Computation of critical groups in resonance problems where the nonlinearity may not be sublinear

Nonlinear Analysis 46 (2001) 777 – 787

www.elsevier.com/locate/na

Computation of critical groups in resonance problems where the nonlinearity may not be sublinear Shujie Lia;1 , Kanishka Pererab; ∗;2 a Institute

b Department

of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA Received 15 October 1999; accepted 2 February 2000

Keywords: Asymptotically linear elliptic boundary value problems; Resonance; Faster than sublinear; Variational methods; Morse theory; Critical groups

1. Introduction Consider the asymptotically quadratic functional F(u) = 12 (Au; u) + G(u)

(1.1)

de8ned on a real Hilbert space H , where A is a bounded self-adjoint operator on H and G∈C 1 (H; R) satis8es |G(u)| u

2

→0

as u → ∞:

(1.2)

Let H− , H0 , H+ denote the negative, zero, positive subspaces of A, respectively, and write any u∈ H as u = u− + u0 + u+ ∈H− ⊕ H0 ⊕ H+ . Assume that A|H± has a bounded inverse on H± , r− := dim H− ;

r0 := dim H0

(1.3)

∗

Corresponding author. E-mail addresses: [email protected] (S. Li), [email protected] (K. Perera). 1

Supported by the Chinese National Science Foundation.

2

Supported by the Chinese National Science Foundation and by the Morningside Center of Mathematics at the Chinese Academy of Sciences. 0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 1 3 3 - 4

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S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

are 8nite, and G has a compact diBerential g satisfying g(u) →0 u

as u → ∞:

(1.4)

If g is bounded and G(u0 ) → −∞ as u0  → ∞;

(1.5)

then this is the well-known Landesman–Lazer setting in an abstract form, and it was shown in Chang [2] that the critical groups of F at in8nity are given by Cq (F; ∞) = qr G;

(1.6)

where r = r− + r0 and G is the coeEcient group. Cq (F; ∞) for nonlinearities satisfying

g(u) ≤ C(u + 1); G(u0 ) u0 

2

→ ± ∞ as u0  → ∞;

(1.7) (1.8)

for some ∈(0; 1), were recently computed in [3]. Note that (1.7) means that g is sublinear at in8nity. In the present paper we obtain compactness and information about Cq (F; ∞) under weaker assumptions that require (1.4), but not (1.7). As an application we obtain new existence results for asymptotically linear elliptic boundary value problems at resonance. We refer the reader to Chang [2] for a wide exposition of Morse theory. The notion of critical groups at in8nity was introduced in [1]. 2. Critical group computations First we assume that (G1 ) G has a compact diBerential g satisfying g(u) ≤ !(u)u

(2.1)

for some nonincreasing positive function ! such that !(t); t! (t) → 0;

!(t)t → ∞ as t → ∞;

(2.2)

(G2± ) G satis8es G(u0 ) !2 (u

0 )u0 

2

→ ± ∞ as u0  → ∞:

(2.3)

The example !(t) = 1=log t, t ¿ 1 shows that g need not satisfy (1.7) for any

∈(0; 1). Set 2

h(u) = !2 (u)u :

(2.4)

S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

779

Lemma 2.1. Given  ¿ 0; there is a C ¿ 0 such that 2

2

|G(u) − G(u0 )| ≤ ( + !(u0 ))(u+  + u−  ) + Ch(u0 ):

(2.5)

Proof. We have |G(u) − G(u0 )|
1 ≤ |(g(tu + (1 − t)u0 )u − u0 )| dt 0


≤

1

0

!(tu + (1 − t)u0 )tu + (1 − t)u0 u − u0  dt

≤ !(u0 )u u+ + u− 

since ! is nonincreasing

2

2

≤ ( + !(u0 ))(u+  + u−  ) +

1 2 2 ! (u0 )u0  : 4

(2.6)

Theorem 2.2. Assume (G1 ). (i) If (G2± ) holds; then F satis:es the Palais–Smale compactness condition (PS) : for any c∈R; F(uj ) → c;

F  (uj ) → 0

(2.7)

implies that {uj } has a convergent subsequence. (ii) If (G2− ) holds; then Cq (F; ∞) = qr G: (iii) If

(G2+ )

(2.8)

holds; then

Cq (F; ∞) = qr− G:

(2.9)

Proof. (i) We have j j j j o(1)(u+  + u− ) = (F  (uj ); u+ − u− ) j j j j j j ; u+ ) − (Au− ; u− ) + (g(uj ); u+ − u− ) = (Au+ 2

2

j j  + u−  ) ≥ m(u+ j j  + u− ) −[!(uj )(u+ j j  + u− ); +!(u0j )u0j ](u+

(2.10)

where m = inf {|(Au± ; u± )|: u± ∈H± ; u±  = 1} ¿ 0:

(2.11)

j

If u  → ∞, then it follows that 2

2

j j  + u−  ≤ C(h(u0j ) + 1); u+

(2.12)

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S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

for suEciently large j. But then 2

2

j j  + u−  ) + G(u0j ) + |G(uj ) − G(u0j )| F(uj ) ≤ 12 A(u+ 2

2

j j ≤ G(u0j ) + C(u+  + u−  + h(u0j ))

by Lemma 2:1

≤ G(u0j ) + C(h(u0j ) + 1) → −∞

(2.13)

if (G2− ) holds, a contradiction, so uj  is bounded, and a standard argument gives (PS). The other case is similar. (ii) Let 2

2

C = {u∈H : u+  − du−  − Kh(u0 ) ≤ M };

(2.14)

where d, K, M ¿ 0 are to be determined. The outward normal vector on @C is n = u+ − du− − K(! (u0 )u0  + !(u0 ))!(u0 )u0 ;

(2.15)

so (F  (u); n) = (Au+ ; u+ ) − d(Au− ; u− ) + (g(u); n) 2

2

≥ m(u+  + du−  ) − !(u0 )u(u+  + du−  +o(1)K!(u0 )u0 ) 2

2

≥ 12 (m + o(1))(u+  + du−  ) − (C + o(1)K 2 )h(u0 ) 2

= 12 (m + o(1))[2du−  + (K − C + o(1)K 2 )h(u0 ) + M ]

(2.16)

as u0  → ∞. Choosing K suEciently large, it follows that there is an R ¿ 0 such that (F  (u); n) ¿ 0

for u0  ¿ R:

(2.17)

On the other hand, for u0  ≤ R,

√ 2 2 (F  (u); n) ≥ m(u+  + du−  ) − !( M )u(u+  + du−  + C) √ 2 2 ≥ 12 (m − C!( M ))(u+  + du−  ) − C √ 2 ≥ 12 (m − C!( M ))(2du−  + M ) − C ¿ 0;

(2.18)

if M is suEciently large. So F has no critical points outside C, and −F  points inwards on @C. Next we show that in the set C, F(u) → −∞ as u− + u0  → ∞

uniformly in u+ ;

(2.19)

S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

781

if d is small. We have for u∈C, F(u) = 12 (Au+ ; u+ ) + 12 (Au− ; u− ) + G(u) 2

2

≤ 12 A u+  − 12 mu−  + G(u0 ) + |G(u) − G(u0 )| 2

≤ Cu+  − ≤−

1

4m

1

4m

 2 − !(u0 ) u−  + G(u0 ) + Ch(u0 )

by Lemma 2:1

 2 − Cd − !(u0 ) u−  + G(u0 ) + C(h(u0 ) + 1):

(2.20)

Choosing d suEciently small, it follows that there is an R ¿ 0 such that (2.19) holds in C ∩ {u0  ¿ R}. Then it is easily seen from (1.2) that (2.19) holds in C ∩ {u0  ≤ R} also. Now we 8x R1 ¿ 0, set CR1 = {u∈C: u− + u0  ≥ R1 }, and take a1 ¡ inf C\CR1 F such that F has no critical points in the sublevel set Fa1 = {u∈H : F(u) ≤ a1 }. It follows from (2.19) that there is an R2 ¿ R1 such that Fa1 ∩ C ⊃ CR2 . Then for any a2 ¡ min {a1 ; inf C\CR2 F}, CR1 ⊃(Fa1 ∩ C) ⊃ CR2 ⊃(Fa2 ∩ C):

(2.21)

Note that there is a geometrically de8ned strong deformation retraction of CR1 onto CR2 . Since F has no critical values in [a2 ; a1 ] and −F  points inwards on @C, the negative gradient Jow of F de8nes a strong deformation retraction of Fa1 ∩ C onto Fa2 ∩ C. Composing them we obtain a strong deformation retraction of CR1 onto Fa2 ∩ C, and hence Cq (F; ∞) = Hq (H; Fa ) ∼ = Hq (C; Fa ∩ C) ∼ = Hq (C; CR ) 2

2

1

∼ = Hq (H− ⊕ H0 ; (H− ⊕ H0 )\BR1 ) = qr G:

(2.22)

(iii) Let 2

2

C = {u∈H : u−  − du+  − Kh(u0 ) ≤ M }:

(2.23)

Arguments similar to those in the proof of (ii) show that F has no critical points outside C and −F  points outwards on @C if K and M are suEciently large, and F(u) → + ∞ as u0 + u+  → ∞

uniformly in u− ;

(2.24)

in C if d is small. In particular, inf C F ¿ − ∞. Taking a ¡ inf C F, we see that the negative gradient Jow of F de8nes a strong deformation retraction of H \C onto Fa , and hence (2.25) Cq (F; ∞) = Hq (H; Fa ) ∼ = Hq (H; H \C) ∼ = Hq (H− ; H− \B) = qr G: −

Next we present in this abstract setting an idea that has already been used in [5] in a diBerent context. Assume that (G3± )

M (u) := G(u) − 12 (g(u); u)

(2.26) j

is bounded below (resp. above), and every unbounded sequence {u } such that u =uj  converges has a subsequence for which M (uj ) → ± ∞:

j

(2.27)

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S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

Note that (G3± ) implies that M (u) → ±∞ as u → ∞ on every 8nite dimensional subspace. Lemma 2.3. (i) If (G3− ) holds; then G ≤ sup M

(2.28)

G(u) → −∞ asu → ∞; u ∈ H− ⊕ H0 :

(2.29)

and (ii) If (G3+ ) holds; then G ≥ inf M: Proof. Since

(2.30)

d 2 dt [G(tu)=t ]
∞ −3

G(u) = 2

t

1

= −t −3 M (tu) and G(tu)=t 2 → 0 as t → ∞,

M (tu) dt:

(2.31)

Eqs. (2.28) and (2.30) follow from this. Since the right-hand side of (2.31) is at most supt≥1 M (tu) and M (u) → −∞ as u → ∞ on H− ⊕ H0 , (2.29) also follows. Theorem 2.4. (i) If (G3± ) holds; then F satis:es the compactness condition of Cerami (C): for any c∈R; F(uj ) → c;

(1 + uj )F  (uj ) → 0

(2.32)

j

implies that {u } has a convergent subsequence. (ii) If (G3− ) holds; then Cr (F; ∞) = 0: (iii) If

(G3+ )

(2.33)

holds; then

Cq (F; ∞) = qr− G:

(2.34)

Proof. (i) We use the equivalent norm induced by the inner product (u; v)A = (|A|u; v) + (u0 ; v0 );

(2.35)

and write 2

2

F(u) = 12 uA + (Au− ; u− ) − 12 u0  + G(u):

(2.36)

If {uj } is bounded, then a standard argument gives a convergent subsequence, so suppose that uj A →∞, and let u˜ j = uj =uj A . Then there is a subsequence for which u˜ j * u˜ and u˜j− + u˜j0 → u˜ − + u˜ 0 . Thus   (F  (uj ); uj ) g(uj ) j j j j 2 = 1 + 2(A u ˜ ; u ˜ ) −  u ˜  + ; u ˜ − − 0 2 uj A uj A 2

→ 1 + 2(Au˜ − ; u˜ − ) − u˜ 0  ;

(2.37)

S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

783

so 2

1 + 2(Au˜ − ; u˜ − ) − u˜ 0  = 0 and for any v∈H , (F  (uj ); v) = (u˜ j ; v)A + 2(Au˜j− ; v) − (u˜j0 ; v) + uj A

(2.38) 

g(uj ) ;v uj A



→ (u; ˜ v)A + 2(Au˜ − ; v) − (u˜ 0 ; v);

(2.39)

so (u; ˜ v)A + 2(Au˜ − ; v) − (u˜ 0 ; v) = 0:

(2.40) j

Taking v = u˜ and comparing with (2.38), we see that u ˜ A = 1, so u˜ → u. ˜ But then F(uj ) = 12 (F  (uj ); uj ) + M (uj ) → ±∞

(2.41)

for a subsequence, a contradiction. (ii) Since F is bounded below on H+ and F(u) → −∞ as u → ∞; u∈H− ⊕ H0 by (i) of Lemma 2.3, (2.33) follows from Proposition 3:8 of Bartsch and Li [1]. (iii) Denote by B the closed unit ball in H with boundary S and set S− = {u∈S: (Au; u) ¡ 0}:

(2.42)

Then for u∈S\S− and t ≥ 0, F(tu) = 12 t 2 (Au; u) + G(tu) ≥ inf M

(2.43)

by Lemma 2.3(ii) so for any a ¡ min{inf M; inf B F}, Fa ⊂ S˜− := {tu: u∈S− ; t ¿ 1}: On the other hand, for u∈S− ,   G(tu) 2 1 F(tu) = t → −∞ as t → ∞ (Au; u) + 2 2 t

(2.44)

(2.45)

by (1.4). Moreover, if F(tu) ≤ a, then t d F(tu) = F(tu) − M (tu) ≤ a − inf M ¡ 0: (2.46) 2 dt It follows that there is a unique t0 = t0 (u) ¿ 1 such that F(tu) ¿ a for 0 ≤ t ¡ t0 ; F(t0 u) = a, and F(tu) ¡ a for t ¿ t0 , i.e., Fa = {tu: u∈S− ; t ≥ t0 (u)}:

(2.47)

Furthermore, the map t0 : S− → (1; ∞) is C 1 by (2.46) and the implicit function theorem. Therefore, we can de8ne a strong deformation retraction r : S˜− × [0; 1] → S˜− of S˜− onto Fa by  [(1 − s)t + st0 (u)]u; 1 ¡ t ¡ t0 (u); r(tu; s) = (2.48) tu; t ≥ t0 (u):

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S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

It follows that Cq (F; ∞) = Hq (H; Fa ) ∼ = Hq (H; S˜− ) ∼ = Hq (H− ; H− \B) = qr− G:

(2.49)

3. Applications to resonance problems We consider the asymptotically linear problem  −Mu = &k u − g(x; u) in N; u=0

(3.1)

on @N;

where N is a bounded domain in Rn with smooth boundary @N; &1 ¡ &2 ¡ · · · denote the distinct Dirichlet eigenvalues of −M on N, and g∈C(NO × R; R) satis8es g(x; t) O = 0 uniformly in N: (3.2) lim t |t|→∞ As is well known, solutions of (3.1) are the critical points of F(u) = 12 (Au; u) + G(u);

u∈H = H01 (N);

where A = I − &k (−M)−1 ;

(3.3)


G(u) =

N


G(x; u);

G(x; t) =

0

t

g(x; s) ds:

(3.4)

G(u) has the compact diBerential g(u) = (−M)−1 g(x; u(x)), and it is easily seen that k (3.2) implies (1.4). Let H− ; H0k ; H+k denote the negative, zero, positive subspaces of k A, respectively, and let dk = dim H− . First we assume that (g1 ) |g(x; t)| ≤ !(|t|) |t| for some nonincreasing positive function ! satisfying (2.2) and !(t) lim inf ¿0 t→∞ !(t p ) for some (and hence all!) p∈(0; 1), (g2± ) G satis8es G(x; t) O → ± ∞ as |t| → ∞ uniformly in N: !2 (|t|)t 2

(3.5)

(3.6)

(3.7)

Note that (3.6) says that ! does not decay too rapidly. In fact, it implies that there is an a ¿ 0 such that 1 !(t) ≥ ; t large: (3.8) (log t)a Example 3.1. If g(x; t)=± t=log|t|; |t| ¿ 2, then (g1 ) and (g2± ) hold with !(t)=1=log t.

S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

785

Lemma 3.2. If (g1 ) and (g2± ) hold; then the conditions (G1 ) and (G2± ) of Section 2 are satis:ed. Proof. We have 2




g(u) =

N

|g(x; u)|2


≤

|u|≥

√

2

u



≤ C(!2 (

2




! (|u|)|u| +

|u|¡

√

u

!2 (|u|)|u|2

2

u)u + u);

(3.9)

so taking

  √ 1 2 ; !˜ (t) = C ! ( t) + t 2

(3.10)

we see that (G1 ) holds with !˜ in place of !. If (g2− ) holds, then for any M ¿ 0, G(x; t) ≤ −M!2 (|t|)t 2 + O(1) so G(u0 ) !˜ 2 (u0 )u0 

2

≤−

≤− ≤−

M

N

O as |t| → ∞ uniformly in N;

!2 (|u0 |)u02

(3.11)

+ o(1)

2

!˜ 2 (u0 )u0 

2

M!2 (u0 L∞ )u0 L2 2

!˜ 2 (u0 )u0 

+ o(1)

M!2 (Cu0 ) + o(1); &k !˜ 2 (u0 )

since dim H0 ¡ ∞, and (G2− ) follows since (3.6) implies that !(Ct) lim inf ¿ 0: t→∞  (t) !

(3.12)

(3.13)

Veri8cation of (G2+ ) is similar. Set


g0 (x; t) = &k t − g(x; t);

G0 (x; t) =

0

t

g0 (x; s) ds:

(3.14)

Theorem 3.3. Assume that (g1 ) holds and 2 1 2 &j t

≤ G0 (x; t) ≤ 12 &j+1 t 2 ;

|t| ≤

(3.15)

for some ¿ 0. Then problem (3:1) has a nontrivial solution in the following cases: (i) (g2− ) holds and j = k; (ii) (g2+ ) holds and j = k − 1.

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S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

Proof. By (3.15), F has a local linking near zero with respect to the decomposition j H = (H− ⊕ H0j ) ⊕ H+j , i.e., F(u) ≤ 0;

j ⊕ H0j ; u ≤ r; u∈H−

(3.16)

F(u) ¿ 0;

u∈H+j ; 0 ¡ u ≤ r

(3.17)

for suEciently small r ¿ 0, and hence Cdj (F; 0) = 0

(3.18)

by Liu [4]. If (g2− ) holds, then Cq (F; ∞) = qdk G

(3.19)

by Lemma 3.2 and Theorem 2.2(ii), so (i) follows. Proof of (ii) is similar. Remark 3.4. Theorem 3.3 extends a result in [3]. Next, we give an application of Theorem 2.4. Assume that (g3± ) O M (x; t) := G(x; t) − 12 g(x; t)t → ± ∞ as |t| → ∞ uniformly in N:

(3.20)

Lemma 3.5. If (g3± ) holds; then (G3± ) is satis:ed. Proof. Since M (x; t) is bounded below (resp. above), so is
M (u) = M (x; u):

(3.21)

N

˜ and let N = {x ∈ N : u˜ = 0}. Then |N |¿0 Suppose that uj  → ∞ and uj =uj  → u, j  and |u | → ∞ a.e. in N , so
M (uj ) ≥ inf M (x; t)|N\N | + M (x; uj ) → ∞ (3.22) N

if (g3+ ) holds. The other case is similar. Theorem 3.6. Assume that |g(x; t1 ) − g(x; t2 )| ≤ C(|t1 |p−2 + |t2 |p−2 + 1)|t1 − t2 |;

t1 ; t2 ∈R

(3.23)

for some p∈(2; 2n=(n − 2)); and either &j ≤

g0 (x; t) ≤ &j+1 − ; t

0 ¡ |t| ≤ ;

(3.24)

g0 (x; t) ≤ &j+1 ; t

0 ¡ |t| ≤

(3.25)

or &j +  ≤

S. Li, K. Perera / Nonlinear Analysis 46 (2001) 777 – 787

787

for some ; ¿ 0. Then problem (3:1) has a nontrivial solution in the following cases: (i) (g3− ) holds and j = k; (ii) (g3+ ) holds and j = k − 1. Proof. It was shown in Perera and Schechter [6] that if (3.23) and either (3.24) or (3.25) hold, then Cq (F; 0) = qdj G: If

(g3− )

(3.26)

holds, then

Cdk (F; ∞) = 0

(3.27)

by Lemma 3.5 and Theorem 2.4(ii), so (i) follows. Proof of (ii) is similar. Acknowledgements The second author gratefully acknowledges the support and hospitality of the Morningside Center of Mathematics, Academia Sinica where this work was completed. References [1] T. Bartsch, S.J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (3) (1997) 419–441. [2] K.C. Chang, In8nite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear DiBerential Equations and their Applications, Vol. 6, BirkhRauser, Boston, MA, 1993. [3] S.J. Li, J.Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear diBerential equations with resonance, Houston J. Math. 25 (3) (1999) 563–582. [4] J.Q. Liu, The Morse index of a saddle point, Systems Sci. Math. Sci. 2 (1) (1989) 32–39. [5] K. Perera, M. Schechter, Double resonance problems with respect to the FuTcUVk spectrum, preprint. [6] K. Perera, M. Schechter, Nontrivial solutions of elliptic semilinear equations at resonance, Manuscripta Math. 101 (3) (2000) 301–311.