The Existence of Multiple Solutions for Nonhomogeneous Kirchhoff Type Equations in ℝ³

Abstract

We are concerned with the existence of multiple solutions to the nonhomogeneous Kirchhoff type equation −(𝑎+𝑏∫_ℝ3|∇𝑢|²)Δ𝑢+𝑢=|𝑢|^𝑝−1𝑢+ℎ(𝑥)inℝ³, where 𝑎,𝑏 are positive constants, 𝑝∈(1,5),0⩽ℎ(𝑥)=ℎ(|𝑥|)∈𝐶¹(ℝ³)∩𝐿²(ℝ³), we can find a constant 𝑚_𝑝>0 such that for all 𝑝∈(1,5) the equation has at least two radial solutions provided |ℎ|₂<𝑚_𝑝.

1. Introduction and Main Result

In this paper, we consider the existence of multiple solutions to the following nonhomogeneous Kirchhoff type equation: −𝑎+𝑏_ℝ3|∇𝑢|²Δ𝑢+𝑢=|𝑢|^𝑝−1𝑢+ℎ(𝑥)inℝ³, (1) where 𝑎,𝑏 are positive constants and 𝑝∈(1,5)⋅ℎ∈𝐶¹(ℝ³)∩𝐿²(ℝ³) satisfies the following conditions: ( h₁)0⩽ℎ(𝑥)=ℎ(|𝑥|)∈𝐿²(ℝ³) and |ℎ|₂⩽𝑚_𝑝, where 𝑚_𝑝=𝑝−12𝑝𝛾₂𝑝+12𝑝𝛾^𝑝+1_𝑝+1^1/(𝑝−1), (2)𝛾_𝑠 is the embedding coefficient of 𝐻¹(ℝ³)↪𝐿^𝑠(ℝ³)and𝑠∈[2,6]; (h₂)(∇ℎ(𝑥),𝑥)∈𝐿²(ℝ³), where (⋅,⋅) denotes the usual inner product in ℝ³.

Recently, there have been many references about the existence of nontrivial solutions to the following Kirchhoff type equation by using variational method [1–5]: −𝑎+𝑏_ℝ3|∇𝑢|²Δ𝑢+𝑉(𝑥)𝑢=𝑓(𝑥,𝑢)inℝ^𝑁, (3) where 𝑎,𝑏 are positive constants. 𝑉∶ℝ^𝑁→ℝ,𝑓∈𝐶(ℝ^𝑁×ℝ,ℝ),𝑁=1,2,3. A main tool to deal with problem (3) is the mountain pass theorem. For this purpose, one usually assumes that 𝑓(𝑥,𝑡) is subcritical, superlinear at the origin, and either 4-superlinear at infinity or satisfies the following global Ambrosetti-Rabinowitz type condition (AR in short): (AR)there exists 𝜇>4 such that 0<𝜇𝐹(𝑥,𝑡)=∫^𝑡₀𝑓(𝑥,𝑠⩽𝑡𝑓(𝑥,𝑡) for all 𝑥∈ℝ^𝑁 and 𝑡∈ℝ. Under the above assumptions, the mountain pass geometry structure and the boundedness of Palais-Smale sequence or Cerami sequence can be obtained.

For example, in [5], when 𝑓 satisfies above assumptions and the potential 𝑉 satisfies the following conditions:(V)𝑉∈𝐶(ℝ^𝑁,ℝ),inf_ℝ^𝑁𝑉>0 and for each 𝑀>0, meas{𝑥∈ℝ^𝑁∶𝑉(𝑥)⩽𝑀}<∞, where meas denotes the Lebesgue measure,

which ensure the compact imbedding of 𝐸={𝑢∈𝐻¹(ℝ^𝑁)∶∫_ℝ𝑁𝑉𝑢²<∞}↪𝐿^𝑞(ℝ^𝑁),𝑞∈[2,2^∗), the author obtained the existence of a nontrivial solution to problem (3).

The existence of infinitely many solutions was considered in [2, 3] respectively, by the fountain theorem and a variant version of the fountain theorem, where 𝑓 is odd on 𝑡∈ℝ and is also subcritical, superlinear at the origin, and either 4-superlinear at infinity or satisfies AR condition or some conditions weaker than AR condition. In [2], 𝑁=2,3,𝑉≡1 and in [3], 𝑁=3,𝑉∈𝐿^∞_loc(ℝ³) satisfies the condition (V).

The existence of ground state solutions to problem (3) was also considered in [1, 4]. In [1], the authors studied (3) under the conditions: 𝑁=3, a positive potential satisfies 𝑉_∞=liminf_|𝑥|→∞𝑉(𝑥)>𝑉₀=inf_𝑥∈ℝ^𝑁>0⋅𝑓(𝑥,𝑡)=𝑓(𝑡)∈𝐶¹(ℝ,ℝ) satisfies (AR), lim_𝑡→0(𝑓(𝑡)/|𝑡|³)=0,lim_𝑡→∞(𝑓(𝑡)/|𝑡|^𝑞)=0 for some 𝑞∈(3,5) and 𝑓(𝑡)/𝑡³ increases for all 𝑡>0. They obtained a positive ground state solution by using the Nehari manifold.

Under the same condition of 𝑉 in [1], the authors in [4] discussed the existence of multiple ground state solutions, where 𝑓(𝑥,𝑡)=𝜆𝑓(𝑡)+|𝑡|⁴𝑡, which contains a critical growth term.

Recently, in [6], the authors studied the existence of a positive solution for the following Kirchhoff equation: 𝑎+𝜆_ℝ3|∇𝑢|²+𝑏𝑢²(−Δ𝑢+𝑏𝑢)=𝑓(𝑢)inℝ³, (4) where 𝑁⩾3,𝑎,𝑏>0,𝑓 is subcritical, superlinear at the origin and infinity. In order to construct the mountain pass geometry structure and obtain the bounded PS sequence, they combined a truncation argument with a monotonicity trick introduced by Jeanjean [7], and obtained that there exists 𝜆₀>0 such that problem (4) has at least one positive solution for 𝜆∈(0,𝜆₀).

Motivated by the aformentioned references, we consider the existence of multiple solutions to the nonhomogeneous Kirchhoff equation (1), where 𝑝∈(1,5). By using the variational method, we obtain that the problem has at least two positive radial solutions. Under proper assumptions on ℎ, the problem has a local minimum around the origin with negative energy by Ekeland variational principle. Note that the term |𝑢|^𝑝−1𝑢 is neither 4-superlinear nor satisfies AR condition for 𝑝∈(1,3]. In order to obtain the bounded PS sequence, we also use the indirect method in [7]. Meanwhile, for 𝑤∈𝐻¹(ℝ³), we take a transform of 𝑤_𝑡(⋅)=𝑡𝑤(𝑡⁻²⋅) to construct the mountain pass geometry structure. Finally, the combination of Pohozaev identity with the method in [7] obtains the bounded PS sequence. Therefore, we obtain the second solution which has positive energy.

Let 𝐻¹(ℝ³) be the usual Sobolev space equipped with the inner product and norm (𝑢,𝑣)=_ℝ3[𝑎∇𝑢⋅∇𝑣+𝑢𝑣],‖𝑢‖=(𝑢,𝑢)^1/2. (5) We denote by |⋅|_𝑠 the usual 𝐿^𝑠(ℝ³) norm. Then, we have that 𝐻¹(ℝ³)↪𝐿^𝑠(ℝ³) continuously for 𝑠∈[2,6]. Hence, there exists 𝛾_𝑠 such that |𝑢|_𝑠⩽𝛾_𝑠‖𝑢‖,𝑢∈𝐻¹ℝ³. (6) Let 𝐻=𝐻¹_𝑟(ℝ³) be the subspace of 𝐻¹(ℝ³) containing only the radial functions. Then the imbedding𝐻↪𝐿^𝑠(ℝ³) is compact for 𝑠∈(2,6) [8, Corollary 1.26, page 18]. Let 𝒟^1,2(ℝ³) be the completion of 𝐶^∞₀(ℝ³) with respect to the norm ‖𝑢‖_𝒟^1,2=(∫_ℝ3|∇𝑢|²)^1/2.

Define the energy functional 𝐽∶𝐻¹(ℝ³)→ℝ by 𝐽(𝑢)=12_ℝ³𝑎|∇𝑢|²+𝑢²+𝑏4_ℝ³|∇𝑢|²²−1𝑝+1_ℝ³|𝑢|^𝑝+1−_ℝ3ℎ𝑢. (7) By 𝑝∈(1,5),ℎ∈𝐿²(ℝ³), we have 𝐽∈𝐶¹(𝐻¹(ℝ³),ℝ). And, for any 𝑢,𝑣∈𝐻¹(ℝ³), 𝐽^′(𝑢),𝑣=_ℝ3[𝑎∇𝑢⋅∇𝑣+𝑢𝑣]+𝑏_ℝ3|∇𝑢|²_ℝ3−∇𝑢⋅∇𝑣_ℝ3|𝑢|^𝑝−1𝑢𝑣−_ℝ3ℎ𝑣. (8)

Furthermore, by (h₁), ℎ(𝑥)=ℎ(|𝑥|), the functional 𝐽 is also a 𝐶¹ functional defined on 𝐻. By standard argument, the weak solution of (1) is corresponding to the critical point of the functional 𝐽 on 𝐻.

Our main result is as follows.

Theorem 1. Let 𝑝∈(1,5) and ℎ satisfy (h₁)-(h₂). Then, problem (1) has at least two nontrivial radial solutions 𝑢₀ and 𝑣₀, satisfying 𝐽(𝑢₀)<0<𝐽(𝑣₀).

The paper is organized as follows. In Section 2, we give the existence of the negative energy solution 𝑢₀. The existence of positive energy solution 𝑣₀ and the proof of Theorem 1 are given in Section 3.

2. Existence of Negative Energy Solution

In this section, we give the existence of the negative energy solution. In order to obtain our first solution, we need the following preliminaries.

Lemma 2. Let 𝑝∈(1,5) and ℎ satisfy (h₁). Then, there exists 𝜌,𝛼>0 such that 𝐽∣_𝜕𝐵𝜌⩾𝛼, where 𝐵_𝜌={𝑢∈𝐻∶‖𝑢‖<𝜌}.

Proof. For 𝑢∈𝐻, by (7), the Hölder inequality and the Sobolev inequality imply that 𝐽(𝑢)⩾12‖𝑢‖²−1𝛾𝑝+1^𝑝+1𝑝+1‖𝑢‖^𝑝+1−ℎ|₂𝑢|₂⩾‖𝑢‖12‖𝑢‖−1𝛾𝑝+1^𝑝+1𝑝+1‖𝑢‖^𝑝−𝛾₂|ℎ|₂. (9) Set 𝑔(𝑡)=121𝑡−𝛾𝑝+1^𝑝+1𝑝+1𝑡^𝑝,𝑡⩾0, (10) since 𝑝>1, by calculating directly, we see that max_𝑡⩾0𝑔(𝑡)=𝑔(𝜌)>0, where 𝜌=((𝑝+1)/2𝑝𝛾^𝑝+1_𝑝+1)^1/(𝑝−1),𝑔(𝜌)=((𝑝−1)/2𝑝)𝜌. Then it follows that, if |ℎ|₂𝛾₂<𝑔(𝜌), that is, |ℎ|₂<𝛾₂⁻¹𝑔(𝜌)≜𝑚_𝑝, there exists 𝛼=𝜌(𝑔(𝜌)−𝛾₂|ℎ|₂)>0 such that 𝐽|_𝜕𝐵𝜌⩾𝛼, where 𝑚_𝑝=𝑝−12𝑝𝛾₂𝑝+12𝑝𝛾^𝑝+1_𝑝+1^1/(𝑝−1). (11)

Lemma 3. Let 𝑝∈(1,5) and ℎ satisfy (h₁). Then 𝑐=inf_𝐵𝜌𝐽<0, where 𝜌 is given by Lemma 2 and 𝐵_𝜌={𝑢∈𝐻∶‖𝑢‖<𝜌}.

Proof. By (h₁), ℎ∈𝐿²(ℝ³),ℎ≠0, then for 𝜀∈(0,|ℎ|₂), there exists 𝜓∈𝐶^∞₀(ℝ³) such that |ℎ−𝜓|₂<𝜀. Then ∫_ℝ3(ℎ²−ℎ𝜓)⩽∫_ℝ3|ℎ²−ℎ𝜓|⩽𝜀|ℎ|₂, so ∫_ℝ3ℎ𝜓⩾|ℎ|²₂−𝜀|ℎ|₂>0. Hence, by (7), for 𝑡>0 small enough, we have 𝐽=𝑡𝑡𝜓²2𝜓²+𝑏𝑡⁴4_ℝ³∇𝜓²²−𝑡^𝑝+1𝑝+1_ℝ³𝜓^𝑝+1−𝑡_ℝ3ℎ𝜓<0. (12) Then, by the definition of 𝐵_𝜌,𝑐=inf_𝐵𝜌𝐽<0.

Lemma 4. Let 𝑝∈(1,5) and ℎ satisfy (h₁). The bounded PS sequence of the functional 𝐽 possesses a convergent subsequence.

Proof. Let {𝑢_𝑛} be a bounded PS sequence of 𝐽, that is {𝑢_𝑛} and {𝐽(𝑢_𝑛)} are bounded, 𝐽^′(𝑢_𝑛)→0 in 𝐻^′, where 𝐻^′ is the dual space of 𝐻. We may assume that, up to a subsequence, 𝑢_𝑛⇀𝑢in𝐻,𝑢_𝑛⟶𝑢in𝐿^𝑝+1ℝ³,𝑢_𝑛⟶𝑢a.e.onℝ³. (13) It follows that _ℝ3𝑢_𝑛^𝑝−1𝑢_𝑛−|𝑢|^𝑝−1𝑢𝑢_𝑛−𝑢⟶0,𝑛⟶∞. (14) By (8), we can obtain that 𝐽^′𝑢_𝑛−𝐽^′(𝑢),𝑢_𝑛=𝑢−𝑢_𝑛−𝑢²+𝑏_ℝ3∇𝑢_𝑛²_ℝ3∇𝑢_𝑛⋅∇𝑢_𝑛−∇𝑢−𝑏_ℝ3|∇𝑢|²_ℝ3∇𝑢⋅∇𝑢_𝑛−−∇𝑢_ℝ3𝑢_𝑛^𝑝−1𝑢_𝑛−|𝑢|^𝑝−1𝑢𝑢_𝑛=𝑢−𝑢_𝑛−𝑢²+𝑏_ℝ3∇𝑢_𝑛²_ℝ3∇𝑢_𝑛−∇𝑢²+𝑏_ℝ3∇𝑢_𝑛²−|∇𝑢|²_ℝ3∇𝑢⋅∇𝑢_𝑛−∇𝑢−_ℝ3𝑢_𝑛^𝑝−1𝑢_𝑛−|𝑢|^𝑝−1𝑢𝑢_𝑛−𝑢. (15) Since {𝑢_𝑛} is also bounded in 𝒟^1,2(ℝ³), then 𝑏_ℝ3∇𝑢_𝑛²−|∇𝑢|²_ℝ3∇𝑢⋅∇𝑢_𝑛−∇𝑢⟶0. (16) Therefore, 0⩽𝑢_𝑛−𝑢²⩽𝑢_𝑛−𝑢²+𝑏∫_ℝ3∇𝑢_𝑛²∫_ℝ3∇𝑢_𝑛−∇𝑢²⟶0,𝑛⟶∞. (17) That is, 𝑢_𝑛→𝑢 in 𝐻.

Theorem 5. Let 𝑝∈(1,5) and ℎ satisfy (h₁). Then, there exists 𝑢₀∈𝐻 such that 𝐽𝑢₀=𝑐=inf_𝐵𝜌𝐽(𝑢)<0, (18) where 𝜌 is given by Lemma 2 and 𝐵_𝜌={𝑢∈𝐻∶‖𝑢‖<𝜌}.

Proof. By Lemma 3, 𝑐=inf{𝐽(𝑢)∶𝑢∈𝐵_𝜌}<0, then by Ekeland variational principle [9], there exists {𝑢_𝑛}⊂𝐵_𝜌 such that 𝑐⩽𝐽𝑢_𝑛1⩽𝑐+𝑛,𝐽(𝜔)⩾𝐽𝑢_𝑛−1𝑛𝜔−𝑢_𝑛∀𝜔∈𝐵_𝜌. (19) Then, by Lemma 2  𝑢_𝑛∈𝐵_𝜌, then {𝑢_𝑛} is a bounded PS sequence of 𝐽. Therefore, Lemma 4 implies that there exists 𝑢₀∈𝐻 such that 𝑢_𝑛→𝑢₀, up to a subsequence. So 𝐽(𝑢₀)=𝑐<0 and 𝐽^′(𝑢₀)=0.

3. Proof of Theorem 1

In this section, we will show the existence of the second solution. Note that 𝑝∈(1,5), when 𝑝∈(1,3],|𝑢|^𝑝−1𝑢 neither satisfies (AR) condition nor is 4-superlinear. So, in order to obtain the bounded PS sequence, following the argument in [6], we also use a direct method in [7]. Firstly, we recall the following main result in [7]. The “monotonicity trick” at the core of this theorem was invented by Struwe (see [9]).

Theorem 6 (see [7]). Let (𝑋,‖⋅‖) be a Banach space and 𝐼⊂ℝ₊ be an interval. Consider the family of 𝐶¹ functionals on 𝑋𝐽_𝜆(𝑢)=𝐴(𝑢)−𝜆𝐵(𝑢),𝜆∈𝐼, (20) with 𝐵 nonnegative and either 𝐴(𝑢)→∞ or 𝐵(𝑢)→∞ as ‖𝑢‖→∞. We assume that there are two points 𝑣₁,𝑣₂ in 𝑋 such that 𝑐_𝜆=inf_{𝛾∈Γ𝜆}max_𝑡∈[0,1]𝐽_𝜆𝛾(𝑡)>max𝐽_𝜆𝑣₁,𝐽_𝜆𝑣₂,∀𝜆∈𝐽, (21) where Γ=𝛾∈𝐶([0,1],𝑋)∶𝛾(0)=𝑣₁,𝛾(1)=𝑣₂. (22) Then, for almost every 𝜆∈𝐼 there is a sequence {𝑢_𝑛(𝜆)}⊂𝑋 such that (i){𝑢_𝑛(𝜆)} is bounded;(ii)𝐽_𝜆(𝑢_𝑛(𝜆))→𝑐_𝜆;(iii)𝐽^′_𝜆(𝑢_𝑛(𝜆))→0 in the dual 𝑋⁻¹ of 𝑋.

In our case, 𝑋=𝐻,𝐼=[1/2,1], and define 𝐽_𝜆∶𝑋→ℝ by 𝐽_𝜆(𝑢)=𝐴(𝑢)−𝜆𝐵(𝑢), (23) where 𝜆∈𝐼, 𝐴(𝑢)=12‖𝑢‖²+𝑏4_ℝ³|∇𝑢|²²−∫_ℝ3(𝑢)=1ℎ𝑢,𝐵∫𝑝+1^ℝ3|𝑢|^𝑝+1. (24) Then {𝐽_𝜆}_𝜆∈𝐼 is a family of 𝐶¹ functionals on 𝐻. For any 𝑢∈𝐻,𝐵(𝑢)⩾0, and 𝐴(𝑢)⩾(1/2)‖𝑢‖²−𝛾₂|ℎ|₂‖𝑢‖→∞ as ‖𝑢‖→∞.

In the following, we verify that the functional 𝐽_𝜆 satisfies the conditions of Theorem 6.

Lemma 7. Let 𝑝∈(1,5) and ℎ satisfy (h₁)-(h₂). Then, the following claims hold:(i)there exist 𝑟,𝑎>0 and 𝑒∈𝐻 such that for all 𝜆∈𝐼𝐽_𝜆∣_𝜕𝐵𝜌(𝑢)⩾𝑎>0,𝐽_𝜆(𝑒)<0𝑤𝑖𝑡ℎ‖𝑒‖>𝑟; (25)(ii)for any 𝜆∈𝐼, 𝑐_𝜆=inf_𝛾∈Γmax_[]𝑡∈0,1𝐽_𝜆𝛾(𝑡)>max𝐽_𝜆(0),𝐽_𝜆(𝑒), (26)where Γ=𝛾∈𝐶([0,1],𝑋)∶𝛾(0)=0,𝛾(1)=𝑒. (27)

Proof. (i) Since for all 𝑢∈𝐻 and 𝜆∈𝐼=[1/2,1],𝐽_𝜆(𝑢)⩾𝐽₁(𝑢). By Lemma 2, there exist 𝑟,𝑎>0 independent of 𝜆∈𝐼 such that 𝐽_𝜆(𝑢)⩾𝑎>0 with ‖𝑢‖=𝑟.
We choose a function 𝑤∈𝐻 and 𝑤≠0. Set 𝑤_𝑡(⋅)=𝑡𝑤(𝑡⁻²⋅) for 𝑡>0. Then, for all 𝜆∈𝐼, by (7) and (h₁), we have 𝐽_𝜆𝑤_𝑡⩽𝑎𝑡⁴2_ℝ³|∇𝑤|²+𝑡⁸2_ℝ³|𝑤|²+𝑏𝑡⁸4_ℝ³|∇𝑤|²²−𝑡^𝑝+72𝑝+1_ℝ³|𝑤|^𝑝+1. (28) Noting 𝑝>1, then there exists 𝑡₀ large enough satisfying ‖𝑤_𝑡0‖>𝑟, which is independent of 𝜆∈𝐼, such that for all 𝜆∈𝐼,𝐽_𝜆(𝑒)<0 with 𝑒=𝑤_𝑡0.
(ii) Since 𝑐_𝜆 is nonincreasing on 𝜆∈𝐼, then by the definition of 𝑐_𝜆 and (i), for all 𝜆∈𝐼, we have 𝑐_1/2⩾𝑐_𝜆⩾𝑐₁⩾𝑎>0.

By Lemma 7 and Theorem 6, for almost every 𝜆∈𝐼, there exists a bounded sequence {𝑢^𝜆_𝑛}⊂𝐻 such that 𝐽_𝜆(𝑢^𝜆_𝑛)→𝑐_𝜆,(𝐽_𝜆)^′(𝑢^𝜆_𝑛)→0. By Lemma 4, there exists 𝑢^𝜆∈𝐻 such that 𝑢^𝜆_𝑛→𝑢^𝜆 in 𝐻. Therefore, (𝐽_𝜆)^′(𝑢^𝜆)=0 and 𝐽_𝜆(𝑢^𝜆)=𝑐_𝜆. It follows from (ii) of Lemma 7 that 𝑢^𝜆≠0.

Therefore, there exists {𝜆_𝑛}⊂𝐼 with 𝜆_𝑛→1⁻ and a nonnegative sequence {𝑢^𝜆𝑛} (denoted by {𝑢_𝑛} for simplicity) satisfying 𝐽_𝜆𝑛𝑢_𝑛=𝑐_𝜆𝑛,𝐽_𝜆𝑛^′𝑢_𝑛=0. (29)

In order to obtain the boundedness of {𝑢_𝑛}, we need the following Pohozaev identity. The proof is similar to the argument in [10].

Lemma 8. Under the conditions of (h₁) and (h₂), if 𝑢∈𝐻 is a weak solution of (1), the following Pohozaev identity holds: 12𝑎+𝑏_ℝ3|∇𝑢|²_ℝ3|∇𝑢|²+32_ℝ³𝑢²=3𝑝+1_ℝ³|𝑢|^𝑝+1+_ℝ3(3ℎ+(∇ℎ(𝑥),𝑥))𝑢. (30)

Lemma 9. Consider {𝑢_𝑛} in (29) is bounded in 𝐻.

Proof. Firstly, since (𝐽_𝜆𝑛)^′(𝑢_𝑛)=0, then by Lemma 8, 𝑢_𝑛 satisfies the following Pohozaev identity: 𝑎2_ℝ³∇𝑢_𝑛²+𝑏2_ℝ³∇𝑢_𝑛²²+32_ℝ³𝑢²_𝑛−𝜆_𝑛3𝑝+1_ℝ³𝑢_𝑛^𝑝+1−3_ℝ3ℎ(𝑥)𝑢_𝑛−_ℝ3(∇ℎ(𝑥),𝑥)𝑢_𝑛=0. (31) On the other hand, by ((𝐽_𝜆𝑛)^′(𝑢_𝑛),𝑢_𝑛)=0 and 𝐽_𝜆𝑛(𝑢_𝑛)=𝑐_𝜆𝑛, we have that 𝑎_ℝ3∇𝑢_𝑛²+𝑏_ℝ3∇𝑢_𝑛²²+_ℝ3𝑢²_𝑛−𝜆_𝑛ℝ3𝑢_𝑛^𝑝+1−_ℝ3ℎ𝑢_𝑛=0,𝑎2_ℝ³∇𝑢_𝑛²+𝑏4_ℝ³∇𝑢_𝑛²²+12_ℝ³𝑢²_𝑛−𝜆_𝑛𝑝+1_ℝ³𝑢_n^𝑝+1−_ℝ3ℎ𝑢_𝑛=𝑐_𝜆𝑛. (32)(33) Then, combining (33) with (31), we can obtain 𝑎_ℝ3∇𝑢_𝑛²+𝑏4_ℝ³∇𝑢_𝑛²²=3𝑐_𝜆𝑛−_ℝ3(∇ℎ(𝑥),𝑥)𝑢_𝑛. (34) Since 𝑐_𝜆𝑛⩽𝑐_1/2 by Lemma 7, and (∇ℎ(𝑥),𝑥)∈𝐿²(ℝ³), so in order to prove the boundedness of {𝑢_𝑛} in 𝐻, we only need to prove that |𝑢_𝑛|₂ is bounded. By contradiction, we assume that |𝑢_𝑛|₂→∞, up to a subsequence. Let 𝑣_𝑛=𝑢_𝑛/|𝑢_𝑛|₂,𝑋_𝑛=𝑎∫_ℝ3|∇𝑣_𝑛|²,𝑌_𝑛=𝑏|𝑢_𝑛|²₂(∫_ℝ3|∇𝑣_𝑛|²)²,𝑍_𝑛=𝜆_𝑛|𝑢_𝑛|₂^𝑝−1∫_ℝ3|𝑣_𝑛|^𝑝+1.
Note that 𝑐_𝜆𝑛 is bounded and ℎ,(∇ℎ(𝑥),𝑥)∈𝐿²(ℝ³). Multiplying (31)–(33) by 1/|𝑢_𝑛|²₂, we have that 𝑋_𝑛2+𝑌_𝑛2−3𝑍𝑝+1^𝑛3=−2(1),𝑋+𝑜_𝑛+𝑌_𝑛−𝑍_𝑛(1),𝑋=−1+𝑜_𝑛2+𝑌_𝑛4−𝑍_𝑛1𝑝+1=−2+𝑜(1), (35) where 𝑜(1) denotes the quantity tends to zero as 𝑛→∞. By calculating, we obtain that 𝑋_𝑛=1−𝑝(1).5−𝑝+𝑜 (36) Since 𝑝∈(1,5) and 𝑋_𝑛⩾0 for 𝑛∈ℕ, so this is a contradiction for 𝑛 large enough. Therefore, {𝑢_𝑛} is bounded in 𝐻.

Proof of Theorem 1. Since 𝐽𝑢_𝑛=𝐽_𝜆𝑛𝑢_𝑛+𝜆_𝑛1−1𝑝+1_ℝ³𝑢_𝑛^𝑝+1,𝐽^′𝑢_𝑛,𝑣=𝐽_𝜆𝑛^′𝑢_𝑛+𝜆,𝑣_𝑛−1_ℝ3𝑢_𝑛^𝑝−1𝑢_𝑛𝑣,𝑣∈𝐻. (37) By Lemma 9 and 𝐻↪𝐿^𝑝+1(ℝ³),|𝑢_𝑛|_𝑝+1 is bounded and |∫_ℝ3|𝑢_𝑛|^𝑝−1𝑢_𝑛𝑣|⩽𝛾^𝑝+1_𝑝+1|𝑢_𝑛|^𝑝_𝑝+1‖𝑣‖. Thus, when 𝜆_𝑛→1⁻, we have that {𝐽(𝑢_𝑛)} is bounded and 𝐽^′(𝑢_𝑛)→0. Therefore {𝑢_𝑛} is a bounded PS sequence of 𝐽. By Lemma 4, {𝑢_𝑛} has a convergent subsequence. We may assume that 𝑢_𝑛→𝑣₀, up to a subsequence. Consequently, 𝐽^′(𝑣₀)=0. According to Lemma 7, we have 𝐽(𝑣₀)=lim_𝑛→∞𝐽(𝑢_𝑛)=lim_𝑛→∞𝐽_𝜆𝑛(𝑢_𝑛)⩾𝑎>0. Thus 𝑣₀ is a positive energy solution to problem (1). Hence, by Theorem 5 problem (1) has two solutions 𝑢₀ and 𝑣₀ satisfying 𝐽(𝑢₀)<0<𝐽(𝑣₀).

Acknowledgments

The authors thank the anonymous referee for the careful reading and some helpful comments, which greatly improve the paper. Projects supported by the National Natural Science Foundation of China (Grant no. 11071149), Science Council of Shanxi Province (2010011001-1, 2012011004-2, 2013021001-4).