Generalized Poincaré–Hopf theorem and application to nonlinear elliptic problem

Journal of Functional Analysis 267 (2014) 3783–3814

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Journal of Functional Analysis www.elsevier.com/locate/jfa

Generalized Poincaré–Hopf theorem and application to nonlinear elliptic problem Chong Li a,b,∗,1 a

Institute of Mathematics, AMSS, Academia Sinica, Beijing 100190, China Beijing Center for Mathematics and Information Interdisciplinary Sciences (BCMIIS), China b

a r t i c l e

i n f o

Article history: Received 21 December 2013 Accepted 12 September 2014 Available online 29 September 2014 Communicated by H. Brezis MSC: 35A15 35P05

a b s t r a c t In this paper we get an extended version of Poincaré– Hopf theorem. Without the assumption that critical point set between two level sets of energy functional is finite, this result actually generalizes Morse inequality. And the isomorphism between Cq (J, ∞) and Cq (J(a,b) , 0) is yielded as (a, b) ∈ / Σ, where Cq (J, ∞) denotes the critical groups of energetic functional J at infinity and Cq (J(a,b) , 0) stands for the critical groups of functional J(a,b) at zero, and Σ is the set of points (a, b) ∈ R2 for which the problem

Keywords: Poincaré–Hopf theorem Nonlinear elliptic problem Non-isolated critical point Multiple solutions



−Δu + αu = au− + bu+ ,

x ∈ Ω,

∂u ∂ν

x ∈ ∂Ω,

= 0,

has a nontrivial solution, u+ = max{u, 0}, u− = min{u, 0}. (Concerning the definitions of J and J(a,b) , see Section 4.) As to application aspects, we are mainly concerned with nonlinear elliptic problem with Neumann boundary condition provided that the origin is a non-isolated critical point and obtain the existence of multiple solutions. © 2014 Elsevier Inc. All rights reserved.

* Correspondence to: Institute of Mathematics, AMSS, Academia Sinica, Beijing 100190, China. 1

E-mail address: [email protected]. The author is supported by NSFC (11471319) and BCMIIS.

http://dx.doi.org/10.1016/j.jfa.2014.09.018 0022-1236/© 2014 Elsevier Inc. All rights reserved.

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1. Introduction The celebrated Poincaré–Hopf theorem shows us the relationship between the indices of zeros of a smooth vector field on a manifold M and the Euler Characteristic of M . The theorem was proven for two dimensions by Henri Poincaré in 1885 and later generalized to higher dimensions by Heinz Hopf [7] in 1926. Infinite dimensional case was studied by E. Rothe [15] and he got the following (1.1) under stronger assumptions. Subsequently, K.C. Chang verified Proposition 1.1 below (see [4] for details): Proposition 1.1. Let E be a real Hilbert space and f ∈ C 2 (E, R) be a function that satisfies the (PS) condition. Assume that df (u) = u − K(u) and u0 is an isolated critical point of f on E. Then we have     deg df, B(u0 , ε), 0 = deg I − K, B(u0 , ε), 0 =

∞ 

(−1)q rank Cq (f, u0 )

(1.1)

q=0

for ε > 0 sufficiently small, where deg(df, B(u0 , ε), 0) stands for the Leray–Schauder degree of df at u0 , B(u0 , ε) := {u ∈ E : u − u0  < ε}, I denotes identity map and K is a compact mapping, and     Cq (f, u0 ) := Hq f c ∩ B(u0 , ε), f c \{u0 } ∩ B(u0 , ε), G stands for the q-th critical group, with coefficient group G of f at u0 , c = f (u0 ), f c := {u ∈ E : f (u) ≤ c}, and H∗ (X, Y, G) represents the singular relative homology groups with the abelian coefficient group G. K.C. Chang also generalized Proposition 1.1 as follows (see [4]): Proposition 1.2. Let E be a real Hilbert space. Suppose f ∈ C 2 (E, R) is a function that satisfies the (PS) condition. Assume that W is a bounded domain in E on which f is bounded. Assume that (a) W− := {u ∈ ∂W : η(t, u) ∈ / W, ∀t > 0} = W ∩ f −1 (a) for some a, where η(t, u) is negative gradient flow of f emanating from u; (b) −df points inward at ∂W \W− , then we have deg(df, W, 0) =

∞  q=0

(−1)q rank Hq (W, W− ).

(1.2)

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Li et al. [11] improved Proposition 1.1 by substituting f ∈ C 1 (E, R) for f ∈ C 2 (E, R) and also got C 1 version of Proposition 1.2 in the case that Kf ∩ W is finite, Kf := {u ∈ E : df (u) = 0}. Actually, Proposition 1.1 alludes to the following consequence: Proposition 1.3. Let E be a real Hilbert space and f ∈ C 1 (E, R) satisfies the (PS)c condition for ∀c ∈ [a, b], where a, b are regular values of f . Assume Kf ∩ f −1 [a, b] = {z1 , z2 , · · · , zl }, f −1 [a, b] := {u ∈ E : a ≤ f (u) ≤ b}. Then there exists δ > 0, s.t. (B(zi , δ)\{zi }) ∩ Kf = ∅ and a<

inf

δ z∈K(a,b) (f )

f (z) ≤

sup

f (z) < b.

(1.3)

δ z∈K(a,b) (f )

Moreover, ∞      δ deg df, K(a,b) (f ), 0 = (−1)q rank Hq f b , f a ,

(1.4)

q=0

where     δ K(a,b) (f ) := x ∈ E : dist x, K(a,b) (f ) < δ , K(a,b) (f ) := Kf ∩ f −1 [a, b]. Proof. Notice that l      δ deg df, K(a,b) (f ), 0 = deg df, B(zi , δ), 0

(1.5)

i=1

Using Poincaré–Hopf formula and Morse inequality l 

l  +∞    deg df, B(zi , δ), 0 = (−1)q rank Cq (f, zi ) i=1 q=0

i=1

=

+∞ l   (−1)q rank Cq (f, zi ) q=0

=

i=1

∞ 

  (−1)q rank Hq f b , f a .

(1.6)

q=0

Combining (1.5) and (1.6) we immediately arrive the conclusion. 2 A natural question which arises is: Does (1.4) still hold provided Kf ∩ f −1 [a, b] is compact?

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Recall the classical Marino–Prodi theorem due to [13]: Proposition 1.4. (See [4]). Suppose that f ∈ C 2 (E, R) has a critical value c with Kc = {p1 , p2 , · · · , pl }. Assume that d2 f (pi ) are Fredholm operators, i = 1, 2, · · · , l. Then ∀ε > 0, there exists a function g ∈ C 2 (E, R) such that (1) (2) (3) (4)

l g = f in E\ j=1 B(pj , ε); g − f C 2 (E) ≤ ε; l g has only nondegenerate critical points, all concentrated in j=1 B(pj , ε); Let ind(f, pj ) = mj , and dim ker d2 f (pj ) = nj , where ind(f, pj ) denotes the Morse index of f at pj ; then the Morse indices of those nondegenerate critical points of g in B(pj , ε) are in [mj , mj + nj ], j = 1, 2, · · · , l. Furthermore, if f satisfies the (PS) condition, then the same is true for g.

Remark 1.5. The assumption Kc finite can be weaken to Kc compact (see [6]). In this paper we will give a positive answer of the question above by presenting a new extension of Proposition 1.4. Notice that the extension of Proposition 1.3 is somewhat of a generalized version of Morse inequality. The paper goes as follows. Section 2 is devoted to the main consequence related to the extensions of Poincaré–Hopf theorem and Marino–Prodi theorem. In Section 3, as an application, theoretic results are used to treat elliptic problems with Neumann boundary condition provided the original is a non-isolated critical point of corresponding energy functional. We derive, in Section 4, the isomorphism between Cq (J, ∞) and Cq (J(a,b) , 0) under the assumption (a, b) ∈ / Σ, which is exploited in Section 5 to complete the proof of application problem. 2. Extensions of Poincaré–Hopf theorem and Marino–Prodi perturbation theorem Let E be a Hilbert space. Denote by  · and ·, · , the norm of E and the inner product of E respectively. Our main consequence concerning the extensions of Poincaré–Hopf theorem and Marino–Prodi perturbation theorem reads: Theorem 2.1. Suppose that ϕ ∈ C 1 (E, R) satisfies the (PS) condition, ∇ϕ = I − F , F is compact, with two regular values c1 < c2 , K(c1 ,c2 ) (ϕ) = ∅, and ∃δ ∗ > 0, s.t. δ∗ ϕ ∈ C 2 (K(c (ϕ), R), then there exist two regular values c1 , c2 , c1 < c1 < c2 < c2 , and 1 ,c2 ) ∗

also exists δ0 ∈ (0, δ2 ), for ∀δ ∈ (0, δ0 ), ∃ε∗ = ε∗ (δ) ∈ (0, 1), ∀ε ∈ [0, ε∗ ], there exists a 2δ δ functional ϕε ∈ C 1 (E, R) ∩ C 2 (K(c (ϕ), R), such that K(c1 ,c2 ) (ϕε ) ⊂ K(c (ϕ) (if 1 ,c2 ) 1 ,c2 ) any) and c1 <

inf

2δ x∈K(c

1 ,c2 )

where ϕ0 := ϕ. Moreover,

(ϕ)

ϕε (x) ≤

ϕε (x) < c2 ,

sup 2δ x∈K(c

1 ,c2 )

(ϕ)

(2.1)

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2δ (i) ϕε = ϕ in E\K(c (ϕ); 1 ,c2 ) c 2 c 2 c 1 (ii) ϕε = ϕ , and ϕε = ϕc 1 . This yields

    Hq ϕc2 , ϕc1 ∼ = Hq ϕcε 2 , ϕcε 1 ;

(2.2)

(iii) ϕε (x) − ϕ(x)C 2 ≤ ε for ∀x ∈ E; (iv) ϕε satisfies the (PS) condition; (v) ϕε has only nondegenerate critical points in ϕ−1 ε [c1 , c2 ]; (vi) ∞      δ deg ∇ϕ, K(c (ϕ), 0 = (−1)q rank Hq ϕc2 , ϕc1 ; 1 ,c2 )

(2.3)

q=0

(vii) If Hl0 (ϕc2 , ϕc1 ) = 0 for some q = l0 , then K(c1 ,c2 ) (ϕε ) = ∅ and mmin (c1 , c2 , ϕ) ≤ ind(ϕε , p ) ≤ Mmax (c1 , c2 , ϕ)

(2.4)

for ∀ p ∈ K(c1 ,c2 ) (ϕε ), where mmin (c1 , c2 , ϕ) :=

min

ind(ϕ, p),

(2.5)

max

M (ϕ, p),

(2.6)

M (ϕ, p) := ind(ϕ, p) + dim ker d2 ϕ(p),

(2.7)

mmin (c1 , c2 , ϕ) ≤ l0 ≤ Mmax (c1 , c2 , ϕ).

(2.8)

p∈K(c1 ,c2 ) (ϕ)

Mmax (c1 , c2 , ϕ) :=

p∈K(c1 ,c2 ) (ϕ)

and this implies

Aiming at the proof of Theorem 2.1, the two consequences below due to [6] are adapted to our needs and we restate the proof of the following Proposition 2.2 for the convenience of later discussion. Proposition 2.2. If K is a compact subset of E, then for every μ > 0 there exists a C ∞ function l : E → [0, 1] with all its derivatives bounded such that l(x) = 1,

for all x ∈ K μ ;

l(x) = 0,

for all x ∈ E\K 2μ ,

where K μ = {x ∈ E : dist(x, K) < μ}. Proof. Fix a finite number of points x1 , x2 , · · · , xm ∈ K such that K ⊂ We get C ∞ -functions li on E with bounded derivatives such that

m i=1

B(xi , μ2 ).

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li (x) = 1



3 as x ∈ B xi , μ ; 2

li (x) = 0 as x ∈ / B(xi , 2μ),

by simply taking a real function α : R →[0, 1] such that α(t) = 1 if t ≤

9 ; 4

α(t) = 0 if t ≥ 4,

i and setting li (x) = α( x−x ). Then we take β : R → [0, 1] such that μ2 2

β(t) = 0 if t ≤ 0;

β(t) = 1 if t ≥ 1.

m The function l : E → R defined by l(u) = β( i=1 li (u)) is the required function. Proposition 2.3. Assume that for every x in the compact set K, d2 ϕ(x) is a Fredholm operator. Then there exist a neighborhood N of K and ε > 0, such that every functional ψ ∈ C 2 , with ϕ − ψC 2 ≤ ε has a gradient ∇ψ that is a proper map on N , where 



ϕ(x) 2 := ϕ(x) + C +

   ∇ϕ(x), v 

sup v∈E,v=1

sup

sup

 2   d ϕ(x)u, v .

(2.9)

u∈E,u=1 v∈E,v=1

Remark 2.4. Assume for every x in the compact set K, d2 ϕ(x) is a Fredholm operator and N is a neighborhood N of K. We say that for every functional ψ ∈ C 2 , with ϕ − ψC 2 ≤ ε has a gradient ∇ψ that is a proper map on N , i.e., for arbitrary sequence {xn } ⊂ N such that ∇ψ(xn ) is convergent, {xn } has a convergent subsequence. Indeed, we can obtain a more general version of Proposition 2.3. For the sake of convenience, we only deal with the case ∇ϕ = I −F , where F is compact. The generalized result is stated as follows: Lemma 2.5. Let ϕ ∈ C 1 (E, R), ∇ϕ = I −F , F is compact. Assume K ⊂ E is compact and N is an open neighborhood of K, s.t., ϕ ∈ C 2 (N, R). Then there exists a neighborhood of K, N ⊂ N , for ∀ε ∈ (0, 1), any ψ ∈ C 1 (E, R) ∩ C 2 (N , R) with max  ϕ(x) − N x∈N . ψ(x)C 2 ≤ ε, has a gradient ∇ψ that is a proper map on N Proof. Notice that N is an open neighborhood of compact K, we claim that there exists δ > 0, Kδ ⊂ N . Indeed, if not, then we can find δn > 0, δn → 0 as n → +∞, s.t. Kδn \N = ∅. Take x∗n ∈ Kδn \N . On the one hand, dist(x∗n , K) ≤ δn → 0. Hence, ∃yn ∈ K, dist(x∗n , yn ) → 0. On the other hand, as K is compact, there exists a subsequence of yn , denoted by ynj , s.t. ynj → y0 ∈ K. Then we have     dist x∗nj , y0 ≤ dist x∗nj , ynj + dist(ynj , y0 ) → 0

(2.10)

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and this yields y0 ∈ ∂N . However, y0 ∈ K ⊂ N = N ◦ , where N ◦ denotes the interior of N . A contradiction! The claim is proved. Once again from the compactness of K, there exists a finite number of points s = s B( x 1 , x 2 , · · · , x s ∈ K, s.t. K ⊂ i=1 B( xi , δ). Take N xi , δ). For ∀ε > 0, i=1 1 2 ∀ψ ∈ C (E, R) ∩ C (N , R), s.t. maxx∈N  ϕ(x) − ψ(x)C 2 ≤ ε. Set ∇ψ(x) := x − F (x) + r(x), d2 ϕ(x) := I − DF (x),

∀x ∈ E,

, ∀x ∈ N

d2 ψ(x) := I − DF (x) + Dr(x),

, ∀x ∈ N

hence we get

Dr(x) =

sup

sup

 2     d ϕ(x) − ψ(x) u, v  ≤ ε,

. ∀x ∈ N

(2.11)

u∈E,u=1 v∈E,v=1

such that ∇ψ(xn ) is convergent. Then there exist Suppose {xn } is a sequence in N i0 ∈ N, 1 ≤ i0 ≤ s, and a subsequence of {xn }, denoted by {xnj }, {xnj } ⊂ B( xi0 , δ). Notice that F is compact, then up to a subsequence, still denoted by {xnj } such that F (xnj ) is convergent. Therefore, ∀η > 0, ∃K ∈ N, for any j, k ∈ N, j, k ≥ K,



∇ψ(xn ) − ∇ψ(xn ) + F (xn ) − F (xn ) ≤ η + η = η. j j k k 2 2

(2.12)

Making use of the integral theorem of the mean, we have

r(xn ) − r(xn ) j k

1

 

= Dr xnk + t(xnj − xnk ) dt(xnj − xnk )

0

1

 

≤ Dr xnk + t(xnj − xnk ) dt · xnj − xnk 

0

and observe that

1

 

Dr xnk + t(xnj − xnk ) dt

0

1         = sup sup  Dr xnk + t(xnj − xnk ) dt u, v    u∈E,u=1 v∈E,v=1 0

(2.13)

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 1        = sup sup  Dr xnk + t(xnj − xnk ) u, v dt  u∈E,u=1 v∈E,v=1 0

1 ≤

 

Dr xn + t(xn − xn ) dt ≤ ε, j k k

(2.14)

0

hence,

r(xnj ) − r(xn ) ≤ εxnj − xn . k k

(2.15)

Take ε ∈ (0, 1). Since r(xnj ) − r(xnk ) = ∇ψ(xnj ) − ∇ψ(xnk ) + F (xnj ) − F (xnk ) − (xnj − xnk ), (2.16) by (2.12), (2.15) and (2.16), we have



(1 − ε)xnj − xnk  ≤ ∇ψ(xnj ) − ∇ψ(xnk ) + F (xnj ) − F (xnk ) ≤ η,

(2.17)

for j, k ∈ N, j, k ≥ K, and this shows that {xnj } is a Cauchy sequence. The conclusion follows. 2 ∗ Proof of Theorem 2.1. Pick δ 0 ∈ (0, δ2 ) sufficiently small so that

c1 <

ϕ(x) ≤

inf

2δ

x∈K(c 0,c 1

2)

(ϕ)

sup 2δ

ϕ(x) < c2 .

(2.18)

x∈K(c 0,c 1

(ϕ) 2)

As K(c1 ,c2 ) (ϕ) is compact, we fix a finite number of points x∗1 , x∗2 , · · · , x∗s ∈ K(c1 ,c2 ) (ϕ) s such that K(c1 ,c2 ) (ϕ) ⊂ i=1 B(x∗i , δ 0 ). Notice that K(c1 ,c2 ) (ϕ) ⊂

s 

  δ 0 δ∗ B x∗i , δ 0 ⊂ K(c (ϕ) ⊂ K(c (ϕ), 1 ,c2 ) 1 ,c2 )

(2.19)

i=1 δ∗ = s B(x∗ , δ 0 ), in terms of the proof of Lemma 2.5, take N = K(c (ϕ), N i i=1 1 ,c2 ) s ∀ε ∈ (0, 1), any ψ ∈ C 1 (E, R) ∩ C 2 ( i=1 B(x∗i , δ 0 ), R) with maxx∈s B(x∗ ,δ 0 ) ϕ(x) − i i=1 s ψ(x)C 2 ≤ ε, has a gradient ∇ψ that is a proper map on i=1 B(x∗i , δ 0 ), and hence, ψ s satisfies the (PS) condition on i=1 B(x∗i , δ 0 ). Once again from the proof of Lemma 2.5, s 2δ there exists δ0∗ ∈ (0, δ 0 ), for ∀δ ∈ (0, δ0∗ ], K(c (ϕ) ⊂ i=1 B(x∗i , δ 0 ). Therefore, ψ 1 ,c2 ) 2δ satisfies the (PS) condition on K(c (ϕ). 1 ,c2 ) Set ρ := inf x∈K(c 2δ δ (ϕ)\K(c ,c ) (ϕ) ∇ϕ > 0. Fix a finite number of points 1 ,c2 ) 1 2 m x1 , x2 , · · · , xm ∈ K(c1 ,c2 ) (ϕ) such that K(c1 ,c2 ) (ϕ) ⊂ i=1 B(xi , 2δ ). Let l(x) be the function given by Proposition 2.2 and also suppose 0 ≤ l(x) ≤ 1 on E, α(t) ∈ C 2 [ 94 , 4], β(t) ∈ C 2 [0, 1].

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Notice that  ∇l(x) = ∇β =β



m 

 li (x)

i=1 m 



i=1



li (x)

m 

·

m  i=1 m 

li (x)



 x − xi 2 li (x) · α · =β δ2 x i=1 i=1  m

m    x − xi 2 2(x − xi ) = β li (x) · α · 2 δ δ2 i=1 i=1 





x − xi 2 δ2

(2.20)

and 

 d2 l(x)u, v m  m



 m 2 2   4    x − xi   x − xj  = 4β li (x) α x − xi , u · α x − xj , v δ δ2 δ2 i=1 i=1 j=1  m m  x − xj 2  4   + 4β li (x) · α x − xj , u · x − xj , v δ δ2 i=1 j=1 m  m

 2  2    x − xj  + 2β · u, v . (2.21) li (x) · α δ δ2 i=1 j=1

Set M1 := maxt∈[ 94 ,4] |α (t)|, M2 := maxt∈[0,1] |β  (t)|, M1∗ := maxt∈[ 94 ,4] |α (t)|, M2∗ := maxt∈[0,1] |β  (t)|. Denote the function ψ by ψ(x) := ϕ(x) + l(x)y, x − x0

(2.22)

2δ for ∀x0 ∈ K(c (ϕ) and we then have 1 ,c2 )

∇ψ(x) = ∇ϕ(x) + y, x − x0 · ∇l(x) + l(x)y,  d ϕ(x) − ψ(x) u = d2 l(x)uy, x − x0 + 2∇l(x)y, u , u = 1. 2



Set M∗ := supx∈K(c 2δ ,c 1

ϕ − ψC 2 

≤ 1 + ∇l(x) +

2)

(ϕ)

(2.24)

2δ x, then for ∀x ∈ K(c (ϕ), 1 ,c2 )

2



 

d l(x)u · y · x − x0  + 1 + 2 ∇l(x) · y

sup

u∈E,u=1



≤ C(x) · y · x − x0  + ≤ Cy,

(2.23)

m 4M1 M2  1+ x − xi  δ2 i=1

 · y (2.25)

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where m 2 m 2M1 M2  4 ∗ 2  C(x) = 1 + x − xi  + 4 M2 M1 x − xi  δ2 δ i=1 i=1 m  4 2 ∗ + 4 M 2 M1 x − xi 2 + 2 mM1 M2 , δ δ i=1

and 8 8 32 mM∗ M1 M2 + 2 mM∗2 M1 M2 + 4 m2 M∗3 M12 M2∗ δ2 δ δ 32 4 + 4 mM∗3 M1∗ M2 + 2 mM∗ M1 M2 . δ δ

C = 1 + 2M∗ +

2δ δ Take ε∗ = ε∗ (δ) ∈ (0, ρ2 ). Combining (2.20) and (2.23), for ∀x ∈ K(c (ϕ)\K(c (ϕ), 1 ,c2 ) 1 ,c2 ) ∗ ∀ε ∈ [0, ε ], we have







∇ψ(x) ≥ ∇ϕ(x) − y, x − x0  · ∇l(x) − l(x)y 4M∗ ≥ 2ε − 2M∗ y · mM1 M2 2 − y δ

 8mM∗2 M1 M2 = 2ε − + 1 y. δ2

(2.26)

δ As ∇ψ(x) = ∇ϕ(x) + y on K(c (ϕ), it follows that all the critical points of 1 ,c2 ) δ ψ in K(c1 ,c2 ) (ϕ) (if any) are nondegenerate and finite in number. Indeed, since ∇ϕ is Fredholm, we may use the Sard–Smale theorem [16] to find y ∈ E ∗ , with y ≤ δ2 ε min{ Cε , 8mM 2 M 2 }, such that −y is a regular value of ∇ϕ. This concludes (v). Us1 M2 +δ ∗ 2δ δ δ ing (2.26) we get ∇ψ(x) ≥ ε on K(c (ϕ)\K(c (ϕ), so K(c1 ,c2 ) (ψ) ⊂ K(c (ϕ) 1 ,c2 ) 1 ,c2 ) 1 ,c2 ) 2δ provided K(c1 ,c2 ) (ψ) = ∅. As ψ(x) = ϕ(x) on E\K(c1 ,c2 ) (ϕ), (2.25) yields (iii) and claims (i), (iv) are thus proved. Notice that inf x∈K(c 2δ δ δ (ϕ)\K(c (ϕ) ∇ϕ = ρ > 0, so inf x∈∂K(c (ϕ) ∇ϕ ≥ ρ. 1 ,c2 ) 1 ,c2 ) 1 ,c2 ) Let ψt (x) = (1 − t)ϕ(x) + tψ(x), then ∇ψt = ∇ϕ + t(∇ψ − ∇ϕ). Hence, for ∀x ∈ δ ∂K(c (ϕ), ∀t ∈ [0, 1], 1 ,c2 )

∇ψt  ≥ ∇ϕ − ty ≥ ρ − tε > (2 − t)ε ≥ ε.

(2.27)

δ δ Since F (x) − y is compact on K(c (ϕ), by (2.27), deg(∇ψt , K(c (ϕ), 0) is well 1 ,c2 ) 1 ,c2 ) defined and by homotopy invariance of topological degree

    δ δ deg ∇ψ, K(c (ϕ), 0 = deg ∇ϕ, K(c (ϕ), 0 . 1 ,c2 ) 1 ,c2 ) Choose c1 < c1 < c2 < c2 such that

(2.28)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

c1 <

inf

2δ x∈K(c

1 ,c2 )

Set a = inf x∈K(c 2δ ,c 1

2)

(ϕ)

(ϕ)

ϕ(x) ≤

3793

ϕ(x) < c2 .

sup

(2.29)

2δ x∈K(c

(ϕ) 1 ,c2 )

ϕ(x), b = supx∈K(c 2δ ,c 1

2)

(ϕ)

ϕ(x). Notice that M∗ depends on δ > 0

and m is independent of δ if and only if K is finite. It is harmless to denote m = mδ , M∗ = Mδ . Here Mδ = supx∈K(c 2δ (ϕ) x. And also observe that ψ is related to ε, write 1 ,c2 ) ψ = ϕε . It suffices to choose sufficiently small δ ∗ ∈ (0, δ 0 ), s.t., ∀δ ∈ (0, δ ∗ ], 0

0

(2Mδ − 1)δ 2 < 8mδ Mδ2 M1 M2 .

(2.30)

Actually, as mδ ≥ 1 and M1 , M2 > 0 are independent of δ, also see the fact that Mδ1 ≤ Mδ2 as 0 < δ1 < δ2 , then for δ > 0 small, (2Mδ − 1)δ 2 < 0 provided K(c1 ,c2 ) (ϕ) = {θ}, and if K(c1 ,c2 ) (ϕ)\{θ} = ∅, then limδ→0 (2Mδ − 1)δ 2 = 0 while lim mδ Mδ2 M1 M2 ≥ lim Mδ2 M1 M2 ≥ M02 M1 M2 > 0,

δ→0

(2.31)

δ→0

where M0 = supx∈K(c ,c ) (ϕ) x. 1 2 c2 − b. Obviously, ϕcε 2 ⊂ ϕc 2 . Indeed, We show that ϕcε 2 = ϕc 2 for ∀δ ∈ (0, δ0∗ ] and ε < c 2 2δ 2δ ∀x ∈ ϕε , if x ∈ / K(c1 ,c2 ) (ϕ), ϕ(x) = ϕε (x) ≤ c2 , and if x ∈ K(c (ϕ), c1 < ϕ(x) < c2 . 1 ,c2 ) c 2 2δ Conversely, ∀x ∈ ϕ , if x ∈ K(c1 ,c2 ) (ϕ), by (2.30) we get   ϕε (x) − ϕ(x) ≤ 2Mδ y ≤

2Mδ δ 2 ε < ε, 8mδ Mδ2 M1 M2 + δ 2

(2.32)

and this yields ϕε (x) < b + ε < b + c2 − b = c2 .

(2.33)

The conclusion follows. Thereby, ϕcε 2 is a strong deformation retract of ϕc2 . And also ϕc1 is a strong deformation retract of ϕcε 1 as ε < a − c1 . This is due to the two facts that c 1 c 1 c 1 c 1 2δ 2δ ϕ ⊂ ϕε and (ϕε \ϕ ) ∩K(c1 ,c2 ) (ϕ) = ∅. Actually, if x ∈ ϕc 1 , then x ∈ / K(c (ϕ), and 1 ,c2 ) ∗ c 1 c 1 2δ hence ϕε (x) = ϕ(x) ≤ c1 . Suppose to the contrary that ∃x ∈ (ϕε \ϕ ) ∩ K(c1 ,c2 ) (ϕ), we get     ϕ x∗ ≤ ϕε x∗ + ε < c1 + a − c1 = a =

inf

2δ x∈K(c

1 ,c2 )

ϕ(x).

(2.34)

(ϕ)

2δ A contradiction! The combination of ϕc 1 ⊂ ϕcε 1 and (ϕcε 1 \ϕc 1 ) ∩ K(c (ϕ) = ∅ alludes 1 ,c2 ) c 1 c 1 to ϕε = ϕ . Above argument indicates ε∗ = ε∗ (δ) < min{ ρ2 , c2 − b, a − c1 } for ∀δ ∈ (0, δ0∗ ], where δ0∗ > 0 sufficiently small such that (2.30) holds for ∀δ ∈ (0, δ0∗ ], and we thereby arrive at (ii).

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Now we are confronted with two cases: (1) K( c1 , c2 ) (ϕε ) = ∅. Combining (2.28) and (ii),     δ δ deg ∇ϕ, K(c (ϕ), 0 = deg ∇ϕε , K(c (ϕ), 0 1 ,c2 ) 1 ,c2 ) =

∞ 

  (−1)q rank Hq ϕcε 2 , ϕcε 1

q=0

=

∞ 

  (−1)q rank Hq ϕc2 , ϕc1 = 0.

(2.35)

q=0

(2) K( c1 , c2 ) (ϕε ) = ∅. As ϕε has finitely many critical points in ϕ−1 c1 , c2 ], denoted ε [ by x 1 , x 2 , · · · , x s . Notice that x j , s depend on the choice of δ > 0, it might be as well set (δ) x j = x j , s = s(δ). Once again from (2.28) and (ii), also by employing Morse inequality, we have ∞ 

∞      (−1)q rank Hq ϕc2 , ϕc1 = (−1)q rank Hq ϕcε 2 , ϕcε 1

q=0

q=0

=

∞ 

(−1)q

q=0

=

s(δ) 

 (δ)  rank Cq ϕε , x j

j=1

s(δ) ∞  

 (δ)  (−1)q rank Cq ϕε , x j

j=1 q=0

=

s(δ) 

  (δ)   deg ∇ϕε , B x j , rj , 0

j=1

=

s(δ) 

(δ)

(−1)ind(ϕε , xj

)

  δ = deg ∇ϕε , K(c (ϕ), 0 1 ,c2 )

j=1

  δ (ϕ), 0 , = deg ∇ϕ, K(c 1 ,c2 ) s(δ)

(2.36)

B( xj , rj ) ⊂ (ϕcε 2 \ϕcε 1 )◦ , B( xi , ri ) ∩ B( xj , rj ) = ∅, i = j, 1 ≤ i, j ≤ s(δ). The combination of (2.35) and (2.36) yields (vi). Suppose Hq0 (ϕc2 , ϕc1 ) = 0 for some q0 ∈ N. Notice that (2.2) holds for ∀δ ∈ (0, δ0∗ ], ∀0 < ε ≤ ε∗ (δ), ε∗ (δ) < min{ ρ2 , c2 − b, a − c1 }, we get K( c1 , c2 ) (ϕε ) = ∅. We claim that ∃δ0 ∈ (0, δ0∗ ], ∀δ ∈ (0, δ0 ], ∀0 < ε ≤ ε∗ (δ), (δ)

(δ)

(δ)

j=1

 (δ)  mmin (c1 , c2 , ϕ) ≤ ind ϕε , x j ≤ Mmax (c1 , c2 , ϕ),

(2.37)

for ∀j = 1, 2, · · · , s(δ). Argue by contradiction. Suppose that ∃δn > 0, ∃εn ∈ (0, ε∗ (δn )], (δ ) ∃ xjnn ∈ K( c1 , c2 ) (ϕεn ), δn → 0, and thus εn → 0, s.t.

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 (δ )  ind ϕεn , x jnn < mmin (c1 , c2 , ϕ).

(2.38)

δn As x jnn ∈ K(c (ϕ), in view of the definition, there is a sequence {x∗n } ∈ K(c1 ,c2 ) (ϕ), 1 ,c2 ) (δ )

 xjnn − x∗n  ≤ δn . Observe that K(c1 ,c2 ) (ϕ) is compact, up to a subsequence, still denoted by x∗n , s.t. ∗ xn → x 0 in E and x 0 ∈ K(c1 ,c2 ) (ϕ). Consequently, we have (δ )

(δn )

(δn )



x −x 0 ≤ x − x∗n + x∗n − x 0 → 0, n → +∞. jn

Set ρn = inf x∈K 2δn

δn (c1 ,c2 ) (ϕ)\K(c1 ,c2 ) (ϕ)

min{ Cε n , 8mδ n

(2.39)

jn

2 δn εn 2 2 n Mδ ∗ M1 M2 +δn 0

∇ϕ > 0. Then we can find yn ∈ X ∗ , with yn  ≤

}, such that −yn is a regular value of ∇ϕ, where

n = 1 + 2Mδ∗ + 8 mδ Mδ∗ M1 M2 + 8 mδ M 2∗ M1 M2 + 32 m2 M 3∗ M 2 M ∗ C 0 δn2 n 0 δn2 n δ0 δn4 δn δ0 1 2 +

32 4 mδn Mδ30∗ M1∗ M2 + 2 mδn Mδ0∗ M1 M2 . 4 δn δn

Denote the function ϕεn by ϕεn (x) := ϕ(x) + l(x)yn , x − x0

(2.40)

for ∀x0 ∈ K(c1 ,c2 ) (ϕ). δn Notice that ∇ϕεn (x) = ∇ϕ(x) + yn , d2 ϕεn (x) = d2 ϕ(x) = I − DF (x) on K(c (ϕ). 1 ,c2 ) (δ )

(δ )

As x jnn → x 0 , DF ( xjnn ) → DF ( x0 ). − Set E := span{Eλ | λ < 0, λ ∈ σ(A)}, A = d2 ϕ( x0 ), l := dim E − . Clearly ind(ϕ, x 0 ) = l. Now we show that ∃N ∈ N, ∀n ≥ N ,  (δ )  ind ϕεn , x jnn ≥ l.

(2.41) (δ )

Indeed, if not, then up to a subsequence, still denoted by x jnn , tn := ind(ϕεn , x jnn ) < l as n sufficiently large. Set E − := span{x(1) , · · · , x(l) } and denote by (δ )

 − E(n) := span Eλ  0 E(n) := span Eλ  + := span Eλ E(n)

   λ < 0, λ ∈ σ(An ) ,    λ = 0, λ ∈ σ(An ) ,    λ > 0, λ ∈ σ(An ) ,

− the negative, zero and positive invariant subspace of An and E = E(n)



0 E(n)



+ E(n) ,

− An := d2 ϕεn ( xjnn ). Since dim E(n) = tn < l, there exist real numbers α1 , · · · , αl , not (n) (j) l − (n) all zero, such that x := j=1 αj x ⊥E(n) . Let  x(n)  = 1. Notice that (δ )

(n)

(n)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

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  (δn )   (n) (n)    DF x − DF ( x0 ) x ,x jn

  (δn )   (n) ≤ DF x − DF ( x0 ) x jn

  (δn )   − DF ( x0 ) → 0, ≤ DF x jn

(2.42)

going if necessary to a subsequence, assume x (n) → x in E − , therefore,      DF ( x0 ) x(n) , x (n) − DF ( x0 ) x, x      (n)   ≤  DF ( x0 ) x(n) , x (n) − x  +  DF ( x0 ) x −x ,x 



(n)  (n)  x0 ) x −x x0 ) x(n) · x + DF ( −x → 0, ≤ DF (

(2.43)

and hence we have   (δn )  (n) (n)  0 ≤ d2 ϕεn x jn x ,x

(n) 2   (δn )   (n) (n)    − DF x = x jn − DF ( x0 ) x ,x − DF ( x0 ) x(n) , x (n)   → d2 ϕ( x0 ) x, x (2.44) which contradicts the definition of E − and this proves (2.41). Combining (2.38) and (2.41), we thereupon obtain an obvious paradox  (δ )  mmin (c1 , c2 , ϕ) ≤ l ≤ ind ϕεn , x jnn < mmin (c1 , c2 , ϕ)

(2.45)

(δ)

for ∀n ≥ N and quite similarly deal with the case ind(ϕε , x j ) ≤ Mmax (c1 , c2 , ϕ). That’s precisely claim (2.37). Applying the q0 -th Morse inequality we get Mq0 ( c1 , c2 , ϕε ) =

s 

rank Cq0 (ϕε , x i ) ≥ βq0 ( c1 , c2 , ϕε )

i=1

  = rank Hq0 ϕcε 2 , ϕcε 1 > 0

(2.46)

and this implies that ∃ xi0 ∈ K( c1 , c2 ) (ϕε ) with Morse index q0 and mmin (c1 , c2 , ϕ) ≤ q0 ≤ Mmax (c1 , c2 , ϕ). We concludes (vii) and Theorem 2.1 is thus proved. 2 3. Application Consider elliptic equation with Neumann boundary condition: 

−Δu + αu = f (u), x ∈ Ω, ∂u x ∈ ∂Ω, ∂ν = 0,

(3.1)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

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on a bounded domain Ω ⊂ RN with smooth boundary and α > 0. Suppose that f ∈ C 1 (R, R), f (0) = 0. Denote by σ(−Δ + α) = {λi | α = λ1 < λ2 < · · · < λk < · · ·} the eigenvalues of the following linear problem 

−Δu + αu = λu, x ∈ Ω, ∂u x ∈ ∂Ω. ∂ν = 0,

(3.2)

and assume: (I) ∃C1 , C2 > 0, ∀s ∈ R,    f (s) ≤ C1 + C2 |s|p−2 , where 2 < p < 2∗ , 2∗ =

2N N −2 , if f (s) lims→0 s = α.

(3.3)

N ≥ 3, and 2∗ = ∞, if N = 1, 2.

(II) f0 = And there exist sequences {ai } and {bi }, ai , bi ∈ R, i = 1, 2, · · · , ai > 0, bi < 0, ai is monotone decreasing and tends to zero while bi is monotone increasing and tends to zero, s.t., f (ai ) = αai , f (bi ) = αbi . This shows that {ai }, {bi } are constant solution sequences of (3.1). Moreover, f (s) < αs as s ∈ (ai+1 , ai ), if i is odd; f (s) > αs as s ∈ (ai+1 , ai ), if i is even; f (s) > αs as s ∈ (bi , bi+1 ), if i is odd; f (s) < αs as s ∈ (bi , bi+1 ), if i is even. (III) f (s) > αs as s ∈ (a1 , +∞) and f (s) < αs as s ∈ (−∞, b1 ). (IV) f  (ai ) > α, f  (bi ) > α for i odd. (V) f (s)s > 0, s ∈ R\{0}. (VI) ∃j, l ∈ N, f  (a1 ) ∈ (λj , λj+1 ), f  (b1 ) ∈ (λl , λl+1 ). (VII) ∃k ∈ N, f∞ = lims→∞ f (s) s ∈ (λk , λk+1 ). And we also assume: (VIII) dj , dl , dk are even, where dj is the dimension of the subspace Nj spanned by the eigenfunctions corresponding to λ1 , · · · , λj . (IX) (−1)dj +dl +dk = −1 and dj , dl , dk are not all odd. (X) (−1)dj +dl +dk = 1 and dj , dl , dk are not all even. Theorem 3.1. Under the hypotheses (I)–(VIII), there exist at least three solutions of (3.1), denoted by u∗1 (x), u∗2 (x) and u∗3 (x), s.t. maxx∈Ω u∗i (x) > a1 or minx∈Ω u∗i (x) < b1 , i = 1, 2, 3. Theorem 3.2. Under the hypotheses (I)–(VII), (IX), there exist at least two solutions of (3.1), denoted by u 1 (x), u 2 (x), s.t. maxx∈Ω u i (x) > a1 or minx∈Ω u i (x) < b1 , i = 1, 2. Theorem 3.3. Under the hypotheses (I)–(VII), (X), there exists at least one solution of (3.1), denoted by u (x), s.t. maxx∈Ω u (x) > a1 or minx∈Ω u (x) < b1 . Indeed, (II) can be substituted by the following hypothesis: ∗ (II) f0 = 0. ∃ε1 ∈ (0, a1 ) sufficiently small, s.t., ∀s ∈ [a1 − ε1 , a1 ], f (s) < αs, and also ∃ε2 ∈ (0, −b1 ) sufficiently small, s.t., ∀s ∈ [b1 , b1 + ε2 ], f (s) ≥ αs.

3798

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

∗

Corollary 3.4. Under the assumptions (I), (II) , (III), (V)–(VIII), there exist at least three solutions of (3.1), denoted by u∗1 (x), u∗2 (x) and u∗3 (x), s.t. maxx∈Ω u∗i (x) > a1 or minx∈Ω u∗i (x) < b1 , i = 1, 2, 3. ∗

Corollary 3.5. Under the assumptions (I), (II) , (III), (V)–(VII), (IX), there exist 1 (x), u 2 (x), s.t. maxx∈Ω u i (x) > a1 or at least two solutions of (3.1), denoted by u minx∈Ω u i (x) < b1 , i = 1, 2. ∗

Corollary 3.6. Under the assumptions (I), (II) , (III), (V)–(VII), (X), there exists at least one solution of (3.1), denoted by u (x), s.t. maxx∈Ω u (x) > a1 or minx∈Ω u (x) < b1 . 4. The computation of critical groups at infinity Let E be a Hilbert space and Φ : E → R be of class C 1 . Write KΦ = {x ∈ E : ∇Φ = 0} for the set of critical points and Φc = {x ∈ E : Φ(x) ≤ c} for the sublevel sets as usual. Definition 4.1. Suppose Φ(KΦ ) is strictly bounded from below by a ∈ R and that Φ satisfies the (PS) condition. Then Cq (Φ, ∞) ∼ = Hq (E, Φa ), q ∈ Z, is the q-th critical group of Φ at infinity. It is independent of the choice of a with (PS) condition. (The readers are referred to [1–3,17].) We work on E = Hν1 (Ω) equipped with the norm

 u =

 |∇u| + α 2

Ω

2

 12

u dx

,

Ω

where Hν1 (Ω) = Cν∞ (Ω), Cν∞ (Ω) = {u ∈ C ∞ (Ω) : ∂u ∂ν |∂Ω = 0}. It is easy to check that Hν1 (Ω) is a Hilbert space. In view of the variational point, solutions of (3.1) are critical points of corresponding functional 1 J(u) = 2



α |∇u| dx + 2

Ω

on E, where F (u) =

u 0



 u dx −

2

2

Ω

F (u)dx Ω

f (s)ds. Denote by ·, · inner product of E and write  u, v =

∇u∇v + αuv Ω

for ∀u, v ∈ E. For the convenience of later argument, it is necessary to present some general hypotheses. Suppose f (x, s) ∈ C 1 (Ω × R, R) and f (x, 0) = 0. A more stronger assumption than (I) is also indispensable for us:

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3799

(I ) ∃C1 , C2 > 0, ∀x ∈ Ω, ∀s ∈ R,    fs (x, s) ≤ C1 + C2 |s|p−2 ,

(4.1)

∗ where 2 < p < 2∗ , 2∗ = N2N −2 , if N ≥ 3, and 2 = ∞, if N = 1, 2. Consider the following equation:



−Δu + αu = t[f (x, u) − au− − bu+ ] + au− + bu+ , x ∈ Ω, ∂u x ∈ ∂Ω, ∂ν = 0,

(4.2)

and define J(a,b,t) (u) =

1 2

 |∇u|2 +

α 2

Ω



 u2 − t

Ω

F (x, u) −

1−t 2

Ω



 2    − 2 au  + bu+  ,

Ω

u where (a, b) ∈ / Σ, t ∈ [0, 1], u+ = max{u, 0}, u− = min{u, 0}, F (x, u) = 0 f (x, s)ds and denote J(a,b,0) (u) by J(a,b) (u). Here Σ denotes by the Fucík spectrum Σ of −Δ and is the set of points (a, b) ∈ R2 for which the problem 

−Δu + αu = au− + bu+ , ∂u ∂ν = 0,

x ∈ Ω, x ∈ ∂Ω,

(4.3)

has a nontrivial solution. Our main result concerning the computation of critical groups at infinity reads: Theorem 4.2. Assume f (x, s) ∈ C 1 (Ω × R, R), satisfies (I ), and f (x, s) − as− − bs+ = 0, s→∞ s lim

uniformly a.e on Ω, (a, b) ∈ / Σ. Then Cq (J, ∞) ∼ = Cq (J(a,b) , 0).

(4.4)

Remark 4.3. Unlike the conclusion ∞) ∼ Cq (J, = Cq (J(a,b) , 0)

(4.5)

yielded by Proposition 5.3.1 of [14] via constructing a cut-off functional  J(u) =

J(u), J(a,b) (u),

as u ≤ R, as u ≥ 2R,

(4.6)

for R > 0 sufficiently large, (4.4) gets further information about Cq (J, ∞), and this is indispensable for us to obtain more solutions in applications.

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3800

Before entering the proof of Theorem 4.2, we first recall a well-known result (see [18]): Proposition 4.4. Assume that |Ω| < ∞, 1 ≤ p, r < ∞, f ∈ C(Ω × R, R) and     f (x, u) ≤ C 1 + |u|p/r .

(4.7)

Then, for every u ∈ Lp (Ω), f (·, u) ∈ Lr (Ω) and the operator A : Lp (Ω) → Lr (Ω) : u −→ f (x, u) is continuous. To complete the proof of Theorem 4.2, we need some lemmas adapted to our needs. Lemma 4.5. Assume f (x, s) ∈ C(Ω × R, R), lims→∞ on Ω, (a, b) ∈ / Σ. Then there exists M > 0, s.t. sup

sup

t∈[0,1] u∈KJ(a,b,t)

f (x,s)−as− −bs+ s

= 0, uniformly a.e

u < M.

(4.8)

Proof. Suppose to the contrary that ∃tk ∈ [0, 1], uk ∈ KJ(a,b,tk ) , uk  → +∞ as  k → +∞, where KJ(a,b,t) = {u ∈ E : J(a,b,t) (u) = 0}, t ∈ [0, 1]. Hence, (4.2) yields  1 = tk Ω

  uk + f (x, uk ) − au− +a k − buk uk 2



Ω

u− k uk 

2



+b Ω

u+ k uk 

2 .

(4.9)

Notice that |f (x, s)| ≤ C(|s| + 1), for ∀x ∈ Ω, ∀s ∈ R, then |f (x, s)s| ≤ C(s2 + |s|). Hence,  |uk |
+ |(f (x, uk ) − au− k − buk )uk | uk 2



≤ ≤

|uk |
+ 2 2 C(u2k + |uk |) + |a| · |u− k | + |b| · |uk |

uk 2 (|a| + |b| + C)Mk2 + CMk |Ω|. uk 2

Set uk  = ρk and take Mk → +∞, s.t. ∀k ≥ K1 ,  |uk |
Mk ρk

(4.10)

→ 0. Therefore, ∀ε > 0, ∃K1 ∈ N,

+ |(f (x, uk ) − au− ε k − buk )uk | < . 2 uk  2

(4.11)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

f (x,s) s

Observe that ∀s ∈ R, |s| ≥ S,

→ a as s → −∞ and

f (x,s) s

3801

→ b as s → +∞, then ∀ε > 0, ∃S > 0,

   f (x, s) − as− − bs+  αε <  . max  s 2 x∈Ω

(4.12)

From Mk → +∞ we obtain that ∃K2 ∈ N, ∀k ≥ K2 , Mk ≥ S, and thus we get  |uk |≥Mk

+ |(f (x, uk ) − au− k − buk )uk | uk 2



= |uk |≥Mk



= uk ≥Mk

≤

αε 2

 +  2  (f (x, uk ) − au− k − buk )  uk    uk 2 uk

  2  f (x, uk )  u  − b k 2 dx +  uk uk 



u2k

|uk |≥Mk

uk 

dx ≤ 2

ε 2

 Ω



uk ≤−Mk

  2  f (x, uk )  u  − a k 2 dx  uk uk 

ε αu2k dx ≤ 2 uk  2

(4.13)

Thereby, ∀ε > 0, take K = max{K1 , K2 }, ∀k ≥ K, 

+ |(f (x, uk ) − au− k − buk )uk | uk 2  + |(f (x, uk ) − au− k − buk )uk | = + 2 uk 

Ω

|uk |
≤



|uk |≥Mk

+ |(f (x, uk ) − au− k − buk )uk | 2 uk 

ε ε + = ε. 2 2

(4.14)

Eq. (4.14) yields  lim

k→+∞

Ω

+ |(f (x, uk ) − au− k − buk )uk | = 0. uk 2

(4.15)

Set u k = uukk  and assume u k  u in E, u k → u in L2 (Ω), combining (4.9) with (4.15) we obtain  1=a Ω

and this yields u = 0.

 − 2 u  +b

 Ω

 + 2 u 

(4.16)

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3802

We claim that u solves (4.3). Indeed, ∀v ∈ E, |

 Ω

+ (f (x, uk ) − au− k − buk )v| ≤ uk 

 |uk |
+ |(f (x, uk ) − au− k − buk )v| uk 



+ |uk |≥Mk

+ |(f (x, uk ) − au− k − buk )v| . uk 

(4.17)

In terms of previous argument, it follows that ∀ε > 0, ∃K1∗ = K1∗ (v) ∈ N, ∀k ≥ K1∗ ,  |uk |
+ |(f (x, uk ) − au− (|a| + |b| + C)Mk + C k − buk )v| ≤ uk  uk 

 |v| Ω

ε ε vL1 = . ≤ 2vL1 2

(4.18)

On the other hand, ∀ε > 0, ∃S ∗ = S ∗ (v) > 0, ∀s ∈ R, |s| ≥ S ∗ ,    f (x, s) − as− − bs+   < αε ,  max  2v s x∈Ω

(4.19)

and hence ∃K2∗ = K2∗ (v) ∈ N, ∀k ≥ K2∗ (v), Mk > S ∗ ,  |uk |≥Mk

+ |(f (x, uk ) − au− k − buk )v| = uk 

 |uk |≥Mk

αε ≤ 2v

 +   f (x, uk ) − au− k − buk   uk | · |v|   · | uk

 | uk | · |v|

Ω

 12   12

 ε 2 2 α | uk | · α |v| ≤ 2v Ω

ε ε v = . ≤ 2v 2

Ω

(4.20)

Therefore, ∀ε > 0, take K ∗ = K ∗ (v) = max{K1∗ (v), K2∗ (v)}, ∀k ≥ K ∗ , (4.17) yields |

 Ω

+ (f (x, uk ) − au− ε ε k − buk )v| ≤ + =ε uk  2 2

(4.21)

so we obtain that ∀v ∈ E,  Ω

+ (f (x, uk ) − au− k − buk )v →0 uk 

(4.22)

and this alludes to the claim. Hence, (a, b) ∈ Σ, which contradicts the assumption. The proof is complete. 2

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3803

Lemma 4.6. Under the hypotheses of Lemma 4.5, δ :=

inf

 J(a,b,t) (u)

t∈[0,1],u∈E\B(0,M )

u

> 0.

(4.23)

Proof. By way of negation, there exist tk ∈ [0, 1], uk ∈ E\B(0, M ), s.t.  J(a,b,t (uk ) k)

uk 

→ 0, k → +∞

(4.24)

i.e.,    1  + uk − (−Δ + α)−1 tk f (x, uk ) + (1 − tk ) au− → 0, k + buk uk 

in E. (4.25)

Repeat above argument, we infer that uk is bounded in E. So we get  J(a,b,t (uk ) → 0. Assume tk → t0 , uk  u0 in E, uk → u0 in L2 (Ω). Once again k) from the fact that |f (x, s)| ≤ C(|s| + 1), for ∀x ∈ Ω, ∀s ∈ R, by Proposition 4.4, 

  f (x, uk ) − f (x, u0 )2 → 0.

(4.26)

Ω

Hence we get    + u0 − (−Δ + α)−1 t0 f (x, u0 ) + (1 − t0 ) au− = 0. 0 + bu0

(4.27)

   +  uk = J(a,b,t (uk ) + (−Δ + α)−1 tk f (x, uk ) + (1 − tk ) au− k + buk k)

(4.28)

Notice that

Combining (4.27) with (4.28), we have     uk − u0 = J(a,b,t (uk ) + (−Δ + α)−1 (tk − t0 )f (x, uk ) + t0 f (x, uk ) − f (x, u0 ) k)     +  − + + (1 − t0 )(−Δ + α)−1 a u− k − u0 + b u k − u0   + (4.29) − (tk − t0 )(−Δ + α)−1 au− k + buk and this yields



uk − u0  ≤ J(a,b,t (uk ) · uk − u0  + (tk − t0 ) k)

 f (x, uk )(uk − u0 )

2

 + t0 Ω

Ω

  f (x, uk ) − f (x, u0 ) (uk − u0 )

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3804

 + (1 − t0 )

  −   +  + a u k − u− (uk − u0 ) 0 + b u k − u0

Ω

− (tk − t0 )



 + au− k + buk (uk − u0 )

Ω

→ 0.

(4.30)

The conclusion follows. 2 Lemma 4.7. Suppose f (x, s) ∈ C(Ω ×R, R), s.t. |f (x, s)| ≤ C(1 +|s|) for ∀x ∈ Ω, ∀s ∈ R. < 0, ∀t ∈ [0, 1], for u ∈ E, if J(a,b,t) (u) ≤ C, then u ≥ M . Then there exists C k → −∞ as k → +∞, then ∃tk ∈ [0, 1] and uk ∈ E, Proof. Suppose the contrary, take C J(a,b,tk ) (uk ) ≤ Ck , uk  < M . Observe that |F (x, s)| ≤ C(|s| + s2 ), then there exist C1∗ , C2∗ > 0, s.t.     |a| + |b| J(a,b,tk ) (uk ) ≥ −C |uk |2 ≥ −C1∗ M − C2∗ M 2 (4.31) |uk | + u2k − 2 Ω

Ω

and this contradicts J(a,b,tk ) (uk ) → −∞. The proof is complete.

2

 −C ∗ M − C ∗ M 2 , Lemma 4.8. Suppose f (x, s) ∈ C 1 (Ω × R, R) and satisfies (I ). If C 1 2 J



(v)

C then ∂t J(a,b,t) (v) J (a,b,t)(v)2 is local Lipschitz for ∀u ∈ J(a,b,t) , ∀t ∈ [0, 1]. (a,b,t)

Proof. Notice that  ∂t J(a,b,t) (v) = −

1 F (x, v) + 2



 2    − 2 av  + bv +  ,

Ω Ω    J(a,b,t) (v) = v − (−Δ + α)−1 tf (x, v) + (1 − t) av − + bv + ,

we write F1 (v) = ∂t J(a,b,t) (v), F2 (v) =

 J(a,b,t) (v)  J(a,b,t) (v)2 .

 −C ∗ M −C ∗ M 2 The hypothesis C 1 2

indicates that u ≥ M so by (4.23) we have inf

 J(a,b,t) (v)

v



C t∈[0,1],v∈J(a,b,t)

≥ δ.

(4.32)

δ , 2

(4.33)

Thus, there exists R > 0 such that inf

 J(a,b,t) (v)

t∈[0,1],v∈B(u,R)

and v ≥

M 2

v

≥

for ∀v ∈ B(u, R). This yields

Mδ



J . (a,b,t) (v) ≥ 4 t∈[0,1],v∈B(u,R) inf

(4.34)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3805

Take v1 , v2 ∈ B(u, R). Notice that v1+ − v2+ ≤ (v1 − v2 )+ , v1− − v2− ≥ (v1 − v2 )− ,

(4.35)

and also observe that (I ) yields that ∃C > 0,     f (x, s) ≤ C 1 + |s|p−1

(4.36)

for all x ∈ Ω and s ∈ R. Using Proposition 4.4, for every u ∈ L(p−1) r1 (Ω), f (·, u) ∈ Lr 1 (Ω). Then we get      F (x, v1 ) − F (x, v2 )   Ω

1         = f x, v2 + t(v1 − v2 ) (v1 − v2 )dxdt   0 Ω

  ≤ max f x, v2 + t(v1 − v2 ) Lr 1 · v1 − v2 Lr 2 t∈[0,1]

 ≤ C max



t∈[0,1]



p−1 r 1  dx 1 + v2 + t(v1 − v2 )

1 r 1

· v1 − v2 Lr 2

Ω

 1 1 + v2 p−1 ≤C + v1 − v2 p−1 · v1 − v2 Lr 2 L(p−1)r 1 L(p−1)r 1   2 1 + v2 p−1 + v1 − v2 p−1 · v1 − v2  ≤C     3 1 + u + R p−1 + (2R)p−1 · v1 − v2  ≤C 4 v1 − v2 , ≤C

(4.37)

∗

∗

2 2 + r 12 = 1, r 1 ∈ ( N2N 2 ∈ ( 2∗ −p+1 , 2∗ ), if N ≥ 3; r 1 , r 2 ∈ (1, +∞), if N = 2; +2 , p−1 ), r r 1 , r 2 ∈ (1, +∞) or r 1 = 1, r 2 = +∞, if N = 1. Hence, 1 r 1

  F1 (v1 ) − F1 (v2 ) 4 v1 − v2  + |a| ≤C 2





 |b| v1− + v2− (v1 − v2 )− + 2

Ω





 v1+ + v2+ (v1 − v2 )+

Ω



  4 v1 − v2  + |a| + |b| v − + v − 2 + v + + v + 2 · v1 − v2 L2 ≤C 1 2 L 1 2 L 2  |a| + |b|  4 v1 − v2  + v1  + v2  · v1 − v2  ≤C α    2(|a| + |b|) 4 + 5 v1 − v2 . ≤ C R + u · v1 − v2  ≤ C α And also observe that for ∀v ∈ E, v = 1,

(4.38)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3806

    +

   v − v + v  ≤ v + − v + 2 · vL2 1 2 1 2 L   Ω

 



=

v1+ − v2+

2



v1+ ≥v2+

≤ α− 2 1



+

v2+ − v1+

1 2

2

· vL2

v2+ ≥v1+

 

 (v1 − v2 )2 +

v1+ ≥v2+

1 2

v2+ ≥v1+

= α− 2 v1 − v2 L2 ≤ 1

(v1 − v2 )2

1 v1 − v2 , α

(4.39)

so we get       −        v − v − v  =  (−v2 )+ − (−v1 )+ v  ≤ 1 v1 − v2 . 1 2    α  Ω

Ω

Notice that p − 2 < ∗

∗

(4.40)

4 N −2 ,

choose q ∈ ( N 4−2 , 2∗ ). Indeed, ∃ > 0 sufficiently small, q

∀q ∈ [2 −, 2 ], < 2. By Proposition 4.4 the operator fs (x, ·) : Lq (Ω) → L p−2 (Ω): q u −→ fs (x, u) is continuous for q ≥ 1 and take r1 , r2 , r3 > 1, r11 + r12 + r13 = 1, r1 = p−2 , N (p−2) q

N (p−2) < 2, q ∈ [2∗ − , 2∗ ], r3 = βr2 , q q r1 = p−2 , q > p − 2, q ∈ [1, +∞), β ∈

β∈(

1 p−2 2N N −2 (1− q )−1

, N2N −2 (1 −

p−2 q )

− 1), if N ≥ 3;

(0, +∞), if N = 1, 2. Therefore,

 

(−Δ + α)−1 f (x, v1 ) − f (x, v2 ) 1 ≤

 

(−Δ + α)−1 fs x, v2 + t(v1 − v2 ) (v1 − v2 ) dt

0

1 =

sup 0

v∈E,v=1

       fs x, v2 + t(v1 − v2 ) (v1 − v2 )v dt   Ω

  6 max fs x, v2 + t(v1 − v2 ) r · v1 − v2  ≤C L 1 t∈[0,1]

7 v1 − v2 . ≤C

(4.41)

Combining (4.39), (4.40) and (4.41),

 

J (a,b,t) (v1 ) − J(a,b,t) (v2 ) 7 v1 − v2  + ≤ v1 − v2  + C 8 v1 − v2 . ≤C Consequently,

 1 |a| + |b| v1 − v2  α (4.42)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3807



F2 (v1 ) − F2 (v2 ) ≤

   J(a,b,t) (v2 )2 · J(a,b,t) (v1 ) − J(a,b,t) (v2 )   J(a,b,t) (v1 )2 · J(a,b,t) (v2 )2

  J(a,b,t) (v1 ) + J(a,b,t) (v2 ) 

 · J(a,b,t) (v1 ) − J(a,b,t) (v2 )   2 J(a,b,t) (v1 ) · J(a,b,t) (v2 )





2 1  = · J(a,b,t) + (v1 ) − J(a,b,t) (v2 )    2 J(a,b,t) (v1 ) J(a,b,t) (v1 ) · J(a,b,t) (v2 )

+

≤

48 

J  (a,b,t) (v1 ) − J(a,b,t) (v2 ) ≤ C9 v1 − v2 . 2 (M δ)

(4.43)

It follows from the assumption (I ) that ∃C > 0,     F (x, s) ≤ C 1 + |s|p

(4.44)

for all x ∈ Ω and s ∈ R. Thereby,

F1 (v1 )F2 (v1 ) − F1 (v2 )F2 (v2 )



   ≤ F1 (v1 ) − F1 (v2 ) F2 (v1 ) + F2 (v1 ) − F2 (v2 ) F1 (v2 ) 

 

 = F1 (v1 ) − F1 (v2 ) · F2 (v1 ) + F1 (v2 ) · F2 (v1 ) − F2 (v2 ) 

5 4C |a| + |b| ∗ p 2 ≤ · v1 − v2  + C|Ω| + C v2  + v2  C9 · v1 − v2  Mδ 2α 

 2  p |a| + |b|  4C5 ∗ 9 · v1 − v2  + C|Ω| + C u + R + u + R C ≤ Mδ 2α 10 v1 − v2  ≤C

(4.45)

and we arrive the desired conclusion. 2 Proof of Theorem 4.2. Consider the following equation on E,  (σ(t, u))   J(a,b,t) d σ(t, u) = −∂t J(a,b,t) σ(t, u) ,  dt J(a,b,t) (σ(t, u))2

σ(0, u) = u, t ∈ [0, 1], (4.46)

C C where C is given by (see [5]) and assume u ∈ J(a,b) , J(a,b) = {u ∈ E : J(a,b) (u) ≤ C}, Lemma 4.7. J (v) Since ∂t J(a,b,t) (v) J (a,b,t)(v)2 is local Lipschitz, there is a t0 > 0 such that the solution (a,b,t)



C of (4.46) exists for any initial value u satisfying u ∈ J(a,b) for t ∈ [0, t0 ].

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3808

Notice that   d J(a,b,t) σ(t, u) = dt

!

 J(a,b,t)

"   d   σ(t, u) , σ(t, u) + ∂t J(a,b,t) σ(t, u) = 0, dt

we have   J(a,b,t) σ(t, u) ≤ C

if and only if J(a,b) (u) ≤ C. −

(4.47)

−bs As f (x, s) ∈ C(Ω × R, R) and lims→∞ f (x,s)−as = 0, uniformly a.e on Ω, we s obtain that ∃C3 > 0, |f (x, s)| ≤ C3 |s| for ∀x ∈ Ω, ∀s ∈ R, and this yields that |F (x, s)| ≤ C4 s2 for ∀x ∈ Ω, ∀s ∈ R. Consequently,

   ∂t J(a,b,t) (u) ≤ C4 + |a| + |b| u2 ≤ C5 u2 (4.48) 2 +

Ω

and by (4.32)

 (σ(t, u))

  J(a,b,t) σ(t, u)2

∂t J(a,b,t) σ(t, u)

≤ C5  

2 J (σ(t, u)) J (σ(t, u)) (a,b,t)

(a,b,t)

C5

σ(t, u) . ≤ δ

(4.49)

Notice that (4.46) derives t

 (σ(s, u))   J(a,b,s) ∂s J(a,b,s) σ(s, u) ds  J(a,b,s) (σ(s, u))2

σ(t, u) − u = − 0

(4.50)

and Hahn–Banach theorem yields that ∃ v ∈ E, v = 1, 

 σ(t, u), v = σ(t, u) ,

(4.51)

by Fubini theorem and (4.49) we obtain

σ(t, u) ≤ u +

t 0

C5 ≤ u + δ

  |∂s J(a,b,s) (σ(s, u))|  · J(a,b,s) σ(s, u) ds  2 J(a,b,s) (σ(s, u)) t



σ(s, u) ds

(4.52)

0

and by Gronwall inequality

C

σ(t, u) ≤ u · e δ5 t .

(4.53)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3809

Repeat above procedure we deduce that (4.53) holds for ∀t ∈ [0, 1]. Now we define a map: Φ : u → σ(1, u),



C which is a homeomorphism between the level sets of J(a,b) and J C . Thereby,

  ∼  C Hq E, J(a,b) = Hq E, J C ,

∀q ∈ N

(4.54)

and the proof is complete. 2 Remark 4.9. The assumptions of Theorem 4.2 differ from those of Theorem 3.2 given by [9]. 5. Proof of Theorem 3.1 In this section we present the proof of Theorem 3.1. Arguing indirectly, assume that under the hypotheses of Theorem 3.1, for any solution u0 (x) of (3.1), b1 ≤ u0 (x) ≤ a1 . Then by (3.3), |f  (u0 (x))| ≤ C1 + C2 |u0 (x)|p−2 ≤ C1 + C2 dp−2 , d = max{a1 , −b1 }. This yields that |f  (s)| ≤ C1 + C2 dp−2 for ∀s ∈ [−d, d]. Obviously, we can find m > 0 such that f (s) + ms is strictly monotone increasing on [−d, d]. An argument quite similar to [8] shows that the operator K = (−Δ + mI)−1 g(u) is strongly order preserving on the order interval [b1 , a1 ] ∈ X, i.e., u > v ⇒ K(u)  K(v), for ∀u, v ∈ [b1 , a1 ], where g(u) = f (u) + mu, X = Cν1 (Ω) = {u ∈ C 1 (Ω) : ∂u ∂ν |∂Ω = 0}. Observe that we usually make the assumption of energy functional with finitely many critical points since aiming at finding infinitely many critical points. However, the hypothesis that J has only finitely many isolated critical points is actually redundant for the proof of mountain pass theorem in order intervals due to [12]. Arbitrarily choose b∗ ∈ (b1 , b2 ) and a∗ ∈ (a2 , a1 ). Set u = b∗ and u = a∗ . Observe that {u, u} is a pair of strict sub-solution and super-solution of ∇J = 0, so [b∗ , a∗ ] is a closed convex set in X and hence deg(∇J, [b∗ , a∗ ], 0) is well defined, ∇J = I − K. In terms of Theorem 1.1 given by [12],     deg ∇J, b∗ , a∗ , 0 = 1.

(5.1)

According to the assumption, b1 ≤ min u(x) ≤ max u(x) ≤ a1 , u∈KJ

u∈KJ

(5.2)

and also observe that by the arbitrariness of b∗ ∈ (b1 , b2 ) and a∗ ∈ (a2 , a1 ), ∂[b∗ , a∗ ] ∩ KJ = ∅, we infer that u1 = b1 and u2 = a2 are isolated critical points of J on X, where ∗ KJ = {u ∈ E : J  (u) = 0}. Define K∗ = [b∗ , a∗ ] ∩KJ , then there exists δ ∗ > 0, K∗δ ∩X ⊂ ∗ ∗ [b∗ , a∗ ]◦ , K∗δ = {x ∈ E : dist(x, K∗ ) < δ ∗ }. Note that ([b∗ , a∗ ]\(K∗δ ∩ X)) ∩ KJ = ∅, we have,

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C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

  ∗ deg ∇J, K∗δ ∩ X, 0 = 1.

(5.3)

Therefore, there exists δ ∈ (0, δ ∗ ),   deg ∇J, K∗δ , 0 = 1.

(5.4)

The combination of (5.2) and the regularity of solutions of elliptic equations shows that KJ is bounded, so there exist c1 < 0 and c2 > 0, c1 < inf J(u) ≤ sup J(u) < c2 . u∈KJ

(5.5)

u∈KJ

∀δ ∈ (0, δ0 ), ∃ε∗ = Since J satisfies the (PS) condition, by Theorem 2.1, ∃δ0 ∈ (0, δ), ∗ 1 2δ (J), R), ε (δ) ∈ (0, 1), ∀ε ∈ [0, ε ], there exists a functional Jε ∈ C (E, R) ∩ C 2 (K(c 1 ,c2 ) δ K(c1 ,c2 ) (Jε ) ⊂ K(c (J). In addition, 1 ,c2 ) ∗

∞      δ deg ∇J, K(c (J), 0 = (−1)q rank Hq J c2 , J c1 . 1 ,c2 )

(5.6)

q=0

Observe that b1 , a1 are all nondegenerate, and ind(J, b1 ) = dl , ind(J, a1 ) = dj are sufficiently small, ∀δ ∈ (0, δ0 ), K δ ∩ B(b1 , δ) ∩ B(a1 , δ) = ∅. even, then for δ0 ∈ (0, δ) ∗ Hence,     deg ∇J, B(b1 , δ), 0 = deg ∇J, B(a1 , δ), 0 = 1.

(5.7)

As dk is even, by using Theorem 4.2, (5.6) yields   δ deg ∇J, K(c (J), 0 = 1. 1 ,c2 )

(5.8)

Consequently, we derive a contradictory equality   δ 1 = deg ∇J, K(c (J), 0 1 ,c2 )       = deg ∇J, B(b1 , δ), 0 + deg ∇J, K∗δ , 0 + deg ∇J, B(a1 , δ), 0 = 1 + 1 + 1.

(5.9)

So there must exist a solution of (3.1), denoted by u∗ (x), s.t. maxx∈Ω u∗ (x) > a1 or minx∈Ω u∗ (x) < b1 . Now we claim that there exist at least two solutions u∗1 (x), u∗2 (x), s.t. maxx∈Ω u∗i (x) > a1 or minx∈Ω u∗i (x) < b1 , i = 1, 2. Choose ai+1 < a∗i+1 < a∗i < ai , i odd, and define the function:

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

⎧ ⎪ ⎨ f (s), fi,− (s) = fi∗ (s), ⎪ ⎩ f (a∗i ),

3811

s ≤ a∗i+1 , a∗i+1 < s < a∗i ,

(5.10)

a∗i ,

s≥

where fi,− (s) ∈ C 1 (R, R), s.t., fi∗ (s) < αs for ∀s ∈ (a∗i+1 , a∗i ), fi∗ (a∗i+1 ) = f (a∗i+1 ), fi∗ (a∗i ) = f (a∗i ). Consider the equation: 

x ∈ Ω, x ∈ ∂Ω,

−Δu + αu = fi,− (u), ∂u ∂ν = 0,

(5.11)

and the corresponding functional denoted by 1 2

Ji,− (u) =

 |∇u|2 +



α 2

 u2 −

Ω

Ω

Fi,− (u), Ω

u Fi,− (u) = 0 fi,− (s)ds. Note that lims→−∞ f (s) s = f∞ ∈ (λk , λk+1 ), lims→+∞ so (f∞ , 0) ∈ / Σ, and an argument analogous to [10] yields Cq (J(f∞ ,0) , 0) ∼ = 0,

∀q ∈ N,

fi,− (s) s

= 0,

(5.12)

and hence, by Theorem 4.2, Cq (Ji,− , ∞) ∼ = 0,

∀q ∈ N.

(5.13) δ∗

δ∗

i i Define Ki,− = [b∗ , a∗i ] ∩ KJ , then there exists δi∗ > 0, Ki,− ∩ X ⊂ [b∗ , a∗i ]◦ , Ki,− =

δ∗

i ∩ X)) ∩ Ki,− = ∅, we have {x ∈ E : dist(x, Ki,− ) < δi∗ }. Since ([b∗ , a∗i ]\(Ki,−

  δi∗ deg ∇Ji,− , Ki,− ∩ X, 0 = 1.

(5.14)

Then there exists δ i ∈ (0, δi∗ ),   δ i deg ∇Ji,− , Ki,− , 0 = 1.

(5.15)

We claim that there exists a solution u∗i,− (x) of (5.11), minx∈Ω u∗i,− (x) < b1 , maxx∈Ω u∗i,− (x) ≤ a∗i . Suppose by contradiction, b1 ≤

min u(x) ≤ max u(x) ≤ a∗i .

u∈KJi,−

u∈KJi,−

(5.16)

As (5.16) and the regularity of solutions of elliptic equations allude to boundedness of KJi,− , there exist ci,1 < 0 and ci,2 > 0, ci,1 <

inf

u∈KJi,−

Ji,− (u) ≤

sup u∈KJi,−

Ji,− (u) < ci,2 .

(5.17)

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C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

It is easy to check that Ji,− satisfies the (PS) condition. Once again from Theorem 2.1, ∈ (0, δ i ), ∀δ ∈ (0, δ0i ), ∃ε∗ = ε∗ (δ) ∈ (0, 1), ∀ε ∈ [0, ε∗ ], there exists a functional 2δ δ Jε ∈ C 1 (E, R) ∩ C 2 (K(c (Ji,− ), R), K(ci,1 ,ci,2 ) (Jε ) ⊂ K(c (Ji,− ). Moreover, i,1 ,ci,2 ) i,1 ,ci,2 ) ∃δ0i

∞     ci,2 ci,1  δ deg ∇Ji,− , K(c (J ), 0 = (−1)q rank Hq Ji,− , Ji,− . i,− i,1 ,ci,2 )

(5.18)

q=0

Resorting to Theorem 4.2, the combination of (5.13) and (5.18) yields   δ deg ∇Ji,− , K(c (Ji,− ), 0 = 0. i,1 ,ci,2 )

(5.19)

Thereby,   δ 0 = deg ∇Ji,− , K(c (Ji,− ), 0 i,1 ,ci,2 )     δ i ,0 = deg ∇Ji,− , B(b1 , δ), 0 + deg ∇Ji,− , Ki,− = 1 + 1.

(5.20)

It is impossible. The claim follows. A similar argument gets a solution u∗i,+ (x) of the following equation 

x ∈ Ω, x ∈ ∂Ω,

−Δu + αu = fi,+ (u), ∂u ∂ν = 0,

(5.21)

s.t., maxx∈Ω u∗i,+ (x) > a1 , minx∈Ω u∗i,+ (x) ≥ b∗i by the cut-off function: ⎧ ∗ ⎪ ⎨ f (bi ), fi,+ (s) = fi∗ (s), ⎪ ⎩ f (s),

s ≤ b∗i , b∗i < s < b∗i+1 ,

(5.22)

s ≥ b∗i+1 ,

bi < b∗i < b∗i+1 < bi+1 , i odd, where fi,+ (s) ∈ C 1 (R, R), s.t., fi∗ (s) > αs for ∀s ∈ (b∗i , b∗i+1 ), fi∗ (b∗i ) = f (b∗i ), fi∗ (b∗i+1 ) = f (b∗i+1 ). The corresponding functional is denoted by Ji,+ (u) =

1 2

 |∇u|2 + Ω

α 2



 u2 −

Ω

Fi,+ (u), Ω

u where Fi,+ (u) = 0 fi,+ (s)ds. Take i → +∞, we get u∗i,+ → u∗+ , u∗i,− → u∗− ,

on X,

(5.23)

C. Li / Journal of Functional Analysis 267 (2014) 3783–3814

3813

s.t. u∗+ (x) ≥ 0, u∗− (x) ≤ 0, x ∈ Ω. As u∗i,+ (x) and u∗i,− (x) are solutions of (3.1), u∗+ (x) and u∗− are also solutions of (3.1). Denoted u∗+ (x), u∗− (x) by u∗1 (x), u∗2 (x) respectively. We claim there exists the third solution, denoted by u∗3 (x), s.t. maxx∈Ω u∗3 (x) > a1 or minx∈Ω u∗3 (x) < b1 . By way of negation, then above argument shows that u∗i,+ (x) = u∗+ (x), u∗i,− (x) = u∗− (x), and there exists r > 0, s.t. B(u∗+ , r)\{u∗+ } ∩ KJ = ∅, B(u∗− , r)\{u∗− } ∩ KJ = ∅. By (5.20),     deg ∇J, B u∗− , r , 0 = −2.

(5.24)

Proceeding along the same lines, we have     deg ∇J, B u∗+ , r , 0 = −2.

(5.25)

Thereby,   δ 1 = deg ∇J, K(c (J), 0 1 ,c2 )     = deg ∇J, B(b1 , δ), 0 + deg ∇J, K∗δ , 0       + deg ∇J, B(a1 , δ), 0 + deg ∇J, B u∗+ , r , 0     + deg ∇J, B u∗− , r , 0 = 1 + 1 + 1 + (−2) + (−2).

(5.26)

A contradiction! The proof is complete. 2 Quite similarly we get Theorem 3.2, Theorem 3.3, Corollary 3.4, Corollary 3.5 and Corollary 3.6. Remark 5.1. Above theorems derive more solutions of (3.1) for ODE case. Acknowledgments I’m greatly indebted to the referees for carefully reading the manuscript, pointing out missing references and giving valuable suggestions. And I also appreciate for strong supports by NSFC (11471319) and BCMIIS (Beijing Center for Mathematics and Information Interdisciplinary Sciences). 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] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris, Sér. I Math. 317 (5) (1993) 465–472. [3] K.C. Chang, H 1 versus C 1 isolated critical points, C. R. Acad. Sci. Paris, Sér. I Math. 319 (5) (1994) 441–446.

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