Infinitely Many Solutions of Superlinear Elliptic Equation

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 769620, 6 pages http://dx.doi.org/10.1155/2013/769620

Research Article Infinitely Many Solutions of Superlinear Elliptic Equation Anmin Mao and Yang Li School of Mathematical Sciences, Qufu Normal University, Jining, Shandong 273165, China Correspondence should be addressed to Anmin Mao; [email protected] Received 4 January 2013; Revised 27 April 2013; Accepted 28 April 2013 Academic Editor: Wenming Zou Copyright © 2013 A. Mao and Y. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Via the Fountain theorem, we obtain the existence of infinitely many solutions of the following superlinear elliptic boundary value problem: −Δ𝑢 = 𝑓(𝑥, 𝑢) in Ω, 𝑢 = 0 on 𝜕Ω, where Ω ⊂ R𝑁 (𝑁 > 2) is a bounded domain with smooth boundary and f is odd in u and continuous. There is no assumption near zero on the behavior of the nonlinearity f, and f does not satisfy the AmbrosettiRabinowitz type technical condition near infinity.

1. Introduction Consider the following nonlinear problem: −Δ𝑢 = 𝑓 (𝑥, 𝑢) 𝑢=0

in Ω,

on 𝜕Ω,

(1)

which has been receiving much attention during the last several decades. Here Ω ⊂ R𝑁 (𝑁 > 2) is a bounded smooth domain and 𝑓 is a continuous function on Ω × 𝑅 and odd in 𝑢. We make the following assumptions on 𝑓: (𝑆1 ) there exist constants 𝑎1 > 0 and 2𝑁/(𝑁 − 2) = 2∗ > ] > 2 such that 󵄨 󵄨󵄨 ]−1 󵄨󵄨𝑓 (𝑥, 𝑢)󵄨󵄨󵄨 ≤ 𝑎1 (1 + |𝑢| ) ,

∀𝑥 ∈ Ω, 𝑢 ∈ 𝑅;

(2)

𝐹 (𝑥, 𝑢) = ∞, |𝑢| → ∞ 𝑢2

uniformly for 𝑥 ∈ Ω,

(3)

−Δ𝑢 = 𝜆𝑢 + 𝑓 (𝑥, 𝑢)

in Ω, 𝑢 = 0 on 𝜕Ω,

(5)

where Ω ⊂ 𝑅𝑁 is a bounded smooth domain, and assumed

𝑢

(𝑓1 ) 𝑓 ∈ 𝐶1 (Ω × 𝑅, 𝑅), (𝑓2 ) 𝑓(𝑥, 0) = 0 = 𝑓𝑢 (𝑥, 0),

where 𝐹(𝑥, 𝑢) = ∫0 𝑓(𝑥, 𝑢)𝑑𝑥; (𝑆3 ) there exists a constant 𝑏 > 0 such that 𝑓 (𝑥, 𝑢) 𝑢 − 2𝐹(𝑥, 𝑢) lim sup < 𝑏, 𝑢2 + 1 |𝑢| → ∞

Theorem 1. Assume that (𝑆1 )–(𝑆3 ) hold and 𝑓(𝑥, 𝑢) is odd in 𝑢. Then problem (1) possesses infinitely many solutions. Problem (1) was studied widely under various conditions on 𝑓(𝑥, 𝑢); see, for example, [6–10]. In 2007, Rabinowitz et al. [6] studied the problem

(𝑆2 ) 𝐹(𝑥, 𝑢) ≥ 0, for all (𝑥, 𝑢) ∈ Ω × 𝑅, and lim

Note that Costa and Magalh˜aes in [1] introduced a condition similar to (𝑆3 ), which also appeared in [2]. In this paper, we will study the existence of infinitely many nontrivial solutions of (1) via a variant of Fountain theorems established by Zou in [3]. Fountain theorems and their dual form were established by Bartsch in [4] and by Bartsch and Willem in [5], respectively. They are effective tools for studying the existence of infinitely many large or small energy solutions. It should be noted that the P.S. condition and its variants play an important role for these theorems and their applications. We state our main result as follows.

uniformly for 𝑥 ∈ Ω. (4)

(𝑓3 ) |𝑓(𝑥, 𝑢)| ≤ 𝐶(1 + |𝑢|𝑝−1 ), 2 < 𝑝 < 2∗ , (𝑓4 ) ∃𝜇 > 2, 𝑀 > 0, s.t. 𝑥 ∈ Ω,

|𝑢| ≥ 𝑀 󳨐⇒ 0 < 𝜇𝐹 (𝑥, 𝑢) ≤ 𝑢𝑓 (𝑥, 𝑢) ,

(6)

2

Abstract and Applied Analysis (𝑓5 ) 𝐹(𝑥, 𝑢) ≥ 0, for all 𝑥 and 𝑢, and 𝑢𝑓(𝑥, 𝑢) > 0 for |𝑢| > 0 small.

They got the existence of at least three nontrivial solutions. (𝑓4 ) was given by Ambrosetti and Rabinowitz [11] to ensure that some compactness and the Mountain Pass setting hold. However, there are many functions which are superlinear but do not necessarily need to satisfy (𝑓4 ). For example, 1 1 𝑢2 𝐹 (𝑥, 𝑢) = 𝑢2 ln (1 + 𝑢) − ( − 𝑢 + ln (1 + 𝑢)) . 2 2 2

(7)

It is easy to check that (𝑓4 ) does not hold. On the other hand, in order to verify (𝑓4 ), it is usually an annoying task to compute the primitive function of 𝑓 and sometimes it is almost impossible, for example, 𝛼

𝑓 (𝑥, 𝑢) = |𝑢| 𝑢 (1 + 𝑒(1+|sin 𝑢|) + |cos 𝑢|𝛼 ) ,

𝑢 ∈ 𝑅, 𝛼 > 0. (8)

More examples are presented in Remark 2. We recall that (𝑓4 ) implies a weaker condition 𝐹 (𝑥, 𝑢) ≥ 𝑐|𝑢|𝜃 −𝑑,

𝑐, 𝑑 > 0, a.e. 𝑥 ∈ Ω, 𝑢 ∈ 𝑅, 𝜃 > 2. (9)

In [12], Willem and Zou gave one weaker condition 𝑐|𝑢|𝜃 ≤ 𝑢𝑓 (𝑥, 𝑢)

for |𝑢| ≥ 𝑅0 , a.e. 𝑥 ∈ Ω, 𝜃 > 2.

(10)

Note that (𝑆2 ) is much weaker than the above conditions. In [13], Schechter and Zou proved that under the hypotheses that (𝑆1 ) holds and 𝐹 (𝑥, 𝑢) 𝐹 (𝑥, 𝑢) = +∞ or lim = +∞, 2 𝑢 → ∞ 𝑢 → −∞ 𝑢 𝑢2

(11)

either lim

problem (1) has a nontrivial weak solution. Recently, Miyagaki and Souto in [2] proved that problem (1) has a nontrivial solution via the Mountain Pass theorem under the following conditions:

lim

𝑓 (𝑥, 𝑢) = 0, 𝑢

(2) 𝑓(𝑥, 𝑢) = 𝛾|𝑢|𝛾−2 𝑢+(𝛾−1)|𝑢|𝛾−3 𝑢 sin2 𝑢+|𝑢|𝛾−1 sin 2𝑢, 𝑢 ∈ 𝑅 \ {0}, 𝛾 > 2, which do not satisfy (𝑓4 ). Example (2) can be found in [3]. So the case considered here cannot be covered by the cases mentioned in [6, 11]. Remark 3. Compared with papers [11, 12], we do not assume any superlinear conditions near zero. Compared with paper [2], we do not impose any kind of monotonic conditions. In addition, although we do not assume (𝑓4 ) holds, we are able to check the boundedness of P.S. (or P.S.∗ ) sequences. So, our result is different from those in the literature. Our argument is variational and close to that in [2, 3, 13, 14]. The paper is arranged as follows. In Section 2 we formulate the variational setting and recall some critical point theorems required. We then in Section 3 complete the proof of Theorem 1.

2. Variational Setting In this section, we will first recall some related preliminaries and establish the variational setting for our problem. Throughout this paper, we work on the space 𝐸 = 𝐻01 (Ω) equipped with the norm ‖𝑢‖ = (∫ |∇𝑢|2 𝑑𝑥) Ω

1/2

.

(13)

Lemma 4. 𝐸 embeds continuously into 𝐿𝑝 , for all 0 < 𝑝 ≤ 2∗ , and compactly into 𝐿𝑝 , for all 1 ≤ 𝑝 < 2∗ ; hence there exists 𝜏𝑝 > 0 such that |𝑢|𝑝 ≤ 𝜏𝑝 ‖𝑢‖ , where |𝑢|𝑝 = (∫Ω |𝑢|𝑝 𝑑𝑥)

1/𝑝

∀𝑢 ∈ 𝐸,

(14)

.

Define the Euler-Lagrange functional associated to problem (1), given by 𝑢 ∈ 𝐸,

(15)

where Ψ(𝑢) = ∫Ω 𝐹(𝑥, 𝑢)𝑑𝑥. Note that (𝑆1 ) implies that

∃ 𝑢0 > 0,

𝑓 (𝑥, 𝑢) s.t. is increasing in 𝑢 ≥ 𝑢0 𝑢

(1) 𝑓(𝑥, 𝑢) = 2𝑢 ln 𝑢 + 𝑢,

1 𝐼 (𝑢) = ‖𝑢‖2 − Ψ (𝑢) , 2

(𝑆1 ) and (𝑆2 ) hold, and |𝑢| → 0

Remark 2. To show that our assumptions (𝑆2 ) and (𝑆3 ) are weaker than (𝑓4 ), we give two examples:

(12)

and decreasing in 𝑢 ≤ −𝑢0 , ∀𝑥 ∈ Ω, and they adapted some monotonicity arguments used by Schechter and Zou [13]. This approach is interesting, but many powerful variational tools such as the Fountain theorem and Morse theory are not directly applicable. In addition, the monotonicity assumption on 𝐹(𝑥, 𝑢)/𝑢2 is weaker than the monotonicity assumption on 𝑓(𝑥, 𝑢)/𝑢. As to the case in the current paper, we make some concluding remarks as follows.

𝐹 (𝑥, 𝑢) ≤ 𝑎1 (|𝑢| + |𝑢|] ) ,

∀ (𝑥, 𝑢) ∈ Ω × 𝑅.

(16)

In view of (16) and Sobolev embedding theorem, 𝐼𝜇 and Ψ are well defined. Furthermore, we have the following. Lemma 5 (see [15] or [16]). Suppose that (𝑆1 ) is satisfied. Then Ψ ∈ 𝐶1 (𝐸, 𝑅) and Ψ󸀠 : 𝐸 → 𝐸∗ is compact and hence 𝐼 ∈ 𝐶1 (𝐸, 𝑅). Moreover Ψ󸀠 (𝑢) V = ∫ 𝑓 (𝑥, 𝑢) V 𝑑𝑥, Ω

𝐼󸀠 (𝑢) V = ∫ ∇𝑢∇V 𝑑𝑥 − Ψ󸀠 (𝑢)V,

(17)

Ω

for all 𝑢, V ∈ 𝐸, and critical points of 𝐼 on 𝐸 are solutions of (1).

Abstract and Applied Analysis

3

Lemma 6 (see [17]). Assume that |Ω| < ∞, 1 ≤ 𝑝, 𝑟 ≤ ∞, 𝑓 ∈ 𝐶(Ω × 𝑅), and |𝑓(𝑥, 𝑢)| ≤ 𝑐(1 + |𝑢|𝑝/𝑟 ). Then for every 𝑢 ∈ 𝐿𝑝 (Ω), 𝑓(𝑥, 𝑢) ∈ 𝐿𝑟 (Ω), and the operator 𝐴 : 𝐿𝑝 (Ω) 󳨃→ 𝐿𝑟 (Ω) : 𝑢 󳨃→ 𝑓(𝑥, 𝑢) is continuous.

3. Proof of Theorem 1

Let 𝐸 be a Banach space equipped with the norm ‖ ⋅ ‖ and 𝐸 = ⨁𝑗∈𝑁𝑋𝑗 , where dim𝑋𝑗 < ∞ for any 𝑗 ∈ 𝑁. Set

𝛼𝑘 (𝜆) :=

Lemma 8. Assume that (𝑆1 )-(𝑆2 ) hold. Then there exists a positive integer 𝑘1 and two sequences 𝑟𝑘 > 𝜌𝑘 → ∞ as 𝑘 → ∞ such that

1 𝑌𝑘 = ⨁𝑘𝑗=1 𝑋𝑗 and 𝑍𝑘 = ⨁∞ 𝑗=𝑘 𝑋𝑗 . Consider the following 𝐶 functional Φ𝜆 : 𝐸 → 𝑅 defined by

Φ𝜆 (𝑢) := 𝐴 (𝑢) − 𝜆𝐵 (𝑢) ,

𝜆 ∈ [1, 2] .

(18)

The following variant of the Fountain theorems was established in [3]. Theorem 7 (see [3, Theorem 2.1]). Assume that the functional Φ𝜆 defined above satisfies the following: (𝐹1 ) Φ𝜆 maps bounded sets to bounded sets uniformly for 𝜆 ∈ [1, 2]; furthermore, Φ𝜆 (−𝑢) = Φ𝜆 (𝑢) for all (𝜆, 𝑢) ∈ [1, 2] × 𝐸; (𝐹2 ) 𝐵(𝑢) ≥ 0 for all 𝑢 ∈ 𝐸; moreover, 𝐴(𝑢) → ∞ or 𝐵(𝑢) → ∞ as ‖𝑢‖ → ∞; (𝐹3 ) there exists 𝑟𝑘 > 𝜌𝑘 > 0 such that 𝛼𝑘 (𝜆) := :=

inf

𝑢∈𝑍𝑘 , ‖𝑢‖=𝜌𝑘

max

𝑢∈𝑌𝑘 , ‖𝑢‖=𝑟𝑘

Φ𝜆 (𝑢) > 𝛽𝑘 (𝜆)

Φ𝜆 (𝑢) ,

∀𝜆 ∈ [1, 2] .

∀𝑘 ≥ 𝑘1 ,

(25)

max 𝐼𝜆 (𝑢) < 0,

∀𝑘 ∈ 𝑁,

(26)

𝑢∈𝑌𝑘 ,‖𝑢‖=𝑟𝑘

where 𝑌𝑘 = ⨁𝑘𝑗=1 𝑋𝑗 = span{𝑒1 , . . . , 𝑒𝑘 } and 𝑍𝑘 = ⨁∞ 𝑗=𝑘 𝑋𝑗 = span{𝑒𝑘 , . . .}, for all 𝑘 ∈ 𝑁. Proof Step 1. We first prove (25). By (16) and (23), for all 𝜆 ∈ [1, 2] and 𝑢 ∈ 𝐸, we have 1 𝐼𝜆 (𝑢) ≥ ‖𝑢‖2 − 2 ∫ 𝑎1 (|𝑢| + |𝑢|] ) 𝑑𝑥 2 Ω 1 = ‖𝑢‖2 − 2𝑎1 (|𝑢|1 + |𝑢|]] ) , 2

(27)

where 𝑎1 is the constant in (16). Let 𝜎] (𝑘) =

(19)

|𝑢|] ,

sup

𝑢∈𝑍𝑘 ,‖𝑢‖=1

∀𝑘 ∈ 𝑁.

(28)

Then

Then 𝛼𝑘 (𝜆) ≤ 𝜁𝑘 (𝜆) := inf max Φ𝜆 (𝛾 (𝑢)) , ∀𝜆 ∈ [1, 2] , 𝛾∈Γ𝑘 𝑢∈𝐵𝑘

∞

𝑘 𝜆 ∈ [1, 2], there exists a sequence {𝑢𝑚 (𝜆)}𝑚=1 such that 󵄩󵄩 󵄩󵄩 𝑘 𝑘 sup 󵄩󵄩󵄩𝑢𝑚 (𝜆)󵄩󵄩󵄩 < ∞, Φ󸀠𝜆 (𝑢𝑚 (𝜆)) 󳨀→ 0, 𝑚 (21) 𝑘 Φ𝜆 (𝑢𝑚 (𝜆)) 󳨀→ 𝜁𝑘 (𝜆) as m 󳨀→ ∞.

In order to apply the above theorem to prove our main results, we define the functionals 𝐴, 𝐵, and 𝐼𝜆 on our working space 𝐸 by 𝐵 (𝑢) = ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥, Ω

(24)

and the corresponding system of eigenfunctions {𝑒𝑗 : 𝑗 ∈ 𝑁}(−Δ𝑒𝑗 = 𝜆 𝑗 𝑒𝑗 ) forming an orthogonal basis in 𝐸. Let 𝑋𝑗 = span{𝑒𝑗 }, for all 𝑗 ∈ 𝑁.

as 𝑘 󳨀→ ∞,

(29)

since 𝐸 is compactly embedded into 𝐿] . Combining (14), (27), and (28), we have 1 𝐼𝜆 ≥ ‖𝑢‖2 − 2𝑎1 𝜏1 ‖𝑢‖ − 2𝑎1 𝜎]] (𝑘) ‖𝑢‖] , 2

(30)

∀ (𝜆, 𝑢) ∈ [1, 2] × 𝑍𝑘 . By (29), there exists a positive integer 𝑘1 > 0 such that 𝜌𝑘 := (16𝑎1 𝜎]] (𝑘))

1/(2−])

> 16𝑎1 𝜏1 ,

∀𝑘 ≥ 𝑘1 ,

(31)

since ] > 2. Evidently, 𝜌𝑘 󳨀→ ∞ as 𝑘 󳨀→ ∞.

(22)

1 𝐼𝜆 (𝑢) = 𝐴 (𝑢) − 𝜆𝐵 (𝑢) = ‖𝑢‖2 − 𝜆 ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥, (23) 2 Ω for all 𝑢 ∈ 𝐸 and 𝜆 ∈ [1, 2]. Note that 𝐼1 = 𝐼, where 𝐼 is the functional defined in (15). From Lemma 5, we know that 𝐼𝜆 ∈ 𝐶1 (𝐸, 𝑅), for all 𝜆 ∈ [1, 2]. It is known that −Δ is a selfadjoint operator with a sequence of eigenvalues (counted with multiplicity) 0 < 𝜆 1 < 𝜆 2 ≤ 𝜆 3 ≤ ⋅ ⋅ ⋅ ≤ 𝜆 𝑗 ≤ ⋅ ⋅ ⋅ 󳨀→ ∞,

𝜎] (𝑘) 󳨀→ 0

(20)

where 𝐵𝑘 = {𝑢 ∈ 𝑌𝑘 : ‖𝑢‖ ≤ 𝑟𝑘 } and Γ𝑘 := {𝛾 ∈ 𝐶(𝐵𝑘 , 𝐸) | 𝛾 is odd, 𝛾|𝜕𝐵𝑘 = 𝑖𝑑}. Moreover, for a.e.

1 𝐴 (𝑢) = ‖𝑢‖2 , 2

𝛽𝑘 (𝜆) :=

𝐼𝜆 (𝑢) > 0,

inf

𝑢∈𝑍𝑘 ,‖𝑢‖=𝜌𝑘

(32)

Combining (30) and (31), direct computation shows 𝛼𝑘 :=

inf

𝑢∈𝑍𝑘 ,‖𝑢‖=𝜌𝑘

𝐼𝜆 (𝑢) ≥

𝜌𝑘2 > 0, 4

∀𝑘 ≥ 𝑘1 .

(33)

Step 2. We then verify (26). We claim that for any finite-dimensional subspace 𝐹 ⊂ 𝐸, there exists a constant 𝜖 > 0 such that 𝑚 ({𝑥 ∈ Ω : |𝑢 (𝑥)| ≥ 𝜖 ‖𝑢‖}) ≥ 𝜖,

∀𝑢 ∈ 𝐹 \ {0} .

(34)

Here and in the sequel, 𝑚(⋅) always denotes the Lebesgue measure in 𝑅.

4

Abstract and Applied Analysis If not, for any 𝑛 ∈ 𝑁, there exists 𝑢𝑛 ∈ 𝐹 \ {0} such that 1 󵄨 1󵄩 󵄩 󵄨 𝑚 ({𝑥 ∈ Ω : 󵄨󵄨󵄨𝑢𝑛 (𝑥)󵄨󵄨󵄨 ≥ 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩}) < . 𝑛 𝑛

Combining (23), (42), (43), and (𝑆2 ), for any 𝑘 ∈ 𝑁 and 𝜆 ∈ [1, 2], we have

(35)

1 |𝑢|2 𝑑𝑥 𝐼𝜆 (𝑢) ≤ ‖𝑢‖2 − ∫ 3 2 Λ𝑘𝑢 𝜖𝑘

Let V𝑛 = 𝑢𝑛 /‖𝑢𝑛 ‖ ∈ 𝐹, for all 𝑛 ∈ 𝑁. Then ‖V𝑛 ‖ = 1, for all 𝑛 ∈ 𝑁, and 1 󵄨 1 󵄨 𝑚 ({𝑥 ∈ Ω : 󵄨󵄨󵄨V𝑛 (𝑥)󵄨󵄨󵄨 ≥ }) < , 𝑛 𝑛

∀𝑛 ∈ 𝑁.

1 1 ≤ ‖𝑢‖2 − 3 𝜖𝑘2 ‖𝑢‖2 𝑚 (Λ𝑘𝑢 ) 2 𝜖𝑘

(36)

1 ≤ ‖𝑢‖2 − ‖𝑢‖2 2

Passing to a subsequence if necessary, we may assume V𝑛 → V0 in 𝐸, for some V0 ∈ 𝐹, since 𝐹 is of finite dimension. Evidently, ‖V0 ‖ = 1. In view of Lemma 4 and the equivalence of any two norms on 𝐹, we have 󵄨 󵄨 ∫ 󵄨󵄨󵄨V𝑛 − V0 󵄨󵄨󵄨 𝑑𝑥 󳨀→ 0 as 𝑛 󳨀→ ∞, Ω

(37)

and |V0 |∞ > 0. By the definition of norm | ⋅ |∞ , there exists a constant 𝛿0 > 0 such that 󵄨 󵄨 𝑚 ({𝑥 ∈ Ω : 󵄨󵄨󵄨V0 (𝑥)󵄨󵄨󵄨 ≥ 𝛿0 }) ≥ 𝛿0 . (38) For any 𝑛 ∈ 𝑁, let 󵄨 1 󵄨 Λ 𝑛 = {𝑥 ∈ Ω : 󵄨󵄨󵄨V𝑛 (𝑥)󵄨󵄨󵄨 < } , 𝑛 Λ𝑐𝑛

󵄨 1 󵄨 = Ω \ Λ 𝑛 = {𝑥 ∈ Ω : 󵄨󵄨󵄨V𝑛 (𝑥)󵄨󵄨󵄨 ≥ } . 𝑛

(39)

𝑚 (Λ 𝑛 ∩ Λ 0 ) = 𝑚 (Λ 0 \ Λ𝑐𝑛 ) 1 𝛿0 ≥ . 𝑛 2

(40)

Consequently, for 𝑛 large enough, there holds 󵄨 󵄨 ∫ 󵄨󵄨󵄨V𝑛 − V0 󵄨󵄨󵄨 𝑑𝑥 ≥ ∫ Ω

Λ 𝑛 ∩Λ 0

≥∫

Λ 𝑛 ∩Λ 0

≥

∀ |𝑢| ≥ 𝑆𝑘 .

then (44) implies 𝛽𝑘 (𝜆) :=

max 𝐼𝜆 (𝑢) ≤ −

𝑢∈𝑌𝑘 ,‖𝑢‖=𝑟𝑘

𝑟𝑘2 < 0, 2

∀𝑘 ∈ 𝑁,

(46)

Proof of Theorem 1. It follows from (16), (23), and Lemma 5 that 𝐼𝜆 maps bounded sets to bounded sets uniformly for 𝜆 ∈ [1, 2]. In view of the evenness of 𝐹(𝑥, 𝑢) in 𝑢, it holds that 𝐼𝜆 (−𝑢) = 𝐼𝜆 (𝑢) for all (𝜆, 𝑢) ∈ [1, 2] × 𝐸. Thus the condition (𝐹1 ) of Theorem 7 holds. Besides, 𝐴(𝑢) = (1/2)‖𝑢‖2 → ∞ as ‖𝑢‖ → ∞ and 𝐵(𝑢) ≥ 0 since 𝐹(𝑥, 𝑢) ≥ 0. Thus the condition (𝐹2 ) of Theorem 7 holds. And Lemma 8 shows that the condition (𝐹3 ) holds for all 𝑘 ≥ 𝑘1 . Therefore, by Theorem 7, for any 𝑘 ≥ 𝑘1 and a.e. 𝜆 ∈ [1, 2], there exists ∞ 𝑘 (𝜆)}𝑚=1 ⊂ 𝐸 such that a sequence {𝑢𝑚 𝑘 𝐼𝜆󸀠 (𝑢𝑚 (𝜆)) 󳨀→ 0,

𝜁𝑘 (𝜆) := inf max 𝐼𝜆 (ℎ (𝑢)) , ℎ∈Γ𝑘 𝑢∈𝐵𝑘

(47)

∀𝜆 ∈ [1, 2] ,

(48)

with 𝐵𝑘 = {𝑢 ∈ 𝑌𝑘 : ‖𝑢‖ ≤ 𝑟𝑘 } and Γ𝑘 := {ℎ ∈ 𝐶(𝐵𝑘 , 𝐸) | ℎ is odd, ℎ|𝜕𝐵𝑘 = 𝑖𝑑}. Furthermore, it follows from the proof of Lemma 8 that

(42)

where Λ𝑘𝑢 := {𝑥 ∈ Ω : |𝑢(𝑥)| ≥ 𝜖𝑘 ‖𝑢‖}, for all 𝑘 ∈ 𝑁, and 𝑢 ∈ 𝑌𝑘 \ {0}. By (𝑆2 ), for any 𝑘 ∈ 𝑁, there exists a constant 𝑆𝑘 > 0 such that |𝑢|2 , 𝜖𝑘3

(45)

as 𝑚 → ∞, where (41)

This is in contradiction to (37). Therefore (34) holds. Consequently, for any 𝑘 ∈ 𝑁, there exists a constant 𝜖𝑘 > 0 such that

𝐹 (𝑥, 𝑢) ≥

𝑆𝑘 }, 𝜖𝑘

𝑘 𝐼𝜆 (𝑢𝑚 (𝜆)) 󳨀→ 𝜁𝑘 (𝜆)

󵄨 󵄨 󵄨 󵄨 (󵄨󵄨󵄨V0 󵄨󵄨󵄨 − 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨) 𝑑𝑥

∀𝑢 ∈ 𝑌𝑘 \ {0} ,

𝑟𝑘 > max {𝜌𝑘 ,

𝑚

𝛿02 > 0. 4

𝑚 (Λ𝑘𝑢 ) ≥ 𝜖𝑘 ,

with ‖𝑢‖ ≥ 𝑆𝑘 /𝜖𝑘 . Now for any 𝑘 ∈ 𝑁, if we choose

󵄩 󵄩 𝑘 sup 󵄩󵄩󵄩󵄩𝑢𝑚 (𝜆)󵄩󵄩󵄩󵄩 < ∞,

󵄨󵄨 󵄨 󵄨󵄨V𝑛 − V0 󵄨󵄨󵄨 𝑑𝑥

1 ≥ (𝛿0 − ) 𝑚 (Λ 𝑛 ∩ Λ 0 ) 𝑛

1 = − ‖𝑢‖2 2

ending the proof.

Set Λ 0 = {𝑥 ∈ Ω : |V0 (𝑥)| ≥ 𝛿0 }. Then for 𝑛 large enough, by (36) and (38), we have

≥ 𝑚 (Λ 0 ) − 𝑚 (Λ𝑐𝑛 ) ≥ 𝛿0 −

(44)

(43)

𝜁𝑘 (𝜆) ∈ [𝛼𝑘 , 𝜁𝑘 ] ,

∀𝑘 ≥ 𝑘1 ,

(49)

where 𝜁𝑘 := max𝑢∈𝐵𝑘 𝐼𝜆 (𝑢) and 𝛼𝑘 := 𝜌𝑘2 /4 → ∞ as 𝑘 → ∞ by (32). ∞

𝑘 Claim 1. {𝑢𝑚 (𝜆)}𝑚=1 ⊂ 𝐸 possesses a strong convergent subsequence in 𝐸, for ∀𝜆 ∈ [1, 2] and 𝑘 ≥ 𝑘1 . ∞ 𝑘 In fact, by the boundedness of the {𝑢𝑚 (𝜆)}𝑚=1 , passing to a subsequence, as 𝑚 → ∞, we may assume 𝑘 𝑢𝑚 (𝜆) ⇀ 𝑢𝑘 (𝜆)

in 𝐸.

(50)

Abstract and Applied Analysis

5

By the Sobolev embedding theorem, 𝑘 𝑢𝑚 (𝜆) 󳨀→ 𝑢𝑘 (𝜆)

in 𝐿] .

(51)

Let Ω ≠ = {𝑥 ∈ Ω : V(𝑥) ≠ 0}. If 𝑥 ∈ Ω ≠ , from (𝑆2 ) it follows that lim

Lemma 6 implies that 𝑘 𝑓 (𝑥, 𝑢𝑚 (𝜆)) 󳨀→ 𝑓 (𝑥, 𝑢𝑘 (𝜆))

in 𝐿]/(]−1) .

(52)

Observe that 󵄩󵄩󵄩𝑢𝑘 (𝜆) − 𝑢𝑘 (𝜆)󵄩󵄩󵄩2 󵄩󵄩 󵄩󵄩 𝑚

Ω

𝑘 (𝑓 (𝑥, 𝑢𝑚

𝑘

(53)

(𝜆)) − 𝑓 (𝑥, 𝑢 (𝜆)))

∫

𝐹 (𝑥, 𝑢𝑛 (𝑥)) 𝑢𝑛 (𝑥)2

as 𝑚 󳨀→ ∞.

Ω

𝑛→∞

< lim sup

󵄨 󵄨 𝑘 ≤ 󵄨󵄨󵄨󵄨𝑓 (𝑥, 𝑢𝑚 (𝜆)) − 𝑓 (𝑥, 𝑢𝑘 (𝜆))󵄨󵄨󵄨󵄨]/(]−1)

as 𝑚 → ∞. Thus by (53), (54), and (55), we have proved that 󵄩󵄩󵄩𝑢𝑘 (𝜆) − 𝑢𝑘 (𝜆)󵄩󵄩󵄩 󳨀→ 0 as 𝑚 󳨀→ ∞, (56) 󵄩󵄩 󵄩󵄩 𝑚 𝑘 that is, 𝑢𝑚 (𝜆) → 𝑢𝑘 (𝜆) in 𝐸. Thus, for each 𝑘 ≥ 𝑘1 , we can choose 𝜆 𝑛 → 1 ∞ 𝑘 (𝜆 𝑛 )}𝑚=1 obtained a convergent such that the sequence {𝑢𝑚 subsequence; passing again to a subsequence, we may assume

(57)

This together with (47) and (49) yields 𝐼𝜆 𝑛 (𝑢𝑛𝑘 ) ∈ [𝛼𝑘 , 𝜁𝑘 ] ,

∀𝑛 ∈ 𝑁, 𝑘 ≥ 𝑘1 . (58)

∞

Claim 2. {𝑢𝑛𝑘 }𝑛=1 is bounded in 𝐸 for all 𝑘 ≥ 𝑘1 . For notational simplicity, we will set 𝑢𝑛 = 𝑢𝑛𝑘 for all 𝑛 ∈ 𝑁 throughout this paragraph. If {𝑢𝑛 } is unbounded in 𝐸, we define V𝑛 = 𝑢𝑛 /‖𝑢𝑛 ‖. Since ‖V𝑛 ‖ = 1, without loss of generality we suppose that there is V ∈ 𝐸 such that in 𝐸, in 𝐿] (Ω) ,

V𝑛 (𝑥) 󳨀→ V (𝑥)

a.e. in Ω.

(59)

2𝑢𝑛2

(63) V𝑛2 = 0

which contradicts (62). Hence {𝑢𝑛 } is bounded. Claim 3. {𝑢𝑛𝑘 } possesses a convergent subsequence with the limit 𝑢𝑘 ∈ 𝐸 for all 𝑘 ≥ 𝑘1 . In fact, by Claim 2, without loss of generality, we have assume 𝑢𝑛𝑘 ⇀ 𝑢𝑘

(55)

in 𝐸, ∀𝑛 ∈ 𝑁, 𝑘 ≥ 𝑘1 .

𝑏 (𝑢𝑛2 + 1)

𝑛→∞

󵄨 󵄨󵄨 󵄨󵄨∫ (𝑓 (𝑥, 𝑢𝑘 (𝜆)) − 𝑓 (𝑥, 𝑢𝑘 (𝜆))) (𝑢𝑘 (𝜆) − 𝑢𝑘 (𝜆)) 𝑑𝑥󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑚 𝑚 󵄨 󵄨 Ω 󵄨 󵄨 𝑘 × 󵄨󵄨󵄨󵄨𝑢𝑚 (𝜆) − 𝑢𝑘 (𝜆)󵄨󵄨󵄨󵄨] 󳨀→ 0

(1/2) 𝑓 (𝑥, 𝑢𝑛 ) 𝑢𝑛 − 𝐹 (𝑥, 𝑢𝑛 ) 2 V𝑛 𝑢𝑛2

(54)

It follows from the H¨older inequality, (51), and (52) that

V𝑛 󳨀→ V

(61)

By (𝑆3 ),

𝑘 𝑘 (𝐼𝜆󸀠 (𝑢𝑚 (𝜆)) − 𝐼𝜆󸀠 (𝑢𝑘 (𝜆)) , 𝑢𝑚 (𝜆) − 𝑢𝑘 (𝜆)) 󳨀→ 0

V𝑛 ⇀ V

1 V𝑛 (𝑥)2 = . 2

𝐼𝜆 (𝑢𝑛 ) (1/2) 𝑓 (𝑥, 𝑢𝑛 ) 𝑢𝑛 − 𝐹 (𝑥, 𝑢𝑛 ) 2 V𝑛 𝑑𝑥 = 𝑛󵄩 󵄩2 > 0. (62) 2 𝑢𝑛 𝜆 𝑛 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩

lim sup

(𝜆 𝑛 ) = 𝑢𝑛𝑘

(60)

We conclude that Ω ≠ has zero measure and V ≡ 0 a.e. in Ω. Moreover, from (49) and (58)

By (47), it is clear that

𝐼𝜆󸀠 𝑛 (𝑢𝑛𝑘 ) = 0,

𝑢𝑛 (𝑥)

V𝑛 (𝑥)2 = ∞.

On the other hand, after a simple calculation, we have lim ∫

𝑘 × (𝑢𝑚 (𝜆) − 𝑢𝑘 (𝜆)) 𝑑𝑥.

lim 𝑢𝑘 𝑚→∞ 𝑚

2

𝑛→∞ Ω

𝑘 𝑘 = (𝐼𝜆󸀠 (𝑢𝑚 (𝜆)) − 𝐼𝜆󸀠 (𝑢𝑘 (𝜆)) , 𝑢𝑚 (𝜆) − 𝑢𝑘 (𝜆))

+ 𝜆∫

𝐹 (𝑥, 𝑢𝑛 (𝑥))

𝑛→∞

as 𝑛 󳨀→ ∞.

(64)

By virtue of the Riesz Representation theorem, 𝐼𝜆󸀠 : 𝐸 󳨃→ 𝐸∗ and Ψ󸀠 : 𝐸 󳨃→ 𝐸∗ can be viewed as 𝐼𝜆󸀠 : 𝐸 󳨃→ 𝐸 and Ψ󸀠 : 𝐸 󳨃→ 𝐸, respectively, where 𝐸∗ is the dual space of 𝐸. Note that 0 = 𝐼𝜆󸀠 𝑛 (𝑢𝑛𝑘 ) = 𝑢𝑛𝑘 − 𝜆 𝑛 Ψ󸀠 (𝑢𝑛𝑘 ) ,

(65)

𝑢𝑛𝑘 = 𝜆 𝑛 Ψ󸀠 (𝑢𝑛𝑘 ) .

(66)

that is

By Lemma 5, Ψ󸀠 : 𝐸 󳨃→ 𝐸 is also compact. Due to the compactness of Ψ󸀠 and (64), the right-hand side of (66) converges strongly in 𝐸 and hence 𝑢𝑛𝑘 → 𝑢𝑘 in 𝐸. Now for each 𝑘 ≥ 𝑘1 , by (58), the limit 𝑢𝑘 is just a critical point of 𝐼1 = 𝐼 with 𝐼(𝑢𝑘 ) ∈ [𝛼𝑘 , 𝜁𝑘 ]. Since 𝛼𝑘 → ∞ as 𝑘 → ∞ in (49), we get infinitely many nontrivial critical points of 𝐼. Therefore (1) possesses infinitely many nontrivial solutions by Lemma 5.

Acknowledgments The authors would like to thank the referee for valuable comments and helpful suggestions. The first author would like to acknowledge the hospitality of Professor Y. Ding of the AMSS of the Chinese Academy of Sciences, where this paper was written during his visit. Anmin Mao was supported by NSFC (11101237) and ZR2012AM006.

6

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