The Existence of a Nontrivial Solution for a 𝑝-Kirchhoff Type Elliptic Equation in ℝ^𝑁

Abstract

Using Mountain Pass lemma, under some appropriate assumptions, we establish the existence of one nontrivial solution for a class of p-Kirchhoff-type elliptic equations in ℝ^𝑁.

1. Introduction and Main Result

In this paper, we consider the existence of solution for the following elliptic problem: −𝑎+𝑏_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥div𝑔(𝑥)|∇𝑢|^𝑝−2(𝑥𝑢|∇𝑢+ℎ)|^𝑝−2𝑢=𝑓(𝑥,𝑢),𝑥∈ℝ^𝑁,𝑢(𝑥)⟶0as|𝑥|⟶∞, (1) where 𝑎>1,𝑏>0,𝑁≥3, 𝑔(𝑥), and ℎ(𝑥) are positive functions, 𝑓(𝑥,𝑡)∶ℝ^𝑁×𝑅→𝑅 is continuous function, and further assumptions will be listed later. Problems like (1) originally came from the stationary problem of a model introduced by Kirchhoff [1]. Due to the existence of the integration over the whole space, problems like (1) are also called nonlocal problems.

In recent years, the Kirchhoff-type equations with 𝑝-Laplacian operator has been considered by many authors; see [2–6]. When ℎ(𝑥)≡0, the authors in [6] considered the following similar Kirchhoff-type elliptic problem on the bounded domain Ω⊂ℝ^𝑁−𝑀_Ω|∇𝑢|^𝑝𝑑𝑥^𝑝−1div|∇𝑢|^𝑝−2()𝑢(𝑥)∇𝑢=𝑓𝑥,𝑢,𝑥∈Ω,=0on𝜕Ω. (2)

The function 𝑀(𝑡) in (2) is required to meet the condition of 𝑚₀≤𝑀(𝑡)≤𝑚₁ for some constants 𝑚₀,𝑚₁>0. The authors proved that problem (2) has at least one positive solution under some other additional conditions. Problem (2) was also considered in [3], where the function 𝑓(𝑥,𝑢) is odd about 𝑢. For 𝑝=2, we refer to [7–9]. The authors in [8] studied the following Kirchhoff type problem: −_Ω𝑎+𝑏|∇𝑢|²𝑑𝑥Δ𝑢+𝑢=𝑓(𝑥,𝑢),𝑥∈ℝ^𝑁,𝑢(𝑥)⟶0as|𝑥|⟶+∞. (3)

By the Fountain theorem, the author proved the existence of infinitely many solutions. Note that one of the assumptions made on the function 𝑓(𝑥,𝑢) in (3) is that 𝑓(𝑥,𝑢)=−𝑓(𝑥,−𝑢) for any (𝑥,𝑢)∈(ℝ^𝑁,ℝ). In the present paper, however, the function 𝑓(𝑥,𝑢) is not required to be odd about 𝑢 as that of [3, 8]. When 𝑎=𝑏=0 and 𝑔(𝑥)≡1, problem (1) becomes the 𝑝-Laplacian elliptic equations without nonlocal term, and this kind of problem is also studied by many authors. For these works, we refer to [10–14] and the references therein. In [15], Liu discussed the following elliptic problem: −div|∇𝑢|^𝑝−2∇𝑢+𝑉(𝑥)|𝑢|^𝑝−2(),𝑢(𝑥)𝑢=𝑓𝑥,𝑢∈𝑊^1,𝑝ℝ^𝑁. (4)

The author proved that problem (4) has at least one ground state. The weight function 𝑉(𝑥) is required to be bounded; more precisely, there exist constants 𝛼,𝛽>0 such that 𝑉(𝑥)∈𝐶ℝ^𝑁,0<𝛼≤𝑉(𝑥)≤𝛽<+∞. (5)

We point out that the weight function ℎ(𝑥) in problem (1) is permitted to tends to infinity.

In this paper, inspired by [4, 8, 15], we consider the existence of solution of problem (1). By the variational method, we will prove that problem (1) has at least one nontrivial weak solution. Since problem (1) is considered in the whole space ℝ^𝑁, the loss of compactness of the Sobolev embedding renders the variational technique more delicate.

In this paper, we make the following hypothesis:(𝐴₁)  𝑔(𝑥)≥1, ℎ(𝑥)≥ℎ₀>0 and ℎ(𝑥)→+∞ as |𝑥|→∞,  (𝐴₂) there exists 𝜇≥2𝑝 such that 0≤𝜇𝐹(𝑥,𝑡)≤𝑡𝑓(𝑥,𝑡), where 𝐹(𝑥,𝑡)=∫^𝑡₀𝑓(𝑥,𝑠)𝑑𝑠,  (𝐴₃) there exist constants 𝑐>0 and 𝑝<𝑞<𝑝^∗ such that |𝑓(𝑥,𝑡)|≤𝑐|𝑡|^𝑞−1.

Remark 1. Throughout this paper, we denote by 𝑐 the constant which may vary from line to line but remains independent of the relevant quantities. Note that there exist many functions 𝑔(𝑥),ℎ(𝑥), and 𝑓(𝑥,𝑡) such that satisfy the assumptions of Theorem 3, for example, 𝑔(𝑥)=1+𝑥⁴, ℎ(𝑥)=𝑒^𝑥2, 𝑓(𝑥,𝑡)=𝑡^5𝑝−1/(1+𝑥⁴).
Let 𝑊₀^1,𝑝(ℝ^𝑁) be the usual Sobolev space with the norm of ‖𝑢‖=_ℝ𝑁|∇𝑢|^𝑝+|𝑢|^𝑝𝑑𝑥^1/𝑝. (6)
Denote 𝐸=𝑢∈𝑊₀^1,𝑝ℝ^𝑁∶_ℝ𝑁|∇𝑢|^𝑝+ℎ(𝑥)|𝑢|^𝑝𝑑𝑥<∞. (7)
Then, 𝐸 is a Sobolev space with the norm of ‖𝑢‖_𝐸=_ℝ𝑁(|∇𝑢|^𝑝+ℎ(𝑥)|𝑢|^𝑝)𝑑𝑥^1/𝑝. (8)
We give another space 𝑋=𝑢∈𝐸∶_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝+ℎ(𝑥)|𝑢|^𝑝𝑑𝑥<∞ (9) endowed with the norm ‖𝑢‖_𝑋=_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝+ℎ(𝑥)|𝑢|^𝑝𝑑𝑥^1/𝑝. (10)
It is not difficult to check that 𝑋 is a Banach space. The Euler functional of problem (1) is 𝐽(𝑢)=𝑎𝑝_ℝ^𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑏𝑑𝑥+2𝑝_ℝ^𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥²+1𝑝_ℝ^𝑁ℎ(𝑥)|𝑢|^𝑝𝑑𝑥−_ℝ𝑁𝐹(𝑥)𝑑𝑥, (11) where 𝐹(𝑢)=∫^𝑢₀𝑓(𝑥,𝑡)𝑑𝑡. Then, the assumptions (𝐴₁)–(𝐴₃) imply that 𝐽(𝑢)∈𝐶¹(𝑋,ℝ¹), and for any 𝜑∈𝑋, there holds 𝐽^′(𝑢),𝜑=𝑎_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝−2∇𝑢∇𝜑𝑑𝑥+𝑏_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝−2+∇𝑢∇𝜑𝑑𝑥_ℝ𝑁ℎ(𝑥)|𝑢|^𝑝−2𝑢𝜑𝑑𝑥−_ℝ𝑁𝑓(𝑥,𝑢)𝜑𝑑𝑥. (12)
Particularly, 𝐽^′(𝑢),𝑢=𝑎_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥+𝑏_ℝ𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥²+_ℝ𝑁ℎ(𝑥)|𝑢|^𝑝𝑑𝑥−_ℝ𝑁𝑓(𝑥,𝑢)𝑢𝑑𝑥. (13)

Definition 2. A function 𝑢∈𝑋 is said to be a weak solution of (1) if and only if (12) holds for any 𝜑∈𝑋.

Our main result is listed later.

Theorem 3. Assume (𝐴₁)–(𝐴₃). Then, problem (1) has at least one nontrivial weak solution in 𝑋.

This paper is organized as follows. In Section 2, we introduce some definitions and prove several lemmas which will be used later. In Section 3, we give the proof of Theorem 3 by making use of the Mountain Pass lemma.

2. Preliminary Results

In this section, we give some important lemmas, which will be needed in the proof of our main result. Particularly, one result of compact embedding on unbounded domain will be proved.

Lemma 4. Assume (𝐴₁) and 𝑝≤𝑞<𝑝^∗. Then, the embedding 𝑋↪𝐿^𝑞(ℝ^𝑁) is compact.

Proof. We split the proof into two cases.
𝐶𝑎𝑠𝑒1 (𝑞=𝑝). Let 𝐵_𝑅=𝑢∈𝑋∶‖𝑢‖_𝑋<𝑅,𝐵^𝑐_𝑅=ℝ^𝑁⧵𝐵_𝑅 (14) and 𝜂=inf_𝐵^𝑐_𝑅ℎ(𝑥); then (𝐴₁) shows that 𝜂→+∞ as 𝑅→∞. Furthermore, one gets that _𝐵𝑐𝑅|𝑢|^𝑝𝑑𝑥≤𝜂⁻¹_𝐵𝑐𝑅ℎ(𝑥)|𝑢|^𝑝𝑑𝑥, (15) which implies that _𝐵𝑐𝑅|𝑢|^𝑝𝑑𝑥^1/𝑝≤𝜂^−1/𝑝_𝐵𝑐𝑅ℎ(𝑥)|𝑢|^𝑝𝑑𝑥^1/𝑝≤𝜂^−1/𝑝_ℝ𝑁ℎ(𝑥)|𝑢|^𝑝𝑑𝑥^1/𝑝≤𝜂^−1/𝑝‖𝑢‖_𝑥. (16)
It follows from (16) that lim_𝑅→∞sup_{{0}𝑢∈𝑋⧵}‖𝑢‖_𝐿^𝑝_(𝐵^𝑐_𝑅)‖𝑢‖_𝑋=0. (17)
Let {𝑢_𝑛} be a bounded sequence of 𝑋 such that ‖𝑢_𝑛‖_𝑋≤𝑐. Then, there exists 𝑢∈𝑋 such that 𝑢_𝑛⇀𝑢. For any 𝜀>0, it follows from (17) that there exists 𝑅_𝜀 large enough such that 𝑢_{𝑛𝐿𝑝(𝐵𝑐𝑅𝜀)}≤𝜀𝑐⁻¹𝑢_𝑛𝑋≤𝜀(𝑛=1,2,…). (18)
Note that the embedding 𝑋↪𝐿^𝑝(𝐵_𝑅𝜀) is compact. Therefore, for these 𝜀 and 𝑅_𝜀, there exists 𝑁_𝜀>0 large enough such that 𝑢_𝑛−𝑢_{𝐿𝑝(𝐵𝑅𝜀)}≤𝜀 (19) for all 𝑛>𝑁_𝜀. Thus, one can get from (18) and (19) that 𝑢_𝑛−𝑢_{𝐿𝑝(ℝ𝑁)}≤𝑢_𝑛−𝑢_{𝐿𝑝(𝐵𝑅𝜀)}+𝑢_{𝑛𝐿𝑝(𝐵𝑐𝑅𝜀)}+‖𝑢‖_𝐿^𝑝_(𝐵^𝑐_𝑅𝜀)<3𝜀, (20) which implies that 𝑢_𝑛→𝑢 strongly in 𝐿^𝑞(ℝ^𝑁).
𝐶𝑎𝑠𝑒2  (𝑝<𝑞<𝑝^∗). By the Hölder and Young inequalities, one can get from (𝐴₁) that _𝐵𝑐𝑅|𝑢|^𝑞𝑑𝑥≤_𝐵𝑐𝑅|𝑢|^𝑝𝑑𝑥^{(𝑝∗−𝑞)/(𝑝∗−𝑝)}×_𝐵𝑐𝑅|𝑢|^𝑝∗𝑑𝑥^{(𝑞−𝑝)/(𝑝∗−𝑝)}≤𝜂^{−(𝑝∗−𝑞)/(𝑝∗−𝑝)}_𝐵𝑐𝑅ℎ(𝑥)|𝑢|^𝑝𝑑𝑥^{(𝑝∗−𝑞)/(𝑝∗−𝑝)}×_𝐵𝑐𝑅|𝑢|^𝑝∗𝑑𝑥^{(𝑞−𝑝)/(𝑝∗−𝑝)}≤𝑐𝜂^{−(𝑝∗−𝑞)/(𝑝∗−𝑝)}_𝐵𝑐𝑅ℎ(𝑥)|𝑢|^𝑝𝑑𝑥^𝑞/𝑝+_𝐵𝑐𝑅|𝑢|^𝑝∗𝑑𝑥^𝑞/𝑝∗≤𝑐𝜂^{−(𝑝∗−𝑞)/(𝑝∗−𝑝)}_𝐵𝑐𝑅ℎ(𝑥)|𝑢|^𝑝𝑑𝑥^𝑞/𝑝+_𝐵𝑐𝑅𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥^𝑞/𝑝≤𝑐𝜂^{−(𝑝∗−𝑞)/(𝑝∗−𝑝)}‖𝑢‖^𝑞_𝑋. (21)
Since 𝑝<𝑞<𝑝^∗ and 𝜂→+∞ as 𝑅→∞, then lim_𝑅→∞sup_{{0}𝑢∈𝑋⧵}‖𝑢‖_𝐿^𝑞_(𝐵^𝑐_𝑅)‖𝑢‖_𝑋=0. (22)
Therefore, similar to the proof of  Case1, the embedding 𝐸↪𝐿^𝑞(ℝ^𝑁) is compact.

Now, we give the definitions of Palais-Smale (simply (PS)_𝑐) sequence and (PS)_𝑐 condition.

Definition 5. Let 𝑐∈𝑅¹, 𝐽(𝑢)∈𝐶¹(𝑋,𝑅), and 𝑋 be a Banach space. The sequence {𝑢_𝑛}⊂𝑋 is said to be a (PS)_𝑐 sequence if there holds 𝐽𝑢_𝑛⟶𝑐,𝐽^′𝑢_𝑛⟶0in𝑋^∗as𝑛⟶∞. (23)
A functional 𝐽(𝑢) is said to satisfy the (PS)_𝑐 condition if any (PS)_𝑐 sequence in 𝑋 contains a convergent subsequence.

Next, we will prove that the functional 𝐽(𝑢) satisfies the (PS)_𝑐 condition.

Lemma 6. Assume   (𝐴₁)–(𝐴₃). Then, 𝐽(𝑢) satisfies the (𝑃𝑆)_𝑐 condition on 𝑋 for any 𝑐∈ℝ¹.

Proof. Let {𝑢_𝑛} be an any (PS)_𝑐 sequence in 𝑋. We divide the proof into two steps.
𝑆𝑡𝑒𝑝1 ({𝑢_𝑛} is bounded in 𝑋). Note that 𝑎>1; then, it follows from (23) and (𝐴₂) that 𝑐+𝑢_𝑛𝑋≥𝐽𝑢_𝑛−1𝐽2𝑝^′𝑢_𝑛,𝑢_𝑛=1𝑝−12𝑝𝑎_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝+1𝑑𝑥𝑝−12𝑝_ℝ^𝑁ℎ(𝑥)𝑢_𝑛^𝑝+𝑑𝑥_ℝ𝑁1𝑓2𝑝𝑥,𝑢_𝑛𝑢_𝑛≥1−𝐹𝑑𝑥2𝑝_ℝ^𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝+1𝑑𝑥2𝑝_ℝ^𝑁ℎ(𝑥)𝑢_𝑛^𝑝1𝑑𝑥=𝑢2𝑝^𝑛𝑝𝑋, (24) which implies that {𝑢_𝑛} is bounded in 𝑋.
𝑆𝑡𝑒𝑝2 ({𝑢_𝑛} converges strongly in 𝑋). Since {𝑢_𝑛} is bounded in the separable 𝑋, there exist 𝑢₀∈𝑋 and a subsequence of {𝑢_𝑛}, still denoted by {𝑢_𝑛}, such that 𝑢_𝑛⇀𝑢₀ in 𝑋. Now, we want to prove that _ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢_𝑛−𝑑𝑥_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢₀𝑑𝑥⟶0,as𝑛⟶∞. (25)
In fact, it follows from (𝐴₃) that _ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑑𝑥−_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢₀≤𝑑𝑥_ℝ𝑁𝑢_𝑛^𝑞−1𝑢_𝑛−𝑢₀𝑑𝑥. (26)
Thus, in order to prove (25), we need only to prove that _ℝ𝑁𝑢_𝑛^𝑞−1𝑢_𝑛−𝑢₀𝑑𝑥⟶0,as𝑛⟶∞. (27)
Here, we divide the proof (27) into two cases:
(i) ()𝑝𝑝≤𝑞<𝑁−1.𝑁−𝑝 (28)
By the Hölder inequality, we obtain that _ℝ𝑁𝑢_𝑛^𝑞−1𝑢_𝑛−𝑢₀≤𝑑𝑥_ℝ𝑁𝑢_𝑛^{𝑝(𝑞−1)/(𝑝−1)}𝑑𝑥^{(𝑝−1)/𝑝}×_ℝ𝑁𝑢_𝑛−𝑢₀^𝑝𝑑𝑥^1/𝑝. (29)
It is easy to check that 𝑝≤𝑝(𝑞−1)/(𝑝−1)<𝑝^∗. Then, the compact embedding in Lemma 4 shows that _ℝ𝑁𝑢_𝑛^{𝑝(𝑞−1)/(𝑝−1)}𝑑𝑥^{(𝑝−1)/𝑝}<∞,_ℝ𝑁𝑢_𝑛−𝑢₀^𝑝𝑑𝑥^1/𝑝⟶0as𝑛⟶∞, (30) which gives (27).
(ii) (𝑁−2)𝑝𝑁−𝑝≤𝑞<𝑝^∗. (31)
Let 𝜆₁=𝑝^∗/(𝑞−2). Since 1<𝑝<𝑁 and 𝑝(𝑁−2)/(𝑁−𝑝)≤𝑞<𝑝^∗, there exist 𝑝≤𝜆₂=𝜆₃<𝑝^∗ such that 1𝜆₁+1𝜆₂+1𝜆₃=1. (32)
The Hölder inequality shows that _ℝ𝑁𝑢_𝑛^𝑞−1𝑢_𝑛−𝑢₀𝑑𝑥≤_ℝ𝑁𝑢_𝑛^{(𝑞−2)𝜆1}𝑑𝑥^1/𝜆1×_ℝ𝑁𝑢_𝑛^𝜆21/𝜆2×_ℝ𝑁𝑢_𝑛−𝑢₀^𝜆3𝑑𝑥^1/𝜆3. (33)
Therefore, the embedding in Lemma 4 implies that _ℝ𝑁𝑢_𝑛^{(𝑞−2)𝜆1}𝑑𝑥^1/𝜆1<∞,_ℝ𝑁𝑢_𝑛^𝜆21/𝜆2<∞,_ℝ𝑁𝑢_𝑛−𝑢₀^𝜆3𝑑𝑥^1/𝜆3⟶0, (34) which also gives (27).
Note that (𝑁−1)𝑝/(𝑁−𝑝)<𝑝^∗ and (𝑁−2)𝑝()𝑝𝑁−𝑝≥𝑝,𝑝≥2,𝑁−2𝑁−𝑝<𝑝,1<𝑝<2. (35)
Therefore, the previous cases (i) and (ii) imply that (25) holds for all 𝑝≤𝑞<𝑝^∗.
Denote 𝐷_𝑛=𝐽^′𝑢_𝑛,𝑢_𝑛−𝐽^′𝑢_𝑛,𝑢₀+_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑑𝑥−_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢₀𝑑𝑥. (36)
Then, it follows from (23) and (25) that 𝐷_𝑛=𝑜(1) and 𝑜(1)=𝐷_𝑛=𝑎_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝−2∇𝑢_𝑛∇𝑢_𝑛−𝑢₀𝑑𝑥+𝑏_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝×𝑑𝑥_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝−2∇𝑢∇𝑢_𝑛−𝑢₀+𝑑𝑥_ℝ𝑁ℎ(𝑥)𝑢_𝑛^𝑝−2𝑢_𝑛𝑢_𝑛−𝑢₀+𝑑𝑥_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑑𝑥−_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢₀𝑑𝑥. (37)
Since 𝑢_𝑛⇀𝑢₀ in 𝑋, one obtains from the Hölder inequality that 𝑜(1)=−𝑎_ℝ𝑁𝑔(𝑥)∇𝑢₀^𝑝−2∇𝑢₀∇𝑢_𝑛−𝑢₀𝑑𝑥−𝑏_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝×𝑑𝑥_ℝ𝑁𝑔(𝑥)∇𝑢₀^𝑝−2∇𝑢₀∇𝑢_𝑛−𝑢₀−𝑑𝑥_ℝ𝑁ℎ(𝑥)𝑢₀^𝑝−2𝑢₀𝑢_𝑛−𝑢₀𝑑𝑥. (38)
Thus, it follows from (37) and (38) that 𝑜(1)=𝑎_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝−2∇𝑢_𝑛−∇𝑢₀^𝑝−2∇𝑢₀∇𝑢_𝑛−𝑢₀𝑑𝑥+𝑏_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝×𝑑𝑥_ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝−2∇𝑢_𝑛−∇𝑢₀^𝑝−2∇𝑢₀∇𝑢_𝑛−𝑢₀+𝑑𝑥_ℝ𝑁ℎ(𝑥)𝑢_𝑛^𝑝−2𝑢_𝑛−𝑢₀^𝑝−2𝑢₀𝑢_𝑛−𝑢₀+𝑑𝑥_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑑𝑥−_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑢₀𝑑𝑥. (39)
Since 𝑎≥1, 𝑏>0, and 𝑔(𝑥), ℎ(𝑥) are positive functions, we get from (25) and (39) that _ℝ𝑁𝑔(𝑥)∇𝑢_𝑛^𝑝−2∇𝑢_𝑛−∇𝑢₀^𝑝−2∇𝑢₀∇𝑢_𝑛−𝑢₀𝑑𝑥⟶0,_ℝ𝑁ℎ(𝑥)𝑢_𝑛^𝑝−2𝑢_𝑛−𝑢₀^𝑝−2𝑢₀𝑢_𝑛−𝑢₀𝑑𝑥⟶0. (40)
Therefore, we can deduce from (40) and the following standard inequalities in [16]: 𝜉−𝜁^𝑝≤𝑐𝜉^𝑝−2𝜉−𝜁^𝑝−2𝜁,𝜉−𝜁for𝑐𝜉𝑝≥2,_𝑝−2𝜉−𝜁^𝑝−2𝜁,𝜉−𝜁^𝑝/2×𝜉^𝑝+𝜁^{𝑝(2−𝑝)/2}for1<𝑝<2 (41) that 𝑢_𝑛−𝑢_0𝑋⟶0as𝑛⟶∞. (42)
Then, we complete the proof.

3. Existence of Solution

In this section, the proof of Theorem 3 is mainly based on the following Mountain Pass lemma [17] (also see [18]).

Lemma 7 (Mountain Pass lemma). Let 𝑋 be a Banach space and 𝐽∈𝐶¹(𝑋,ℝ) satisfies (𝑃𝑆)_𝑐 condition. Suppose that 𝐽(0)=0 and (𝐼₁) there are constant 𝜌,𝛼>0 such that 𝐽_𝜕𝐵𝜌≥𝛼,(𝐼₂) there is an 𝑒∈𝑋⧵𝐵_𝜌 such that 𝐽(𝑒)≤0.Then, 𝐽 possesses a critical value 𝑐≥𝛼. Moreover, 𝑐 can be characterized as 𝑐=inf_𝑔∈Γmax_{[]𝑢∈𝑔0,1}𝐽(𝑢)≥𝛼>0, (43) where Γ=𝑔∈𝐶([0,1],𝑋)∶𝑔(0)=0,𝑔(1)=𝑒. (44)

In view of Lemma 6, the functional 𝐽(𝑢) satisfies the (PS)_𝑐 condition. It is obvious that 𝐽(0)=0; then, in order to apply Lemma 7, we need only to prove that 𝐽(𝑢) satisfies the geometric conditions of the Mountain Pass lemma.

Lemma 8. Assume (𝐴₁)–(𝐴₂). Then, (𝐼₁) there exist 𝜌,𝛼>0 such that 𝐽(𝑢)≥𝛼>0 with ‖𝑢‖_𝑋=𝜌,(𝐼₂) there is an 𝑒∈𝑋 such that ‖𝑒‖_𝑋>𝜌 and 𝐽(𝑒)≤0.

Proof. It follows from (𝐴₂), (𝐴₃), and the embedding in Lemma 4 that 𝐽(𝑢)≥𝑎𝑝_ℝ^𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑏𝑑𝑥+2𝑝_ℝ^𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥²+1𝑝_ℝ^𝑁ℎ(𝑥)|𝑢|^𝑝1𝑑𝑥−𝜇_ℝ^𝑁|𝑢|^𝑞≥1𝑑𝑥𝑝_ℝ^𝑁𝑔(𝑥)|∇𝑢|^𝑝+ℎ(𝑥)|𝑢|^𝑝𝑑𝑥−𝑐‖𝑢‖^𝑞_𝑋=1𝑝‖𝑢‖^𝑝_𝑋−𝑐‖𝑢‖^𝑞_𝑋. (45) Since 𝑝<𝑞, there exists a sufficient small 𝜌>0 and 𝛼>0 such that 𝐽(𝑢)≥𝛼>0 with ‖𝑢‖_𝑋=𝜌. On the other hand, for fixed 𝑢₀∈𝑋 and 𝑡>0, it follows from (𝐴₂) that 𝐽𝑡𝑢₀≤𝑎𝑡^𝑝𝑝_ℝ^𝑁𝑔(𝑥)|∇𝑢|^𝑝+𝑑𝑥𝑏𝑡^2𝑝2𝑝_ℝ^𝑁𝑔(𝑥)|∇𝑢|^𝑝𝑑𝑥²+𝑡^𝑝𝑝_ℝ^𝑁ℎ(𝑥)|𝑢|^𝑝𝑑𝑥−𝑡^𝜇_ℝ𝑁𝑢₀^𝜇𝑑𝑥⟶−∞as𝑡⟶+∞. (46) Then, there is a large 𝑡₀>0 such that ‖𝑡₀𝑢₀‖_𝑋>𝜌 and 𝐽(𝑡₀𝑢₀)<0. One may choose 𝑒=𝑡₀𝑢₀; then, ‖𝑒‖_𝑋>𝜌 and 𝐽(𝑒)<0. Thus, the proof of Lemma 8 is complete.

Proof of Theorem 3. It follows from Lemmas 7 and 8 that the functional 𝐽(𝑢) has a critical point 𝑢 such that 𝐽(𝑢)=𝑐>0; that is, problem (1) has at least one weak solution. On the other hand, 𝐽(0)=0; then, problem (1) has at least one nontrivial weak solution.

Acknowledgment

The author is very grateful to the anonymous reviewer for valuable comments and suggestions.