Multiplicity of solutions for a class of elliptic boundary value problems

Nonlinear Analysis 71 (2009) 2606–2613

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Multiplicity of solutions for a class of elliptic boundary value problems Xiaoming He a,∗ , Wenming Zou b a

College of Science, Central University for Nationalities, Beijing 100081, PR China

b

Department of Mathematical Sciences, Tsinghua University Beijing 100084, PR China

article

abstract

info

Article history: Received 17 February 2008 Accepted 15 January 2009

In this paper we study the existence of infinitely many solutions for a class of elliptic boundary value problems. The existence results include both superlinear case and an asymptotically linear case. Our arguments are based on a recently given new Fountain Theorem due to Zou [W. Zou, Variant Fountain Theorems and their applications, Manuscripta Math. 104 (2001) 343–358]. © 2009 Elsevier Ltd. All rights reserved.

MSC: 35J60 35J25 Keywords: Elliptic boundary value problems New fountain theorems Superlinear and asymptotically linear Minimax method

1. Introduction In this paper we are concerned with the existence for multiplicity of solutions of the elliptic problem



−∆u + a(x)u = g (x, u) u=0

in Ω on ∂ Ω ,

(1.1)

¯ × R, R), a ∈ Lp (Ω ), p > N /2. where Ω is a bounded domain of RN (N ≥ 3) whose boundary is a smooth manifold, g ∈ C (Ω Problems of the form (1.1) have been extensively studied via the critical point theory in recent years. For example, in [1], Li and Willem proved the existence of a nontrivial solution of (1.1) by application of a local linking method. Jiang and Tang [2] used the Li–Willem local linking theorem [1] to obtain a nontrivial solution for problem (1.1) under a superquadratic condition. In [3], the linking theorem of Rabinowitz is applied to obtain a nontrivial solution for (1.1). Brezis and Nirenberg [4] investigated problem (1.1) with g (x, u) ≡ λg (u) and obtained two nontrivial solutions for λ large enough by searching for critical points in the presence of splitting, which is related to the results of Liu and Li [5]. We note that in that paper Liu and Li studied a problem similar to problem (1.1) via an earlier version of a local linking theorem. When a(x) ≡ 0, problem (1.1) simplifies as 

−∆u = g (x, u) u=0

in Ω on ∂ Ω .

(1.2)

Many existence results for (1.2) have appeared in the literature during the past thirty years; we refer the reader to [6,3] for a survey. The existence of infinitely many solutions for (1.2) is generally obtained using the Symmetric Mountain Pass Theorem [7,6] and the Fountain Theorem [3]. For example, in [3], the following existence theorem is given for (1.2).

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (X. He).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.111

X. He, W. Zou / Nonlinear Analysis 71 (2009) 2606–2613

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Theorem A ([3, Theorem 3.7]). Assume that |Ω | < ∞ and

¯ × R) and for some 2 < p < 2∗ = (f1 ) g ∈ C (Ω |g (x, u)| ≤ c (1 + |u|

p−1

2N N −2

, c > 0,

);

(f2 ) there exists α > 2 and R > 0 such that |u| ≥ R ⇒ 0 < α G(x, u) ≤ ug (x, u); (f3 ) g (x, −u) = −g (x, u), ∀x ∈ Ω , ∀u ∈ R. Then problem (1.2) has a sequence of solutions (uk ) such that ϕ(uk ) → ∞, k → ∞, where ϕ(u) := and G(x, u) :=

Ru 0

R

g (x, s)ds.

Ω

(|∇ u|2 /2 − G(x, u))dx

Theorem A is proved using the classical Fountain Theorem [3, Theorem 3.6]; it can also be proved using the Symmetric Mountain Theorem [6, Theorem 9.38]. Usually, in order to obtain the existence of infinitely many solutions for superlinear problems, the nonlinearity g is generally required to satisfy the superquadraticity condition (f2 ) due to Ambrosetti and Rabinowitz [7]. However, there are many functions which are superlinear but for which it is not possible to satisfy (f2 ) for any α > 2. For example g (x, u) ≡ θ |u|θ −2 u + (θ − 1)|u|θ−3 u cos2 u − |u|θ−1 sin 2u, u ∈ R \ {0}

(1.3)

with θ > 2. In the present paper, we shall revisit problem (1.1) and study the weak superlinear cases without (f2 ) and the asymptotically linear case. It is well known that the main role of (f2 ) is to guarantee the boundedness of all (PS) (or (PS)∗ ) sequences of the corresponding functional. So, without (f2 ), the problem becomes quite different and complex, in particular, if a(x) 6≡ 0 and may be sign-changing in Ω . Then the spectrum of the operator −∆ + a with the Dirichlet boundary condition has a negative part and a positive part, and 0 can be an eigenvalue of σ (−∆ + a). So, the main obstacle is giving the proof of the boundedness of the (PS) (i.e., Palais–Smale) sequence for the corresponding energy functional with the strong indefinite nature. It is also not easy to derive a (PS) sequence for the asymptotically linear case. However, by virtue of the new theory established in [8] we can easily obtain a bounded (PS) sequence directly from the new Fountain Theorem for the modified functional and thereby have a sequence of critical points which provides a nontrivial solution of (1.1). To the best of the author’s knowledge, this method has not yet been applied to problem (1.1). 1.1. The superlinear case Here and in the sequel, we denote by the letter c the various positive constants where the exact values are irrelevant. We adopt the following assumptions.

(A1 ) (A2 ) (A3 ) (A4 )

There exists µ > 2 such that c |u|µ ≤ zg (x, z ), ∀(x, z ) ∈ Ω × R. −2 |g (x, z )| ≤ c (1 + |z |p ) for all (x, z ) ∈ Ω × R, where 1 < p < NN − . 2

There exist constants τ > N2N p and L > 0 such that zg (x, z ) − 2G(x, z ) ≥ c |z |τ for all |z | ≥ L and all x ∈ Ω . +2 0 is an eigenvalue of −∆ + a with the Dirichlet boundary condition.

Theorem 1.1. Assume that (A1 )–(A4 ) hold and G is even in z. Then problem (1.1) has infinitely many nontrivial solutions. Next, we consider the potential satisfying some local conditions at zero and at infinity.

(S1 ) There exist ν > p > 2, ν < N2N , and constants d1 , d2 , d3 > 0 such that d1 |z |ν ≤ g (x, z )z ≤ d2 |z |ν + d3 |z |p for all −2 (x, z ) ∈ Ω × R. (S2 ) zg (x, z ) − 2G(x, z ) > 0 for all (x, z ) 6= (0, 0). (S3 ) There exists γ0 > 2 such that lim inf

|z |→∞

zg (x, z ) G(x, z )

≥ γ0

uniformly for x ∈ Ω . (S4 ) There exists an α > p and a d0 > 0 such that lim inf

zg (x, z ) − 2G(x, z )

z →0

|z |α

≥ d0

uniformly for x ∈ Ω . Theorem 1.2. Assume that (S1 )–(S4 ) hold and G is even in z. Then (1.1) has infinitely many solutions.

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X. He, W. Zou / Nonlinear Analysis 71 (2009) 2606–2613

1.2. The asymptotically linear case We make the following assumptions.

(B1 ) G(x, z ) = 12 β|z |2 + H (x, z ), where β 6∈ σ (−∆ + a) a constant; σ denotes the spectrum. (B2 ) There exist δi ∈ (1, 2), i = 1, 2, and b1 , b2 > 0 such that b1 |z |δ1 ≤ H (x, z ), H (x, 0) ≡ 0, |Hz (x, z )| ≤ b2 |z |δ2 −1 , for all ( x, z ) ∈ Ω × R . Theorem 1.3. Assume (A4 ), (B1 ) and (B2 ) hold and that H is even in z. Then problem (1.1) has infinitely many nontrivial solutions. Remark 1.1. In [8], the existence of infinitely many solutions of the following nonlinear Schrödinger equations:

− ∆u + b(x)u = f (x, u),

x ∈ RN

(1.4)

and



−∆u = f (x, u) + g (x, u) u=0

in Ω on ∂ Ω ,

(1.5)

is considered. But the nonlinearities are required to satisfy some monotonicity conditions. i.e., (cf. [8, Theorem 3.1]), or assumptions.

f (x,u) u

,

g (x,u) u

f (x,u) u

is an increasing function

are decreasing functions ([8, Theorem 3.2]). However, in this paper we do not need these

Remark 1.2. The results of this paper improve and extend Theorem 2.18 of [3] and Theorem 3.7 of [3]. On one hand, in Theorem 1.1 we replace the superquadraticity growth condition (f2 ) by a weak form. In fact, if we consider



G(x, u) = V (x) |u|µ + (µ − 2)|u|µ−ε sin2



|u|ε ε



,

¯ ), and V (x) > 0 for all x ∈ Ω ¯ . Then it is easy to see that G satisfies the conditions (A1 )–(A3 ) where ε ∈ (0, µ − 2), V ∈ C (Ω in Theorem 1.1 with µ = 3, p = 2, ε = 0.1, τ = 2.9, N = 5, but G does not satisfy (f2 ). On the other hand, Theorem 1.3 deals with the asymptotically linear case which is not contained in [3]. Remark 1.3. Our results can be viewed as the complements of Theorem 1 of [2]. In fact, in [2] only one nontrivial solution of (1.1) is obtained using the local linking theorem [1]. In this paper infinitely many nontrivial solutions are obtained via the new Fountain Theorem. Moreover, Theorem 1.2 gives a new superquadraticity condition which is quite a bit weaker than (f2 ). And in Theorem 1.3 we consider the case where the nonlinearity is asymptotically linear at infinity which is not included in [2]. The paper is organized as follows. In Section 2 we investigate problem (1.1) in the superlinear cases and prove Theorems 1.1 and 1.2. In Section 3 we deal with the asymptotically linear case and prove Theorem 1.3. 2. Proofs of Theorems 1.1 and 1.2 We first introduce some preliminary notation. Let E be a Banach space with the norm k · k and E =

L

j∈N

Xj with

j=k Xj and Bk = {u ∈ Yk : kuk ≤ ρk }, Nk = {u ∈ Zk : kuk = rk } for j=0 Xj , Zk := ρk > rk > 0. Consider the C 1 -functional Iλ : E → R defined by

dim Xj < ∞ for any j ∈ N. Set Yk = Iλ (z ) = A(z ) − λB(z ),

Lk

L∞

λ ∈ [1, 2].

Assume that: (C1 ) Iλ maps bounded sets to bounded sets uniformly for λ ∈ [1, 2]. Furthermore, Iλ (−z ) = Iλ (z ) for all (λ, z ) ∈ [1, 2] × E . (C2 ) B(z ) ≥ 0 for all z ∈ E ; A(z ) → ∞ or B(z ) → ∞ as kz k → ∞; or (C02 ) B(z ) ≤ 0 for all z ∈ E ; B(z ) → −∞ as kz k → ∞. For k ≥ 2, define Γk := {γ ∈ C (Bk , E ) : γ is odd; γ |∂ Bk = id}, ck (λ) := infγ ∈Γk maxz ∈Bk Iλ (γ (z )), bk (λ) := infz ∈Zk ,kz k=rk Iλ (z ), ak (λ) := maxz ∈Yk ,kz k=ρk Iλ (z ). Theorem 2.1 ([8, Theorem 2.1, p.345]). Assume (C1 ) and (C2 ) (or (C02 )). If bk (λ) > ak (λ) for all λ ∈ [1, 2], then ck (λ) ≥ bk (λ) 0 k k for all λ ∈ [1, 2]. Moreover, for a.e. λ ∈ [1, 2], there exists a sequence {znk (λ)}∞ n=1 such that supn kzn (λ)k < ∞, Iλ (zn (λ)) → 0 and Iλ (znk (λ)) → ck (λ) as n → ∞.

X. He, W. Zou / Nonlinear Analysis 71 (2009) 2606–2613

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Now, we turn to problem (1.1). Define a functional I on the E := H01 (Ω ) by

Z

Z (|∇ u|2 + a(x)u2 )dx − G(x, u)dx 2 Ω Ω Z 1 := (ku+ k2 − ku− k2 ) − G(x, u)dx, 1

I (u) =

2

Ω

where u = u + u0 + u+ ∈ E with u− ∈ E − , u0 ∈ E 0 , u+ ∈ E + , and E − , E 0 , and E + are the spaces spanned by the eigenvectors corresponding to negative, zero, and positive eigenvalues, respectively. Under the conditions of Theorems 1.1–1.3, we know that I is continuously differentiable and finding solutions of (1.1) is equivalent to finding critical points of I in E and −

hI (u), vi =

Z

Z

(∇ u∇v + a(x)uv)dx − g (x, u)v dx Ω Z = (u+ − u− , v) − g (x, u)v dx

0

Ω

Ω

for all u, v ∈ E . Now we define a class of functionals on E by 1

Iλ (z ) :=

2

kz + k2 − λ



1 2

kz − k2 +



Z Ω

G(x, z )dx

:= A(z ) − λB(z ),

λ ∈ [1.2],

where z − ∈ E − , z + ∈ E + . Denote by Xj := span{ej }, j ∈ N, and by |·|s the norm of Ls (Ω ). Evidently, we see that Iλ ∈ C 1 (E , R) for all λ ∈ [1, 2] and the critical points of I1 correspond to the solutions of (1.1). It follows from (A2 ) and (A3 ) that 2N

1+p<

N −2

,

τ 2N < . τ −p N −2

(∗)

Therefore, by the Sobolev inequality there exists a constant C > 0 such that

kukL1 (Ω ) ≤ C kuk,

kukL2 (Ω ) ≤ C kuk,

kukLp+1 ≤ C kuk,

k uk

τ L τ −p

≤ C k uk

(∗∗)

for all u ∈ E . Lemma 2.1. There exists ρk large enough such that ak (λ) := maxz ∈Yk ,kz k=ρk Iλ (z ) ≤ 0 for all λ ∈ [1, 2]. Proof. By (A1 ), we have G(x, z ) ≥ c |z |µ for all (x, z ). Let z ∈ Yk , z = z − + z 0 + z + ; noting that dim Yk < ∞, it is easy to see that Iλ (z ) ≤ 0 for kz k = ρk large enough.  Lemma 2.2. There exist rk > 0, bek → ∞ such that bk (λ) := infz ∈Zk ,kz k=rk Iλ (z ) ≥ bek for all λ ∈ [1, 2]. Proof. Set βk := supz ∈Zk ,kz k=1 |z |p+1 . Then βk → 0 as k → ∞ (cf. [3, Lemma 3.8]). Choose k large enough such that Zk ⊂ E + . By (A2 ), we see that, for any ε > 0, there exists Cε > 0 such that |G(x, z )| ≤ ε|z |2 + Cε |z |p+1 for all (x, z ). Therefore, for ε small enough, Iλ ( z ) ≥

 Let rk :=

1 2

1

1 kz k2 − λε|z |22 − λCε |z |pp+ +1 ≥

1

1/(p−1)

p+1

8c βk

for λ as k → ∞.

4

kz k2 − c βkp+1 kz kp+1 .

. Then for z ∈ Zk with kz k = rk , we have that Iλ (z ) ≥

1 8



1 p+1

8c βk

2/(p−1) := bek → ∞, uniformly



0 e e Lemma 2.3. There exist λn → 1 as n → ∞, {zn (k)}∞ n=1 ⊂ E such that Iλn (zn (k)) = 0, Iλn (zn (k)) ∈ [bk , ck ], where cek := supz ∈Bk I1 (z ).

Proof. It follows immediately from Lemmas 2.1 and 2.2 and Theorem 2.1.



Proof of Theorem 1.1. By Lemma 2.3 it suffices to prove the boundedness of {zn (k)}∞ n=1 . Assume, by contradiction, that we suppose that, up to a subsequence, kzn (k)k → ∞ as n → ∞. In view of (A3 ), we have that zg (x, z ) − 2G(x, z ) ≥ c |z |τ − c ,

for all (x, z ) ∈ Ω × R.

Therefore, 2Iλn (zn (k)) − hIλn (zn (k)), zn (k)i = λn

Z Ω

Z ≥c Ω

[g (x, zn (k))zn (k) − 2G(x, zn (k))] dx

|zn (k)|τ dx − c |Ω |,

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X. He, W. Zou / Nonlinear Analysis 71 (2009) 2606–2613

which yields that

R

Ω

|zn (k)|τ dx → 0 as n → ∞. kzn (k)k

(2.1)

Write zn (k) = zn (k)− + zn (k)0 + zn (k)+ ∈ E − ⊕ E 0 ⊕ E + . It follows from (A2 ), (∗), (∗∗), and the Hölder inequality, that

Z g (x, zn (k))zn (k)+ dx hIλ0 n (zn (k)), zn (k)+ i = kzn (k)+ k2 − λn Ω Z + 2 ≥ kzn (k) k − 2 |g (x, zn (k))| · |zn (k)+ |dx Ω Z Z ≥ kzn (k)+ k2 − c |zn (k)+ |dx − c |zn (k)|p |zn (k)+ |dx Ω

Ω

≥ kzn (k) k − c |zn (k) |1 − c + 2

+

Z

 τp Z  τ −τ p τ + τ− p (|zn (k)| ) · |zn (k) | dx τ p p

Ω

Ω

≥ kzn (k)+ k2 − c kzn (k)+ k − c |zn (k)|pτ · kzn (k)+ k for any n ∈ N. Since τ >

2N p N +2

and N ≥ 3, we have p < τ . So, by (2.1) we get

kzn (k)+ k → 0 as n → ∞. kzn (k)k

(2.2)

By a similar argument, we have

kzn (k)− k → 0 as n → ∞. kzn (k)k

(2.3)

Again by (A3 ), one has zg (x, z ) − 2G(x, z ) ≥ c4 |z | − c5

for all (x, z ) ∈ Ω × R

which implies that 2Iλn (zn (k)) −

hIλ0 n (zn (k)), zn (k)i

Z

[g (x, zn (k))zn (k) − 2G(x, zn (k))] dx = λn Z Ω ≥c |zn (k)|dx − c |Ω | ZΩ ≥ c (|zn (k)0 | − |zn (k)+ | − |zn (k)− |)dx − c |Ω | Ω

≥ c kzn (k)0 k − c (kzn (k)− k + kzn (k)+ k) − c keeping in mind that dim E 0 < ∞ and (∗). Hence, by (2.2) and (2.3), we have

kzn (k)0 k → 0 as n → ∞. kzn (k)k Then we arrive at 1=

kzn (k)k kzn (k)− k + kzn (k)0 k + kzn (k)+ k ≤ → 0 as n → ∞ kzn (k)k kzn (k)k

which is a contradiction. So, {zn (k)} is bounded in E. Then we can assume that zn (k) * wk in E. Note that

kzn (k) − wk k2 = hIλ0 n (zn (k)) − Iλ0 n (wk ), zn (k) − wk i − + λn

Z Ω

Z Ω

a(zn (k) − wk )2 dx

(g (x, zn (k)) − g (x, wk ))(zn (k) − wk )dx.

(2.4)

Taking n → ∞ we have lim kzn (k) − wk k = 0.

n→∞

This means that zn (k) → wk in E and I 0 (wk ) = 0. Hence, I has a critical point wk with I (wk ) ∈ [bek , cek ]. Consequently, we obtain infinitely many solutions since bek → ∞.

X. He, W. Zou / Nonlinear Analysis 71 (2009) 2606–2613

2611

Proof of Theorem 1.2. Under the assumptions (S1 )– (S4 ), the conclusions of Lemmas 2.1–2.3 are still true. To complete the proof, we need only to prove the boundedness of {zn (k)}∞ n=1 . Now by (S3 ), there exists R0 > 0 and ε0 > 0 such that γ0 − ε0 > 2 and g (x, zn (k))zn (k) ≥ (γ0 − ε0 )G(x, zn (k))

for |zn (k)| ≥ R0 .

(2.5)

By (S2 ) and (S4 ), we have that zn (k)g (x, zn (k)) − 2G(x, zn (k)) ≥ c |zn (k)|α

for |zn (k)| ≤ R0 .

(2.6)

On the other hand, Iλn (zn (k)) − 12 Iλ0 n (zn (k)) ≤ c . Hence, by (2.5) and (2.6), and (S1 ), we see that

Z  c ≥

1 2

Ω



zn (k)g (x, zn (k)) − G(x, zn (k)) dx

Z ≥c |zn (k)|≤R0

|zn (k)|α dx,

(2.7)

and

Z  c ≥

1 2

Ω



zn (k)g (x, zn (k)) − G(x, zn (k)) dx



Z ≥c

2

|zn (k)|≥R0

Z ≥c |zn (k)|≥R0

γ0 − ε0



− 1 G(x, zn (k))dx

|zn (k)|ν dx.

(2.8)

Consequently,

Z |zn (k)|≥R0

|zn (k)|p dx ≤ c .

Assumptions (S1 ) and (S4 ) imply that either R ν > α > p or α ≥ ν > p. If p < α < ν , then by (2.7), we have |z (k)|≤R |zn (k)|p dx ≤ c, and for t small enough, n

Z |zn (k)|≤R0

|zn (k)|p dx =

Z |zn (k)|≤R0

0

|zn (k)|(1−t )p |zn (k)|tp dx

Z

α

≤ |zn (k)|≤R0

 (1−αt )p Z

|zn (k)| dx

Z

|zn (k)| dx tpq

≤c |zn (k)|≤R0

|zn (k)| dx tpq

|zn (k)|≤R0

 1q

 1q

≤ c kzn (k)ktp ,

(2.9)

(1−t )p α

where + = 1. If p < ν ≤ α , then by (2.7) and (2.8), we have that 1 q

Z |zn (k)|≤R0

Z

|zn (k)| dx ≤ c kzn (k)k , p

tp

|zn (k)|≤R0

|zn (k)|ν dx ≤ c kzn (k)kt ν .

Combining (2.7)–(2.10), we have the following estimates:

Z kzn (k)+ k2 = λn zn (k)+ g (x, zn (k))dx Z Ω ≤ c (|zn (k)|ν−1 + |zn (k)|p−1 )|zn (k)+ |dx Ω

≤ c kzn (k)+ k

Z

+ c kzn (k)+ k

|zn (k)|≥R0

|zn (k)|ν dx +

Z |zn (k)|≥R0

Z |zn (k)|
|zn (k)|p dx +

|zn (k)|ν dx

Z |zn (k)|≤R0

≤ c kzn (k)+ k(c + kzn (k)kt (ν−1) + kzn (k)kt (p−1) ).

 ν−ν 1

|zn (k)|p dx

 p−p 1

(2.10)

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X. He, W. Zou / Nonlinear Analysis 71 (2009) 2606–2613

By a similar argument, we get

kzn (k)− k2 ≤ c kzn (k)− k(c + kzn (k)kt (ν−1) + kzn (k)kt (p−1) ). 0 0 2 0 Since t can be arbitrarily small, if E 0 = {0}, then {zn (k)}∞ n=1 is bounded. If E 6= {0}, then |zn (k) |2 = (zn (k) , zn (k))2 ≤ |zn (k)0ν · |zn (k)|ν 0 , where 1/ν + 1/ν 0 = 1, ν is from (S1 ). Note that dim E 0 < ∞; we observe that

kzn (k)0 k2 ≤ c |zn (k)|2ν ≤ c kzn (k)k2t + c ≤ c + c kzn (k)0 k2t + c kzn (k)+ k2t + c kzn (k)− k2t . Since t > 0 can be small enough, we have that {zn (k)}∞ n=1 is bounded. 3. Proof of Theorem 1.3 We recall the following abstract result which will be used to prove Theorem 1.3. Theorem 3.1 ([8, Theorem 2.2, p.349]). The C 1 -functional Iλ : E → R defined by Iλ (z ) = A(z ) − λB(z ), λ ∈ [1, 2], satisfies the following.

(T1 ) Iλ maps bounded sets to bounded sets uniformly for λ ∈ [1, 2]. Moreover, Iλ (−z ) = Iλ (z ) for all (λ, z ) ∈ [1, 2] × E . (T2 ) B(z ) ≥ 0; B(z ) → ∞ as kz k → ∞ on any finite-dimensional subspace of E. (T3 ) There exist ρk > rk > 0 such that ak (λ) := infz ∈Zk ,kz k=ρk Iλ (z ) ≥ 0 > bk (λ) := maxz ∈Yk ,kz k=rk Iλ (z ) for all λ ∈ [1, 2] and dk (λ) := infz ∈Zk ,kz k≤ρk Iλ (z ) → 0 as k → ∞ uniformly for λ ∈ [1, 2]. Then there exist λn → 1, z (λn ) ∈ Yn such that Iλ0 n |Yn (z (λn )) = 0, Iλn (z (λn )) → ck ∈ [dk (2), bk (1)] as n → ∞. In particular, if {z (λn )} has a convergent subsequence for every k, then I1 has infinitely many nontrivial critical points {zk } ⊂ E \ {0} satisfying I1 (zk ) → 0− as k → ∞. The following two lemmas are preliminaries to proving Theorem 1.3. Lemma 3.1. Under the assumptions of Theorem 1.3, there exists rk small enough such that bk (λ) := maxz ∈Yk Iλ (z ) < 0 uniformly for λ ∈ [1, 2]. Proof. Let z ∈ Yk with z = z − + z 0 + z + ∈ E = E − ⊕ E 0 ⊕ E + . Then Iλ (z ) =

1 2 1

kz + k2 −

λ 2

λ

kz − k2 − λ

Z

λβ

Ω

G(x, z )dx

Z |z |2 dx − λ H (x, z )dx 2 2 2 Ω Ω Z 1 λ λβ 2 ≤ kz + k2 − kz − k2 − |z |2 − λc |z |δ1 dx =

≤

2 1 2

kz + k2 −

kz − k2 −

2

Z

2

Ω

δ1

kz k − c kz k − c kz k − c kz k , + 2

− 2

2

since dim Yk < ∞. So, for kz k = rk small enough, we get Iλ (z ) < 0 for all λ ∈ [1, 2].



Lemma 3.2. ak (λ) := infz ∈Zk ,kz k=ρk Iλ (z ) ≥ 0, where ρk is small enough.

{Rej } ⊂ E + and λk > 2β . Define by ηk := supz ∈Zk ,z 6=0 |z |δ2 /kz k. Then ηk → 0 as k → ∞. Note that for any z ∈ Zk , λk |z | ≤ kzk k2 . So, we have Z Z 1 λβ Iλ (z ) = kz k2 − |z |2 dx − λ H (x, z )dx 2 2 Ω   Ω Z 1 β ≥ − kz k2 − c |z |δ2 dx 2 λk Ω   1 β δ2 2 ≥ − kz k − c ηk kz kδ2 2 λk Proof. Without loss of generality, we assume that k is large enough such that Zk :=

L∞

j =k

2 2

δ

= c ηk2 (ρk )δ2  1/(2−δ2 ) δ for kz k = ρk = 4c λk ηk2 /(λk − 2β) . Obviously, ρk → 0 as k → ∞.



X. He, W. Zou / Nonlinear Analysis 71 (2009) 2606–2613

2613

Proof of Theorem 1.3. Under the assumptions of Theorem 1.3, it is easy to see that (T1 ), (T2 ) of Theorem 3.1 hold. By Lemmas 3.1 and 3.2 we see that Condition (T3 ) is also satisfied. Hence, by Theorem 3.1 we have that there exist λn → 1, z (λn ) := zn such that Iλ0 n |Yn (zn ) = 0, Iλn (zn ) → ck ∈ [dk (2), bk (1)] as n → ∞. We shall prove that {zn } is bounded. Assume ±

z0

by contradiction that, up to a subsequence, kzn k → ∞. Then we can suppose that kzzn k * w, kzz k * w ± , kzn k * w 0 . n n n Since

hzn , ϕn i − λn hzn , ϕn i − λn β +

−

Z Ω

zn ϕn dx − λn

Z Ω

Hz (x, zn )ϕn dx = 0,

(3.1)

where ϕn = ϕ|Yn , ϕ = i=1 ti ei ∈ E . By (B2 ) and the Hölder inequality, we have

P∞

1

kzn k

λn

Z Ω

Hz (x, zn )ϕn dx ≤

≤

Z

c

kzn k

Ω

|zn |δ2 −1 · |ϕn |dx δ

c

Z

kzn k

Ω δ −1

=c

|zn |δ22 

=c

|zn |

2 (δ2 −1) δ − 2 1 dx

 δ2δ−1 Z 2

Ω

 δ1 2 |ϕn |δ2 dx

|ϕn |δ2

kzn k  |zn |δ2 δ2 −1 |ϕn |δ2 · →0 kzn k kzn k2−δ2

(3.2)

as n → ∞. It follows from (3.1), (3.2) that

hw + , ϕi − hw− , ϕi − β

Z Ω

wϕ dx = 0,

∀ϕ ∈ E .

If w 6= 0, then β ∈ σ (−∆ + a), which is a contradiction. If w = 0, by Iλ0 n |Yn (zn ) = 0, we have that 1=

1

kz n k2

Z Ω

Hz (x, zn )(λn zn+ − zn− − zn0 )dx + kzn0 k2



.

But (B2 ) and w = 0 imply that the right side of the above equation tends to zero, which gives a contradiction. Therefore, {zn } is bounded. By a standard argument, this yields a critical point z k of I such that I (z k ) ∈ [dk (2), ck (1)]. Since dk (2) → 0− , we can obtain infinitely many critical points. References [1] [2] [3] [4] [5] [6]

S. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995) 6–32. Q. Jiang, C. Tang, Existence of a nontrivial solution for a class of superquadratic elliptic problems, Nonlinear Anal. 69 (2008) 523–529. M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. XLIV (1991) 939–963. J. Liu, S. Li, Some existence theorems on multiple critical points and their applications, Kexue Tongbao 17 (1984) 1025–1027 (in Chinese). P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Reg. Conf. Series Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986. [7] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [8] W. Zou, Variant Fountain Theorems and their applications, Manuscripta Math. 104 (2001) 343–358.