1: +

(23) +

For each element u ∈ A4 we have I (u ) 6 −M then I (tu ) with t ¿ 1 is decreasing by (17). Choose T5 large enough so that I (T5 u+ ) 6 −M − C5

for all u ∈ A4 :

(24) +

−

De9ne a Now )5 : A4 × [0; 1] → I−M by )5 (u; t) = ((1 − t) + tT5 )u + u . Let A5 = )5 (A4 ; 1). We have A5 ⊂ I−M ∩ {u|I (u+ ) 6 −M − C5 }. Step 6: Deform to a subset of nonnegative functions. For each element u ∈ A5 we have I (u+ + su− ) = I (u+ ) + I (su− ) 6 −M − C5 + C5 = −M

for all s ∈ [0; 1]: (25)

Now we de9ne a Now )6 : A5 × [0; 1] → I−M by )6 (u; t) = u+ + (1 − t)u− : Let A6 = )6 (A5 ; 1). We have A6 ⊂ I−M and each element of A6 is nonnegative. Step 7: Deform to singleton on the “sphere”. Let B = {u|I (u) 6 −M; u ¿ 0}; B1 = {u| u = 1; u ¿ 0}:

(26)

B is contractible. In fact, for u ∈ B let t(u) ¿ 0 satisfy I (t(u)u) = −M , from Step 1 we have B = {tu|u ∈ B1 ; t ¿ t(u)}:

(27)

Thus, B is homotopic to B1 . The map tu0 + (1 − t)u (u; t) → tu0 + (1 − t)u deforms B1 to the singleton {u0 }, where {u0 } is a 9xed point in B1 . Therefore, the subset A6 of B is contractible.

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Summing up all these steps we see that the compact set A is deformed into a singleton in the sublevel set I−M . The proposition is proved. Proof of Theorem 2.1. By Proposition 2.1 we have H˜ q (I−M ) = 0

for all q = 0; 1; : : : :

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Also H˜ q (E) = 0; q = 0; 1; : : : . Hence, by the exact sequence · · · → H˜ q (I−M ) → Hq (E; I−M ) → H˜ q−1 (E) → · · ·

(29)

we have Hq (E; I−M ) = 0

for all q = 0; 1; : : : :

(30)

Remark. In fact the set I−M itself is contractible. We need only to modify the above proof, using the technique of unit decomposition if necessary, so that all the constants we choose are replaced by some continuous functions. 2.2. The case 0 ¡ p ¡ 1 From (15), the functional I now has no positive critical value and zero is the unique critical point at the zero level. Furthermore we show that a counterpart of Proposition 2.1 holds: Proposition 2.2. Any compact subset A of I0 ∩ B \ {0} is contractible; where B is a small ball centered at zero. Proof. We shall make the deformations step by step to shrink A into a singleton in I0 ∩ B \ {0}. Step 1: Reduce the radius of the ball. Suppose u ∈ A ⊂ I0 ∩ B \ {0}. If I (u) ¿ 0, we have

d I (tu)|t=1 = [|∇u|2 − u2 ] − (u+ )p+1 dt  
1−p = 2I (u) + (u+ )p+1 1+p  ¿ 2I (u) ¿ 0:

(31)

It implies that if I (u) 6 0, then I (tu) 6 0 as 0 ¡ t ¡ 1. So de9ne a Now )1 : A × [0; 1] → I0 ∩ B \ {0} by )1 (u; t) = (1 − 34 t)u: Let A1 = )1 (A; 1). Then A1 ⊂ I0 ∩ B(1=4) \ {0}. Step 1: Decrease the functional. Let )2 be the negative gradient Now determined by I  (u) (or by a pseudo-gradient vector 9eld) as usual. Then there exists an * ¿ 0 such that )2 : A1 ×[0; 1] → I0 ∩B(1=2) \ {0} and A2 ⊂ I−2* ∩ B(1=2) \ {0}, where A2 = )2 (A1 ; 1).

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Step 3: Deform to a subset of smooth functions. Now we smooth the elements in the set A2 as in the step 2 of Proposition 2.1 by means of the decomposition of unit. We have a continuous mapping u ∈ A2 → u∗ ∈ C01 . De9ne a Now )3 : A2 × [0; 1] → I−* ∩ B(3=4) \ {0} by )3 (u; t) = (1 − t)u + tu∗ : )3 is well de9ned provided u∗ is suHciently close to u. Let A3 = )3 (A2 ; 1). Then A3 ⊂ I−* ∩ B(3=4) \ {0} and u ∈ A3 is smooth. Step 4: Deform to a subset of functions with nonzero positive part. Now add a small function ’* with large derivative at boundary to the element in A3 as in the step 3 of Proposition 2.2 so that the element u + ’* has a nonzero positive part and remains in the sublevel set I0 . We de9ne a Now )4 : A3 ×[0; 1] → I0 ∩B \{0} by )4 (u; t) = u + t’* provided ’*  1. Let A4 = )4 (A3 ; 1). Then the positive part of the element in A4 is not zero. Step 5: Make I (u+ ) nonpositive. De9ne a Now )5 : A4 × [0; 1] → I0 by )5 (u; t) = ((1 − t) + tt ∗ )u; where the continuous function t ∗ of u is de9ned as follows:  1 if I (u+ ) 6 0; ∗ t (u) = sup{t ¡ 1|I (tu+ ) = 0} if I (u+ ) ¿ 0:

(32)

Since u+ = 0; t ∗ ¿ 0 is well de9ned and continuous by (31). Let A5 = )5 (A4 ; 1). We have A5 ⊂ I0 ∩ B ∩ {u|I (u+ ) 6 0} \ {0}: Step 6: Deform to a subset of nonnegative functions. Now we try to vanish the negative part of the element of A5 . Just de9ne a Now )6 : A5 × [0; 1] → I0 ∩ B \ {0} by )6 (u; t) = (1 − t)u− + u+ : By (31), if I (u− ) 6 0; I (su− ) 6 0 for all s ∈ [0; 1]; if I (u− ) ¿ 0; I (su− ) is decreasing as s is decreasing. In present case I (su− ) = s2 I (u− ), the above claim is obvious. Anyway we have I ((1 − t)u− + u+ ) = I ((1 − t)u− ) + I (u+ ) 6 max{0; I (u)} 6 0:

(33)

Let A6 = )6 (A5 ; 1). We have A6 ⊂ B = {u|u ¿ 0; I (u) 6 0; 0 ¡ u 6 }: Step 7: Contract B to a singleton. The set B is contractible. In fact, B = {tu|u ¿ 0; u = ; 0 ¡ t 6 t ∗ (u)};

(34)

∗

where t 6 1 is a continuous function of u ∈ B as de9ned in (32). Then the set B is deformed to the set B∗ = {t ∗ u|u ¿ 0; u = };

(35)

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L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

which is homeomorphic to B+ = {u|u ¿ 0; u = }: +

De9ne a Now )7 : B × [0; 1] → B tu0 + (1 − t)u )7 (u; t) = : tu0 + (1 − t)u

(36) +

by

Then )7 (·; 1) deforms B+ to a singleton u0 in B+ , hence B is contractible. Summing up, we see the subset A is contractible in I0 ∩ B \ {0}. The proposition is proved. As a consequence of this proposition we have a counterpart of the Theorem 2.1. Theorem 2.2. The critical groups of the functional I at zero are all trivial. Proof. The set I0 ∩ B contracts to {0} by the mapping (u; t) → (1 − t)u. So H˜ q (I0 ∩ B ) = 0

for all = 0; 1; : : : :

(37)

By Proposition 2.2 H˜ q (I0 ∩ B \ {0}) = 0

for all q = 0; 1; : : : :

(38)

Again the following exact sequence gives the desired result: · · · → H˜ q (I0 ∩ B \ {0}) → Hq (I0 ∩ B ; I0 ∩ B \ {0}) → H˜ q−1 (I0 ∩ B ) → · · ·

for all q = 0; 1; : : : :

3. Calculating the critical groups, generalizations In this section, we consider the problem −2u = g(x; u) in ; u = 0 on @;

(39)

where the function g(x; t) is a Caratheodory function in  × R1 and satis9es the following subcritical conditions: (g0 ) There exist constants C0 ¿ 0 and p ∈ (1; (n + 2)=(n − 2)) such that |g(x; t)| 6 C0 (1 + |t|p+1 ): First we assume that g has a jumping at the in9nity: (g1 ) There exists a constant t− ¡ 0 such that sg(x; t) ¿ 2G(x; t) as t 6 t− ; (g2 ) There exist constants t+ ¿ 0 and + ¿ 2 such that sg(x; t) ¿ +G(x; t) ¿ 0

as t ¿ t+ ;

where G(x; t) is the primitive of the function g(x; t) with respect to t.

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Both conditions (g1 ) and (g2 ) are related to the behavior of the function g at the in9nity, the typical case is g= t +(t + )p , with p ∈ (1; (n+2)=(n−2)). Another possible example is g = |t|p−1 t, the superlinear nonlinearity, which is considered by Z.Q. Wang in [12]. Denote the corresponding functional to problem (31) on the space E by

1 J (u) = |∇u|2 − G(x; u); (41) 2   which is diLerentiable on E and the Frechet’s derivative is

 J (u); v = ∇u∇v − g(x; u)v: 



(42)

From (g1 ) and (g2 ), it follows there is a constant C1 ¿ 0 such that for all t ∈ R1 :

C1 + tg(x; t) ¿ 2G(x; t)

(43)

Proposition 3.1. Under the conditions (g0 ); (g1 ) and (g2 ); any compact subset A of the sublevel set J−M = {u ∈ H01 ()|J (u) 6 −M } is contractible in J−M for M ¿ C1 . Proof. It suHces to show the 9rst step is still true as in the proof of Proposition 2.1 under the assumptions (g1 ) and (g2 ). In fact, for any given u ∈ A ⊂ J−M ,

d |∇u|2 − g(x; u)u J (tu)|t=1 = dt  
= 2J (u) + 2G(x; u) − ug(x; u) 

6 2J (u) + C1 6 −M:

(44)

Then, we de9ne a Now )1 : A × [0; 1] → J−M by )1 (u; t) = (1 + t)u: Let A1 = )1 (A; 1). We have A1 ⊂ J−2M . The other steps are the same as ones in the proof of Proposition 2.1. Similarly we can generalize the case p ¡ 1 by assuming: (g3 )

2G(x; t) − tg(x; t) ¿ 0

for all x

and

t 6 0;

(g4 ) There exist constants 0 ¡ / ¡ 2 and t+ ¿ 0 such that /G(x; t) ¿ tg(x; t) ¿ 0

for all x

and

0 ¡ t ¡ t+ :

Proposition 3.2. Under conditions (g0 ); (g3 ) and (g4 ); any compact subset A of U \ {0} is contractible in U \ {0}; where U = J0 ∩ B . Proof. Recall that in the proof of Proposition 2.2 estimate (31) plays a role, in particular, which implies that I (tu) 6 0 for t ∈ [0; 1], provided I (u) 6 0. Now this estimate no longer holds true in general. Instead we have the estimates for J (u+ ) and J (u− ),

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respectively. For J (u− ) by (g3 ) we have

d g(x; u− )u− |∇u− |2 − J (tu− )|t=1 = dt  
= 2 J (u− ) + [2G(x; u− ) − u− g(x; u− )] 

¿ 2 J (u− ):

(45)

+

For J (u ) by (g4 ) we have

d g(x; u+ )u+ |∇u+ |2 − J (tu+ )|t=1 = dt  

/

+ 2 + = 1− |∇u | + /J (u ) + /G(x; u+ ) − u+ g(x; u+ ) 2  

/

+ 2 + ¿ /J (u ) + 1 − |∇u | − C (u+ )p+1 2   ¿ /J (u+ ) + 0u+ 2

(46)

for some constants 0 ¿ 0, as u+  6  with  suHciently small, where we have used the fact that /G(x; t) − tg(x; t) ¿ −Ct p+1 , for t ¿ 0 and a constant C ¿ 0. As in the proof of Proposition 2.2, we 9rst reduce the norms of the elements of A. Several substeps are needed. Substep 1.1: Make J (u+ ) negative. By (g0 ) and (g4 ), we have G(t) ¿ Ct /+1 − Ct p+1 Thus, J (tu+ ) =

1 2

1 6 2




t 2 |∇u+ |2 −




1 = t2 2

2



for all t ¿ 0:

+ 2

t |∇u | −




+ 2









|∇u | − Ct

(47)

G(x; tu+ ) [C(tu+ )/+1 − C(tu+ )p+1 ]

/+1




 +



+ /+1

(u )

+ Ct

p+1




(u+ )p+1 :

(48)

If u+ = 0, then I (tu ) ¡ 0 as t ¿ 0 small enough, since the leading term of the coeHcients of (48) is negative. We de9ne a function t1∗ : u → [0; 1] as in (32):  1 if J (u+ ) 6 0; ∗ (49) t1 (u) = sup{t ¡ 1|J (tu+ ) = 0} if J (u+ ) ¿ 0: By (48) t1∗ is well de9ned. And by (46) t1∗ is continuous in u if u+ = 0. But in any case t1∗ u+ is continuous in u. De9ne a Now )1 : A × [0; 1] → U \ {0} by )1 (u; t) = (1 − t)u+ + tt1∗ u+ + u− : Let A1 = )1 (A; 1). Then J (u+ ) 6 0 as u ∈ A1 .

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Substep 1.2: Make J (u+ ) strictly negative, if u+  ¿ 12 . Let u ∈ A1 . Take * ¿ 0 such that −/* + 14 02 ¿ 0, where /; 0 are given in (46). We de9ne a function t2∗ as follows: if u+  6 =2 or J (u+ ) 6 −*, de9ne t2∗ = 1, if u+  ¿ =2 and J (u+ ) ¿ − * for all t ∈ [=2u; 1], de9ne t2∗ = =2, if u+  ¿ =2, J (u+ ) ¿ − * and there is t ∈ [=2u; 1] with J (tu+ ) 6 −*, de9ne t2∗ ∈ [=2u; 1] the biggest one, that is J (t2∗ u+ ) = −* and J (tu+ ) ¿ − * for t ∈ (t2∗ ; 1]. By (46), t2∗ is continuous in u. Now de9ne a Now )2 : A1 × [0; 1] → U \ {0} by )2 (u; t) = ((1 − t) + tt2∗ )u+ + u− : Let A2 = )2 (A1 ; 1). Then either u+  6 =2 or J (u+ ) 6 −* as u ∈ A2 . Substep 1.3: Make J (u) strictly negative, if u ¿ 34 . Let u ∈ A2 . Take 1 6 14  suHciently small such that J (u) 6 12 * for u 6 1. By (45) and the fact that J (u+ ) 6 0 we can de9ne a Now )3 : A2 × [0; 1] → U \ {0} by )3 (u; t) = ((1 − t) + t1)u− + u+ : Let A3 = )3 (A2 ; 1). Since J (u+ ) 6 −* for u ¿ 12 , we have either u 6 u+  + u−  6 12  + 1 6 43  or J (u) = J (u+ ) + J (u− ) 6 −* + 12 * = − 12 *

(50)

as u ∈ A3 . Substep 1.4: Deform the subset into a smaller ball. Let u ∈ A3 . Since J (u) 6 − 12 * as u ¿ 34 , there is a suHciently small s0 ¿ 0 such that J ((1 − ts0 )u) 6 0 for t ∈ [0; 1], u ∈ A3 and u ¿ 34 . Now we retract the set A3 a little bit. De9ne a Now )4 by  as u 6 34 ;   u;   )4 (u; t) = 3  ts0 u; as u ¿ 34 : u − 1 −  4u Let A4 = )4 (A3 ; 1). Then u 6 (1 − 14 s0 ) = 1 , and J (u) 6 0 as u ∈ A4 . Now the proof of the step 1 is completed and we proceed from the set A4 step by step as in the proof of Proposition 2.2 to complete the proof of Proposition 3.2. With the help of Propositions 3.1 and 3.2 we have the following descriptions about the critical groups, the proof is the same as one for Theorems 2.1 and 2.2. Theorem 3.1. Under assumptions (g0 ); (g1 ) and (g2 ); Cq (∞; J ) = Hq (E; J−M ) = 0

for all q = 0; 1; : : : :

Theorem 3.2. Under assumptions (g0 ); (g3 ) and (g4 ); Cq (0; J ) = Hq (J0 ∩ B ; J0 ∩ B \ {0}) = 0

for all q = 0; 1; : : : :

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Remark. In Theorem 3.2, condition (g3 ) is a global one, while we are concerned with the local property of I . In the appendix of this paper we return to this problem, where (g3 ) is replaced by a local one: (g3 )

2G(x; t) − tg(x; t) ¿ 0

for all x

and

t− 6 t ¡ 0:

4. Existence theorems In order to make use of the Morse theory to get the existence of solutions, we need to verify the Palais–Smale condition 9rst. It is worth to point out that in Propositions 2.1 and 2.2 we do not assume this condition. In fact in the proof of Proposition 2.1 we do use the negative gradient Now at all. Meanwhile, for Proposition 2.2 we do use this Now. However, with a subcritical nonlinearity the functional satis9es automatically the Palais–Smale condition in a bounded domain. One can list many possible conditions suHcient for the veri9cation of P.S. condition. But in this paper we are mostly interested in calculating the critical groups, as for the existence theory we are satis9ed with the following “abstract” theorems and give all details only for the special case of the Section 2. De"nition 4.1. We say a critical point u is homologically nontrivial, if all the critical groups are not trivial. Meanwhile we say the functional I is not homologically trivial at the in9nity, if all the homology groups Hq (E; I−M ) are not trivial. De"nition 4.2. We say an isolated critical point u is homologically nondegenerate, if its critical groups Cq (u; I ) = qm

for all q = 0; 1; : : : ;

(51)

where m is an integer. Meanwhile, we say the functional I is homologically nondegenerate at the in9nity, if the homology groups Hq (E; I−M ) = qm

for all q = 0; 1; : : : ;

(52)

where m is an integer. Obviously, a nondegenerate critical point is homologically nondegenerate, and a homologically nondegenerate critical point is homologically nontrivial. For the in9nity we have similar relationship. Some nontrivial examples can be found in [11] for homologically nondegenerate critical point. With these terms we have the following theorems. Theorem 4.1. Suppose conditions (g0 ); (g1 ) and (g2 ) hold; and the corresponding functional I satis?es P.S. condition. Suppose moreover g(x; 0) = 0 for all x ∈ 

L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

791

and u = 0 is a homologically nontrivial critical point. Then problem (P) will have at least one nonzero solution. Theorem 4.2. Suppose conditions (g0 ); (g3 ) and (g4 ) hold; and the corresponding functional I satis?es P.S. condition. Suppose I is homologically nontrivial at the in?nity. Then problem (P) will have at least one nonzero solution. Proof of Theorems 4.1 and 4.2. Suppose on the contrary u = 0 is the unique critical point of I . We have the Morse relation M (t; 0) = P(t; ∞) + (1 + t)Q(t):

(53)

For Theorem 4.2 M (t; 0) = 0; P(t; ∞) = 0. We arrive at a contradiction by (53). For Theorem 4.1 M (t; 0) = 0, P(t; ∞) = 0. More arguments are needed. In this case the series Q(t) = 0, see [3]. In fact suppose I has only one critical point 0. By the deformation theorem I0 is a deformation retract of E and I−M is a deformation retract of I0 \ {0}. Hence Cq (∞; I ) = Hq (E; I−M ) = Hq (I0 ; I0 \ {0}) = Cq (0; I )

for all q = 0; 1; : : : ;

which implies Q(t) = 0, again we get a contradiction. Now we are going to verify the assumptions of Theorems 4.1 and 4.2 for the special case g = t + (t + )p . The following Lemma 4.1 is well known: Lemma 4.1. The functional

1 1 [|∇u|2 − |u|2 ] − (u+ )p+1 I (u) = 2  p+1  veri?es the P.S. condition on the space E as of −2 with Dirichlet boundary condition.

=

1,

where

1

is the ?rst eigenvalue

Lemma 4.2. Suppose 1 ¡ p ¡ (n + 2)=(n − 2). Suppose the trivial critical point u = 0 of the functional I is isolated. Then it is homologically nondegenerate. Proof. Let i be the ith eigenvalue of the Laplacian with multiplicity mi , and m = m1 + m2 + · · · + mi . As ∈ ( i ; i+1 ); u = 0 is a nondegenerate critical point with Morse index m. We have Cq = qm G

for all q = 0; 1; : : : ;

(54)

where G is the coeHcient group and qm is the conventional Kronecker symbol. While = i (i ¿ 1), u = 0 is a degenerate critical point. By Theorem (2:1) of [1] we still have Cq (I; 0) = qm G. Thus in any case the trivial critical point u = 0 is homologically nondegenerate. Lemma 4.3. Suppose 0 ¡ p ¡ 1. Suppose I has not any critical value less than −M0 . Then the functional I is homologically nondegenerate at the in?nity.

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L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

Proof. The derivative of I is asymptotically linear at the in9nity. If I is nonresonant, and for M ¿ M0 , Cq (∞; I ) = Hq (E; I−M ) = qm G

for all q = 0; 1; : : : :

∈ ( i;

i+1 ),

(55)

While = i (i ¿ 1); I is resonant at the in9nity. By the (2) of Theorem 1:3 in [11] we still have Cq (E; I−M ) = qm G. Summing all of these we recover Theorem 4.3. For all 1 ¡ p ¡ (n + 2)=(n − 2); p = 1 and , the elliptic equation has at least one nontrivial solution: −2u = u + (u+ )p u=0

in ;

on @:

Appendix. H01 versus C01 isolated critical points Consider the semilinear elliptic equation 2u + g(x; u) = 0 in ; u=0

on @;

(A.1)

where  is a smooth, bounded domain in R ; g is a Caratheodary function in 7 × R1 , satisfying n+2 |g(x; t)| 6 C(1 + |t|p ); 1 ¡ p ¡ : (A.2) n−2 The corresponding functional is 
 1 2 |∇u| − G(x; u) ; (A.3) f(u) =  2 n

where G is the primitive of g. If g(x; 0)=0, then u=" is a solution of (1), equivalently, a critical point of f. Suppose " is an isolated critical point of f, then we can de9ne its critical groups Cq = Hq (f0 ∪ U; f0 ∪ U \ {"})

for all q = 0; 1; : : : ;

where f0 is the sublevel set {u|f(u) 6 0} and U is any neighborhood of ", provided " is the unique critical point in U . By the excision property, Cq is independent of the neighborhood U . So far we have not mentioned where the functional f is de9ned. A natural choice is the Sobolev space H01 . But f can be regarded as well as de9ned on C01 , the set of C 1 functions on 7 with zero boundary value. A problem rises: are the critical groups independent of the choice of the working spaces? The answer is YES, if either 1. " is a local minimizer (Brezis–Nirenberg [2]), or 2. " is a general critical point and g is continuously diLerentiable in t (Chang [3]). In this note we prove that this property still holds even if g is only continuous in t.

L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

793

Theorem. Let U and U˜ be neighborhoods of " in H01 and C01 , respectively. De?ne Cq = Hq (f0 ∪ U; f0 ∪ U \ {"});

(A.4)

C˜ q = Hq (f0 ∪ U˜ ; f0 ∪ U˜ \ {"})

(A.5)

for all q = 0; 1; 2; : : : ; then Cq = C˜ q . The idea is very simple. Using a GalXerkin approximation, we relate these groups to ones of a functional de9ned on a 9nite-dimensional space. But for a 9nite-dimensional space all topologies are equivalent, and the theorem follows. Step 1: The pseudo-gradient vector 9eld. Let B2 be a small ball in H01 with center at " and radius 2 . Suppose " is the unique critical point of f in the ball B2 . Take 0 ¡ 1 ¡ 2 . Suppose df(u) ¿ *1 ¿ 0 for u ∈ B2 \ B1 . Take 0 ¡  ¡ 1 such that |f(u)| ¡ a =

1 10 *1 (2

− 1 )

for u ∈ B :

(A.6)

We construct a pseudo-gradient vector 9eld as follows. For a given u ∈ B2 \ {"}, we choose a smooth function u∗ in H01 such that 1 g(·; u) − u∗ Lq ¡ df(u) for u ∈ B2 \ {"}; (A.7) 10m where 1 ¡ q = 2n=(n − 2)p ¡ (n + 2)=(n − 2), and m is the Sobolev embedding constant for H01 ,→ Lq . For a given u, the inequality holds in a neighborhood U (u). There is a locally 9nite-re9nement of this covering {U (u)|u ∈ B2 \ {"}} and a corresponding unit decomposition +i . De9ne  w(u) = (A.8) +i (u)ui∗ ; i

V (u) = u − Kw(u);

(A.9)

where K is the inverse of −2 with the Dirichlet boundary condition. Then df − V  = Kg(x; u) − Kw(u)      6 m g(x; u) − +i (u)ui∗ 

Lq

6

1 10 df;

(A.10)

which makes V a pseudo-gradient vector 9eld of f in B2 \ {"}. Step 2: The Now and the Gromoll–Mayer pairs. We de9ne the Now by d ) = −7()2 )V ()); dt )(0) = u ∈ B ;

(A.11)

where 7 is a smooth function: 7(t) = t if t ¡ 1, and 7(t) = 1 if t ¿ 1. Due to the special form of V , ) is a Now in C01 , too.

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L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

Choose b ∈ (2a; 3a). De9ne W = )(B ; R) ∩ f−1 [ − b; b];

(A.12)

W − = )(B ; R) ∩ f−1 (−b);

(A.13)

where )(B ; R) = {)(u; t)|u ∈ B ; t ∈ R}. By the choice of a and , we have W and W − ⊂ B2 . In fact, while a point Nows from B1 to @B2 , the value of f increases at least by 2b. (W; W − ) is a Gromoll–Meyer pair of f, and we have ([4]) Cq (f; ") = Hq (W; W − );

(A.14)

C˜ q (f; ") = Hq (Wc ; Wc− );

(A.15)

where Wc = W ∩ C01 ; Wc− = W − ∩ C01 . Step 3: The GalXerkin approximation. Let {Hn } be an increasing sequence of 9nite-dimensional subspaces and ∪Hn = H01 , for example, the subspace spanned by the eigenfunctions of (−2)−1 corresponding the 9rst n eigenvalues. Hn consists of smooth functions. De9ne

1 |∇u|2 − G(x; Pn u) fn = 2  


1 1 = |∇vn |2 + |∇wn |2 − G(x; wn ); (A.16) 2  2   where wn = Pn u; vn = u − Pn u and Pn is the orthogonal projection to Hn . Lemma. We have fn → f;

uniformly in B ;

dfn → df; Proof.

(A.17)

uniformly in B :

(A.18)

G(x; Pn u) − G(x; u):

(A.19)




fn − f =



G : H01 ,→ L1 ; u → G(x; u) is weakly continuous, so the convergence fn → f is uniformly. In fact, suppose on the contrary for some * ¿ 0, we have a sequence {un } ⊂ B such that |fn (un )−f(un )| ¿ *. Up to a subsequence, un weakly converges to u ∈ B . Meanwhile Pn un weakly converges to u too. We have
|G(x; Pn un ) − G(x; un )| * ¡ |fn (un ) − f(un )| 6 


6



→ 0;

|G(x; Pn un ) − G(x; u)| + as n → ∞:




|G(x; un ) − G(x; u)| (A.20)

L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

795

A contradiction. Similarly

dfn − df; ’ = g(Pn u)Pn ’ − g(u)’ 




=



(g(Pn u) − g(u))’ +




g(u)(Pn ’ − ’)

6 g(Pn u) − g(u)Lq ’Lq
+ g(u)Lq Pn ’ − ’Lq
;

(A.21)

where 1=q + 1=q = 1; 1 ¡ q ¡ 2n=(n − 2). Now g(Pn u) uniformly converges to g(u) in Lq norm and Pn ’ − ’Lq
6 *n ’, where *n → 0 as n → ∞. In fact since
Pn ’ − ’L2 6 n−1=2 ’, by the interpolation we have the estimate for Lq norm. We complete the proof. Similar to V , we construct a pseudo-gradient vector 9eld Vn for fn so that the Now generated by Vn is a Now in C01 too. We assume 1 dfn : dfn − Vn  ¡ 10

(A.22)

Let 7 is a cut-oL function: 7(t) = 1, if t ¿ 122 ; 7(t) = 0, if t 6 121 . De9ne h(u) = 7(u2 )f(u) + (1 − 7(u2 ))fn (u);

(A.23)

Vh (u) = 7(u2 )V (u) + (1 − 7(u2 )) dfn :

(A.24)

Since V , Vn are the pseudo-gradient vector 9elds for f and fn respectively, for u ∈ B12 \ B11 ; Vh is a pseudo-gradient vector 9eld for h. Suppose df(u) ¿ * for 11 6 u 6 12 . Then for 11 6 u 6 12 as n → ∞, df − dh df − dfn  + C|f − fn | 6

1 10 *

+

1 10 *

6 15 df;

(A.25)

dh − Vh  6 df − V  + dfn − Vn  + C|f − fn | 6

1 10 df

+

1 10 dfn 

+ C|fn − f|

6 15 df + Cdfn − df + C|fn − f| 6 15 df + 15 * 6 25 df 6 12 dh;

(A.26)

Vh is a pseudo-gradient vector 9eld of h. The critical set of h in W is the same of the functional fn , hence is contained in an arbitrary small ball, provided n is large enough. Let Sn be this critical set of fn . Notice that fn has a separate form with respect to the variables vn and wn we have Sn ⊂ Hn . Let fn = f|Hn and (Wn ; Wn− ) is a Gromoll– Mayer pair for the critical set Sn of fn . Set D = B1 ∩ Hn⊥ , then (D × Wn ; D × Wn− ) is a Gromoll–Mayer pair of fn for the critical set Sn ⊂ Hn ⊂ H . On the other hand (W; W − ) is a Gromoll–Mayer pair for Sn too, since outside of B12 f = h, V = Vh and the Nows are the same. By the Gromoll–Mayer theory, being the Gromoll–Mayer pairs

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L. Jiaquan, Wu Shaoping / Nonlinear Analysis 49 (2002) 779 – 797

of the same critical set, Hq (W; W − ) = Hq (D × Wn ; D × Wn− ) = Hq (Wn ; Wn− )

for all q = 0; 1; : : : : (A.27)

But the same is true for the pair (Wc ; Wc− ). Set Dc = D ∩ C01 , we have Hq (Wc ; Wc− ) = Hq (Dc × Wn ; Dc × Wn− ) = Hq (Wn ; Wn− )

for all q = 0; 1; : : : : (A.28)

The proof is completed. As an application we consider Proposition 3.2 again. But now instead of (g3 ) we assume (g3 ). There is a constant t− ¡ 0 such that 2G(x; t) − tg(x; t) ¿ 0

for all x

and

t− 6 t ¡ 0:

Proposition 3.2 . Conditions (g0 ); (g3 ) and (g4 ), any compact subset A of U c \ {0} is contractible in U c \ {0}, where U c = J0 ∩ Bc and Bc is a ball neighborhood of 0 in C01 . Notice that condition (g0 ) is only used in estimate (45). If we assume  ¡ |t− |, then for u ∈ Bc estimate (45) remains true under condition (g3 ) and we proceed as before. Finally we have Theorem 3.2 . Under assumptions (g0 ); (g3 ) and (g4 ), Cq (0; J ) = Hq (J0 ∩ B ; J0 ∩ B \ {0}) = 0

for all q = 0; 1; : : : :

Remark. For the problem discussed in this appendix, see Palais [7] in a much more general setting. Here we present an analytic proof for our special case through the Gromoll–Mayer theory and the GalXerkin approximation. References [1] A. Ambrosetti, G. Prodi, On the inversion of some diLerentiable mappings with singularities between Banach spaces, Ann. Math. Pure Appl. 93 (1973) 231–247. [2] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris 317 (1993) 465–475. [3] K.C. Chang, In9nite Dimensional Morse Theory and its Applications, BirkhXauser, Boston, 1993. [4] K.C. Chang, N. Ghoussoub, The Conley index and the critical groups via an extension of Gromoll– Meyer theory, Topological Methods Nonlinear Anal. 7 (1996) 77–94. [5] D.G. De Figueiredo, On superlinear elliptic problems with nonlinearities interacting only with higher eigenvalues, Rocky Mountain, J. Math. 18 (1988) 287–303. [6] J.Q. Liu, S.P. Wu, A note on a class of sublinear elliptic equation, Research Report, Peking University, No. 84, Beijing, China, 1997. [7] R.S. Palais, Homotopy theory of in9nite dimensional manifolds, Topology 5 (1966) 115–132. [8] K. Perera, Multiplicity results for some elliptic problems with concave nonlinearities, J. DiLerential Equations 140 (1997) 133–141.

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[9] B. Ruf, P.N. Srikanth, Multiplicity results for superlinear elliptic problems with partial interference with the spectrum, J. Math. Anal. Appl. 118 (1986) 15–23. [10] A.J. Rumbos, A multiplicity result for strongly nonlinear perturbations of elliptic BVPs, J. Math. Anal. Appl. 199 (1997) 859–873. [11] Shujie Li, J.Q. Liu, Computations of critical group at degenerate critical point and applications to nonlinear diLerential equations at resonance, Houston J. Math. 25 (1999) 563–582. [12] Z.Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. PoincarZe Anal. NonlinZeaire 8 (1991) 43–57.