Existence of Nontrivial Solutions for Perturbed 𝑝-Laplacian Equation in ℝ^𝑁 with Critical Nonlinearity

Abstract

We consider a perturbed 𝑝-Laplacian equation with critical nonlinearity in ℝ^𝑁. By using variational method, we show that it has at least one positive solution under the proper conditions.

1. Introduction and Main Results

In this paper, we are concerned with the existence of nontrivial solutions for the following nonlinear perturbed 𝑝-Laplacian equation with critical nonlinearity: −𝜀^𝑝Δ_𝑝𝑢+𝑉(𝑥)|𝑢|^𝑝−2𝑢=𝐾(𝑥)|𝑢|^𝑝∗−2𝑢+𝑓(𝑥,𝑢),𝑥∈ℝ^𝑁,𝑢(𝑥)𝑢(𝑥)>0,⟶0as|𝑥|⟶∞, (1) where Δ_𝑝𝑢=div(|∇𝑢|^𝑝−2∇𝑢) is the 𝑝-Laplacian operator with 1<𝑝<𝑁, 𝑁≥3, 𝑝^∗=𝑁𝑝/(𝑁−𝑝) denotes the Sobolev critical exponent, 𝑉(𝑥) is a nonnegative potential, 𝐾(𝑥) is a bounded positive function, and 𝑓(𝑥,𝑢) is a superlinear but subcritical function.

For 𝑝=2, (1) turns into the following Schrödinger equation of the form −𝜀²Δ𝑢+𝑉(𝑥)𝑢=𝐾(𝑥)|𝑢|^2∗−2𝑢+𝑓(𝑥,𝑢),𝑥∈ℝ^𝑁. (2) The equation (2) has been studied extensively under various hypotheses on the potential and nonlinearity by many authors including Ambrosetti and Rabinowitz [1], Bartsch and Wang [2], Brézis and Lieb [3], Brėzis and Nirenberg [4], and Del Pino and Felmer [5] in bounded domains. Meanwhile, we recall some works in unbounded domains which contain Cingolani and Lazzo [6], Clapp and Ding [7], Ding and Lin [8], Floer and Weinstein [9], Grossi [10], Jeanjean and Tanaka [11], Kang and Wei [12], Oh [13], Pistoia [14], Rabinowitz [15], and Tang [16].

For general 𝑝>1, most of the work (see [17–19] and the reference therein) dealt with (1) with 𝜀=1, 𝐾(𝑥)≡0 and a certain sign potential 𝑉(𝑥). Liu and Zheng [20] considered the above mentioned problem with sign-changing potential and subcritical 𝑝-superlinear nonlinearity. Cao et al. [21] also studied the similar problem. However, to our best knowledge, it seems that there is almost no work on the existence of semi-classical solutions to the equation in ℝ^𝑁 with critical nonlinearities. This paper will study the critical nonlinearity case in whole space.

Throughout the paper, we make the following assumption:(𝐻₁)𝑉∈𝐶(ℝ^𝑁), 𝑉(0)=inf_𝑥∈ℝ^𝑁𝑉(𝑥)=0 and there exists 𝑏>0 such that the set 𝜈^𝑏∶={𝑥∈ℝ^𝑁∶𝑉(𝑥)<𝑏} has finite Lebesgue measure;(𝐻₂)𝐾(𝑥)∈𝐶(ℝ^𝑁,ℝ⁺),  0<inf𝐾≤sup𝐾<∞;(𝐻₃)𝑓∈𝐶(ℝ^𝑁×ℝ,ℝ) and 𝑓(𝑥,𝑡)=𝑜(|𝑡|^𝑝−2𝑡) uniformly in 𝑥 as 𝑡→0;(𝐻₄)there are 𝑎₀>0 and 𝑝<𝑞<𝑝^∗ such that |𝑓(𝑥,𝑡)|≤𝑎₀(1+|𝑡|^𝑞−1) for all (𝑥,𝑡);(𝐻₅)there exist 𝑏₀>0, 𝛼>𝑝 and 𝜇∈(𝑝,𝑝^∗) such that 𝐹(𝑥,𝑡)≥𝑏₀|𝑡|^𝛼 and 𝜇𝐹(𝑥,𝑡)≤𝑓(𝑥,𝑡)𝑡 for all (𝑥,𝑡), where 𝐹(𝑥,𝑡)=∫^𝑡₀𝑓(𝑥,𝑠)𝑑𝑠.

Our main result reads as follows.

Theorem 1. Assume that (𝐻₁)--(𝐻₅) hold. Then for any 𝜎>0, there exists 𝜀_𝜎>0 such that if 𝜀≤𝜀_𝜎, (1) has at least one positive solution 𝑢_𝜀 of least energy which satisfied the following estimate: 𝜇−𝑝𝑝𝜇_ℝ^𝑁𝜀^𝑝∇𝑢_𝜀^𝑝+𝑉(𝑥)𝑢_𝜀^𝑝≤𝜎𝜀^𝑁. (3) The main tool used in the proof of Theorem 1 is variational method which was mainly developed in [8]. The main difficulty in the case is to overcome the loss of the compactness of the energy functional related to (1) because of unbounded domain ℝ^𝑁 and critical nonlinearity. Although the energy functional does not satisfy the (𝑃𝑆) condition, we can prove that it possesses (𝑃𝑆)_𝑐 condition at some energy level 𝑐.

This outline of the paper is organized as follows. In Section 2, we give the variational settings and preliminary results. In Section 3, we show that the corresponding energy functional satisfies (PS)_𝑐 condition at the levels less than 𝛼₀𝜆^{1−𝑁/𝑝} with some 𝛼₀>0 independent of 𝜆. Furthermore, it possesses the mountain geometry structure. Section 4 is devoted to the proof of the main result.

2. Preliminaries

Let 𝜆=𝜀^−𝑝 in (1). The equation (1) reads as −Δ_𝑝𝑢+𝜆𝑉(𝑥)|𝑢|^𝑝−2𝑢=𝜆𝐾(𝑥)|𝑢|^𝑝∗−2𝑢+𝜆𝑓(𝑥,𝑢),𝑥∈ℝ^𝑁,𝑢(𝑥)𝑢(𝑥)>0,⟶0,as|𝑥|⟶∞. (4) In order to prove Theorem 1, we are going to prove the following result.

Theorem 2. Assume that (𝐻₁)--(𝐻₅) is satisfied. Then for any 𝜎>0, there exists 𝜆_𝜎>0 such that if 𝜆>𝜆_𝜎, (4) has at least one positive solution 𝑢_𝜆 satisfying the following estimate: 𝜇−𝑝𝑝𝜇_ℝ^𝑁∇𝑢_𝜆^𝑝+𝜆𝑉(𝑥)𝑢_𝜆^𝑝≤𝜎𝜆^{1−𝑁/𝑝}. (5)

Next, we introduce the space 𝐸_𝜆ℝ^𝑁=,𝑉𝑢∈𝑊^1,𝑝ℝ^𝑁∶∫_ℝ𝑁𝜆𝑉(𝑥)|𝑢|^𝑝<∞,𝜆>0 (6) equipped with the norm ‖𝑢‖_𝐸𝜆=_ℝ𝑁|∇𝑢|^𝑝+𝜆𝑉(𝑥)|𝑢|^𝑝1/𝑝. (7) Note that the norm ‖⋅‖_𝐸1 is equivalent to the one ‖⋅‖_𝐸𝜆 for any 𝜆>0. It follows from (𝐻₁) that 𝐸_𝜆(ℝ^𝑁,𝑉) continuously is embedded in 𝑊^1,𝑝(ℝ^𝑁). To prove Theorem 2, one considers the 𝐶¹ functional 𝐼∶𝑊^1,𝑝(ℝ^𝑁)→ℝ defined by 𝐼_𝜆(𝑢)=1𝑝_ℝ^𝑁|∇𝑢|^𝑝+𝜆𝑉(𝑥)|𝑢|^𝑝−𝜆𝑝^∗_ℝ^𝑁𝐾(𝑥)|𝑢|^𝑝∗−𝜆_ℝ𝑁𝐹()=1𝑥,𝑢𝑝‖𝑢‖^𝑝_𝐸𝜆−𝜆_ℝ𝑁𝐺(𝑥,𝑢), (8) where 𝐺(𝑥,𝑢)=(1/𝑝^∗)𝐾(𝑥)|𝑢|^𝑝∗+𝐹(𝑥,𝑢).

Under the assumptions of Theorem 2, standard arguments [22] show that 𝐼_𝜆∈𝐶¹(𝐸_𝜆,ℝ) and its critical points are weak solutions of (4).

3. Necessary Lemmas

This section will show some lemmas which are important for the proof of the main result.

Lemma 3. Assume that (𝐻₁)–(𝐻₅) is satisfied. For the (𝑃𝑆)_𝑐 sequence {𝑢_𝑛}⊂𝐸_𝜆 for 𝐼_𝜆, we get that 𝑐≥0 and {𝑢_𝑛} is bounded in the space 𝐸_𝜆.

Proof. By direct computation and the assumptions (𝐻₂) and (𝐻₅), one has 𝐼_𝜆𝑢_𝑛−1𝜇𝐼^′_𝜆𝑢_𝑛𝑢_𝑛=1𝑝−1𝜇𝑢_𝑛^𝑝_𝐸𝜆+1𝜇−1𝑝^∗𝜆_ℝ𝑁𝐾(𝑥)𝑢_𝑛^𝑝∗+𝜆_ℝ𝑁1𝜇𝑓(𝑥,𝑢)𝑢−𝐹(𝑥,𝑢). (9) Together with 𝐼_𝜆(𝑢_𝑛)→𝑐 and 𝐼^′_𝜆(𝑢_𝑛)→0 as 𝑛→∞, we easily get that the (PS)_𝑐 sequence is bounded in 𝐸_𝜆 and the energy level 𝑐≥0.

By Lemma 3, there is 𝑢∈𝐸_𝜆 such that 𝑢_𝑛⇀𝑢 in 𝐸_𝜆. Furthermore, passing to a subsequence, we have 𝑢_𝑛→𝑢 in 𝐿^𝑏_loc(ℝ^𝑁) for any 𝑏∈[𝑝,𝑝^∗) and 𝑢_𝑛→𝑢𝑎.𝑒. in ℝ^𝑁.

Lemma 4. For any 𝑠∈[𝑝,𝑝^∗), there is a subsequence {𝑢_𝑛𝑖} such that, for any 𝜀>0, there exists 𝑟_𝜀>0 with lim_𝑖→∞sup_{𝐵𝑖⧵𝐵𝑟}𝑢_𝑛𝑖^𝑠≤𝜀𝑓𝑜𝑟𝑎𝑛𝑦𝑟≥𝑟_𝜀, (10) where 𝐵_𝑟∶={𝑥∈ℝ^𝑁∶|𝑥|≤𝑟}.

Proof. From 𝑢_𝑛→𝑢 in 𝐿^𝑠_loc(ℝ^𝑁), we have _𝐵𝑖𝑢_𝑛^𝑠⟶_𝐵𝑖|𝑢|^𝑠as𝑛⟶∞. (11) Thus, there exists 𝑛_𝑖∈ℕ such that _𝐵𝑖𝑢_𝑛^𝑠−|𝑢|^𝑠<1𝑖,∀𝑛=𝑛_𝑖+𝑗,𝑗=1,2,…. (12) In particular, for 𝑛_𝑖=𝑛_𝑖+𝑖, we have _𝐵𝑖𝑢_𝑛𝑖^𝑠−|𝑢|^𝑠<1𝑖. (13) Note that there exists 𝑟_𝜀>0 satisfying _{ℝ𝑁⧵𝐵𝑟}|𝑢|^𝑠<𝜀∀𝑟≥𝑟_𝜀. (14) Then _{𝐵𝑖⧵𝐵𝑟}𝑢_𝑛𝑖^𝑠=_{𝐵𝑖⧵𝐵𝑟}𝑢_𝑛𝑖^𝑠−|𝑢|^𝑠+_{𝐵𝑖⧵𝐵𝑟}|𝑢|^𝑠=_𝐵𝑖𝑢_𝑛𝑖^𝑠−|𝑢|^𝑠+_𝐵𝑟|𝑢|^𝑠−𝑢_𝑛𝑖^𝑠+_{𝐵𝑖⧵𝐵𝑟}|𝑢|^𝑠≤1𝑖+_{ℝ𝑁⧵𝐵𝑟}|𝑢|^𝑠+_𝐵𝑟|𝑢|^𝑠−𝑢_𝑛𝑖^𝑠≤𝜀,as𝑖⟶∞. (15) This completes the proof of Lemma 4.

Let 𝜂∈𝐶^∞(ℝ⁺) be a smooth function satisfying 0≤𝜂(𝑡)≤1, 𝜂(𝑡)=1 if 𝑡≤1 and 𝜂(𝑡)=0 if 𝑡≥2. Define 𝑢_𝑖(𝑥)=𝜂(2|𝑥|/𝑖)𝑢(𝑥). It is clear that 𝑢−𝑢_𝑖𝐸𝜆⟶0as𝑖⟶∞. (16)

Lemma 5. One has lim_𝑖→∞sup_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑖−𝑓𝑥,𝑢_𝑛𝑖−𝑢_𝑖−𝑓𝑥,𝑢_𝑖𝜑=0 (17) uniformly in 𝜑∈𝐸_𝜆 with ‖𝜑‖_𝐸𝜆≤1.

Proof. From (16) and the local compactness of Sobolev embedding, for any 𝑟≥0, we have lim_𝑖→∞sup_𝐵𝑟𝑓𝑥,𝑢_𝑛𝑖−𝑓𝑥,𝑢_𝑛𝑖−𝑢_𝑖−𝑓𝑥,𝑢_𝑖𝜑=0 (18) uniformly in ‖𝜑‖_𝐸𝜆≤1. For any 𝜀>0, it follows from (14) that lim_𝑖→∞sup_{𝐵𝑖⧵𝐵𝑟}𝑢_𝑖^𝑠≤_{ℝ𝑁⧵𝐵𝑟}|𝑢|^𝑠≤𝜀 (19) for all 𝑟≥𝑟_𝜀. By Lemma 4 and (𝐻₃)-(𝐻₄), we obtain lim_𝑖→∞sup_ℝ𝑁𝑓𝑥,𝑢_𝑛𝑖−𝑓𝑥,𝑢_𝑛𝑖−𝑢_𝑖−𝑓𝑥,𝑢_𝑖𝜑=lim_𝑖→∞sup_{𝐵𝑖⧵𝐵𝑟}𝑓𝑥,𝑢_𝑛𝑖−𝑓𝑥,𝑢_𝑛𝑖−𝑢_𝑖−𝑓𝑥,𝑢_𝑖𝜑≤𝑎₁lim_𝑖→∞sup_{𝐵𝑖⧵𝐵𝑟}𝑢_𝑛𝑖^𝑝−1+𝑢_𝑖^𝑝−1𝜑+𝑎₂lim_𝑖→∞sup_{𝐵𝑖⧵𝐵𝑟}𝑢_𝑛𝑖^𝑞−1+𝑢_𝑖^𝑞−1𝜑≤𝑎₁lim_𝑖→∞sup𝑢_𝑛𝑖𝐿^𝑝−1_{𝑝𝐵𝑖⧵𝐵𝑟}+𝑢_𝑖𝐿^𝑝−1_{𝑝𝐵𝑖⧵𝐵𝑟}𝜑_{𝐿𝑝(𝐵𝑖⧵𝐵𝑟)}+𝑎₂lim_𝑖→∞sup𝑢_𝑛𝑖𝐿^𝑞−1_{𝑞𝐵𝑖⧵𝐵𝑟}+𝑢_𝑖𝐿^𝑞−1_{𝑞𝐵𝑖⧵𝐵𝑟}×𝜑_{𝐿𝑞(𝐵𝑖⧵𝐵𝑟)}≤𝑎₃𝜀^{(𝑝−1)/𝑝}+𝑎₄𝜀^{(𝑞−1)/𝑞}. (20) This shows that the desired conclusion holds.

Lemma 6. One has along a subsequence 𝐼_𝜆𝑢_𝑛−𝑢_𝑛⟶𝑐−𝐼_𝜆(𝑢),𝐼^′_𝜆𝑢_𝑛−𝑢_𝑛⟶0𝑖𝑛𝐸_𝜆⁻¹𝑡ℎ𝑒𝑑𝑢𝑎𝑙𝑠𝑝𝑎𝑐𝑒𝑜𝑓𝐸_𝜆. (21)

Proof. By Lemma 2.1 of [23] and the arguments of [24], we have 𝐼_𝜆𝑢_𝑛−𝑢_𝑛=1𝑝_ℝ^𝑁∇𝑢_𝑛−∇𝑢_𝑛^𝑝+𝜆𝑉(𝑥)𝑢_𝑛−𝑢_𝑛^𝑝−𝜆𝑝^∗_ℝ^𝑁𝐾(𝑥)𝑢_𝑛−𝑢_𝑛^𝑝∗−𝜆_ℝ𝑁𝐹𝑥,𝑢_𝑛−𝑢_𝑛=𝐼_𝜆𝑢_𝑛−𝐼_𝜆𝑢_𝑛+𝜆𝑝^∗_ℝ^𝑁𝐾(𝑥)𝑢_𝑛^𝑝∗−𝑢_𝑛−𝑢_𝑛^𝑝∗−𝑢_𝑛^𝑝∗+𝜆_ℝ𝑁𝐹𝑥,𝑢_𝑛−𝐹𝑥,𝑢_𝑛−𝑢_𝑛−𝐹𝑥,𝑢_𝑛+𝑜(1). (22) By (16) and the similar idea of proving the Brézis-Lieb Lemma [3], we easily get lim_{𝑛→∞ℝ𝑁}𝐾(𝑥)𝑢_𝑛^𝑝∗−𝑢_𝑛−𝑢_𝑛^𝑝∗−𝑢_𝑛^𝑝∗=0,lim_{𝑛→∞ℝ𝑁}𝐹𝑥,𝑢_𝑛−𝐹𝑥,𝑢_𝑛−𝑢_𝑛−𝐹𝑥,𝑢_𝑛=0. (23) Together with the fact 𝐼_𝜆(𝑢_𝑛)→𝑐 and 𝐼_𝜆(𝑢_𝑛)→𝐼_𝜆(𝑢), one has 𝐼_𝜆𝑢_𝑛−𝑢_𝑛⟶𝑐−𝐼_𝜆(𝑢). (24) Next, we will check the fact 𝐼^′_𝜆(𝑢_𝑛−𝑢_𝑛)→0 in 𝐸_𝜆⁻¹. For any 𝜑∈𝐸_𝜆, we have 𝐼^′_𝜆𝑢_𝑛−𝑢_𝑛𝜑=𝐼^′_𝜆𝑢_𝑛𝜑−𝐼^′_𝜆𝑢_𝑛𝜑+𝜆_ℝ𝑁𝐾(𝑥)𝑢_𝑛^𝑝∗−2𝑢_𝑛−𝑢_𝑛−𝑢_𝑛^𝑝∗−2𝑢_𝑛−𝑢_𝑛−𝑢_𝑛^𝑝∗−2𝑢_𝑛𝜑+𝜆_ℝ𝑁𝑓𝑥,𝑢_𝑛−𝑓𝑥,𝑢_𝑛−𝑢_𝑛−𝑓𝑥,𝑢_𝑛𝜑+𝑜(1). (25) By the standard argument, it follows that lim_{𝑛→∞ℝ𝑁}𝐾(𝑥)𝑢_𝑛^𝑝∗−2𝑢_𝑛−𝑢_𝑛−𝑢_𝑛^𝑝∗−2𝑢_𝑛−𝑢_𝑛−𝑢_𝑛^𝑝∗−2𝑢_𝑛𝜑=0 (26) uniformly in ‖𝜑‖_𝐸𝜆≤1. Together with Lemma 5, we get the desired conclusion.

Set 𝑢¹_𝑛=𝑢_𝑛−𝑢_𝑛; then, 𝑢_𝑛−𝑢=𝑢¹_𝑛+(𝑢_𝑛−𝑢). From (16), it shows that 𝑢_𝑛→𝑢 in 𝐸_𝜆 if and only if 𝑢¹_𝑛→0 in 𝐸_𝜆.

Furthermore, we have 𝐼_𝜆𝑢¹_𝑛−1𝑝𝐼^′_𝜆𝑢¹_𝑛𝑢¹_𝑛=1𝑝−1𝑝^∗𝜆_ℝ𝑁𝐾(𝑥)𝑢¹_𝑛^𝑝∗+𝜆_ℝ𝑁1𝑝𝑢¹_𝑛𝑓𝑥,𝑢¹_𝑛−𝐹𝑥,𝑢¹_𝑛≥𝜆𝑁_ℝ^𝑁𝐾(𝑥)𝑢¹_𝑛^𝑝∗≥𝜆𝑁𝐾_min𝑢¹_𝑛^𝑝∗_𝑝∗, (27) where 𝐾_min=inf_𝑥∈ℝ^𝑁𝐾(𝑥)>0.

By the facts that 𝐼_𝜆(𝑢¹_𝑛)→𝑐−𝐼_𝜆(𝑢) and 𝐼^′_𝜆(𝑢¹_𝑛)→0in𝐸_𝜆⁻¹, one has 𝑢¹_𝑛^𝑝∗_𝑝∗≤𝑁𝑐−𝐼_𝜆(𝑢)𝜆𝐾_min+𝑜(1). (28) Let 𝑉_𝑏(𝑥)∶=max{𝑉(𝑥),𝑏}, where 𝑏 is the positive constant in the assumption (𝐻₁). Since the set 𝜈_𝑏 has finite measure and 𝑢¹_𝑛→0 in 𝐿^𝑝_loc(ℝ^𝑁), we get _ℝ𝑁𝑉(𝑥)𝑢¹_𝑛^𝑝=_ℝ𝑁𝑉_𝑏(𝑥)𝑢¹_𝑛^𝑝+𝑜(1). (29) From (𝐻₂)–(𝐻₅) and Young inequality, there exists 𝐶_𝑏>0 such that _ℝ𝑁𝐾(𝑥)𝑢¹_𝑛^𝑝∗+𝑢¹_𝑛𝑓𝑥,𝑢¹_𝑛≤𝑏𝑢¹_𝑛^𝑝_𝑝+𝐶_𝑏𝑢¹_𝑛^𝑝∗_𝑝∗. (30) Next, we consider the energy level of the functional 𝐼_𝜆 below which the (PS)_𝑐 condition held.

Lemma 7. Assume that the assumptions of Theorem 2 are satisfied. There exists 𝛼₀>0 (independent of 𝜆) such that, for any (𝑃𝑆)_𝑐 sequence {𝑢_𝑛}⊂𝐸_𝜆 for 𝐼_𝜆 with 𝑢_𝑛⇀𝑢, either 𝑢_𝑛→𝑢 in 𝐸_𝜆 or 𝑐−𝐼_𝜆(𝑢)≥𝛼₀𝜆^{1−(𝑁/𝑝)}.

Proof. Assume that 𝑢_𝑛↛𝑢; then, liminf_𝑛→∞𝑢¹_𝑛𝐸𝜆>0,𝑐−𝐼_𝜆(𝑢)>0. (31) By the Sobolev inequality, (29), and (30), we get 𝑆𝑢¹_𝑛^𝑝_𝑝∗≤_ℝ𝑁∇𝑢¹_𝑛^𝑝=_ℝ𝑁∇𝑢¹_𝑛^𝑝+𝜆𝑉(𝑥)𝑢¹_𝑛^𝑝−𝜆_ℝ𝑁𝑉(𝑥)𝑢¹_𝑛^𝑝=𝜆_ℝ𝑁𝐾(𝑥)𝑢¹_𝑛^𝑝∗+𝑢¹_𝑛𝑓𝑥,𝑢¹_𝑛×−𝜆_ℝ𝑁𝑉_𝑏(𝑥)𝑢¹_𝑛^𝑝(1)𝑢+𝑜≤𝜆𝑏¹_𝑛^𝑝_𝑝+𝜆𝐶_𝑏𝑢¹_𝑛^𝑝∗_𝑝∗−𝜆𝑏𝑢¹_𝑛^𝑝_𝑝+𝑜(1)=𝜆𝐶_𝑏𝑢¹_𝑛^𝑝∗_𝑝∗+𝑜(1), (32) where 𝑆 is the best Sobolev constant of the immersion 𝑆‖𝑢‖^𝑝_𝑝∗≤_ℝ𝑁|∇𝑢|^𝑝∀𝑢∈𝑊^1,𝑝ℝ^𝑁. (33) This gives 𝑆≤𝜆𝐶_𝑏𝑢¹_𝑛^𝑝∗_𝑝^−𝑝_∗+𝑜(1)≤𝜆𝐶_𝑏𝑁(𝑐−𝐼_𝜆(𝑢))𝜆𝐾_min^𝑝/𝑁+𝑜(1)=𝜆^{1−𝑝/𝑁}𝐶_𝑏𝑁𝐾_min^𝑝/𝑁𝑐−𝐼_𝜆(𝑢)^𝑝/𝑁+𝑜(1). (34)
Set 𝛼₀=𝑆^𝑁/𝑝𝐶_𝑏^−𝑁/𝑝𝑁⁻¹𝐾_min; then, 𝛼₀𝜆^{1−𝑁/𝑝}≤𝑐−𝐼_𝜆(𝑢)+𝑜(1). (35) This proof is completed.

From Lemma 7, we will show that 𝐼_𝜆 satisfies the following local (PS)_𝑐 condition.

Lemma 8. Assume that (𝐻₁)–(𝐻₅) is satisfied. There exists a constant 𝛼₀>0 (independent of 𝜆) such that, if a (𝑃𝑆)_𝑐 sequence {𝑢_𝑛}⊂𝐸_𝜆 for 𝐼_𝜆 satisfies 𝑐≤𝛼₀𝜆^{1−𝑁/𝑝}, the sequence {𝑢_𝑛} has a strongly convergent subsequence in 𝐸_𝜆.

Proof. By Lemma 7, we easily obtain the required conclusion.

Now, we consider 𝜆≥1. The following standard arguments show that the energy functional 𝐼_𝜆 possesses the mountain-pass structure.

Lemma 9. Under the assumptions of Theorem 2, there exist 𝛼_𝜆,𝜌_𝜆>0 such that 𝐼_𝜆(𝑢)>0𝑖𝑓0<‖𝑢‖_𝐸𝜆<𝜌_𝜆,𝐼_𝜆(𝑢)≥𝛼_𝜆𝑖𝑓‖𝑢‖_𝐸𝜆=𝜌_𝜆. (36)

Proof. By (30) and (𝐻₅), for any 𝛿>0, there is 𝐶_𝛿>0 such that _ℝ𝑁𝐺(𝑥,𝑢)≤𝛿‖𝑢‖^𝑝_𝑝+𝐶_𝛿‖𝑢‖^𝑝∗_𝑝∗. (37) Thus 𝐼_𝜆(𝑢)=1𝑝‖𝑢‖^𝑝_𝐸𝜆−𝜆_ℝ𝑁𝐺()≥1𝑥,𝑢𝑝‖𝑢‖^𝑝_𝐸𝜆−𝜆𝛿‖𝑢‖^𝑝_𝑝−𝜆𝐶_𝛿‖𝑢‖^𝑝∗_𝑝∗. (38) Observe that ‖𝑢‖^𝑝_𝑝≤𝑎₅‖𝑢‖^𝑝_𝐸𝜆. Choosing 𝛿≤(2𝑝𝜆𝑎₅)⁻¹, 𝐼_𝜆(𝑢)≥1‖𝑢‖2𝑝^𝑝𝐸𝜆−𝜆𝐶_𝛿‖𝑢‖^𝑝∗_𝑝∗. (39) The fact 𝑝^∗>𝑝 implies the desired conclusion.

Lemma 10. For any finite dimensional subspace 𝐹⊂𝐸_𝜆, we have 𝐼_𝜆(𝑢)⟶−∞,𝑢∈𝐸_𝜆𝑎𝑠‖𝑢‖_𝐸𝜆⟶∞. (40)

Proof. By the assumption (𝐻₅), one has 𝐼_𝜆(𝑢)≤1𝑝‖𝑢‖^𝑝_𝛼−𝜆𝑏₀‖𝑢‖^𝛼_𝛼∀𝑢∈𝐸_𝜆. (41) Since all norms in a finite-dimensional space are equivalent and 𝛼>𝑝, this implies the desired conclusion.

Lemma 8 shows that 𝐼_𝜆 satisfies (PS)_𝑐𝜆 condition for 𝜆 large enough and 𝑐_𝜆 small sufficiently. In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small minimax levels.

Define the functional Φ_𝜆(𝑢)=1𝑝_ℝ^𝑁|∇𝑢|^𝑝+𝜆𝑉(𝑥)|𝑢|^𝑝−𝜆𝑏_0ℝ𝑁|𝑢|^𝛼. (42) It is apparent that Φ_𝜆∈𝐶¹(𝐸_𝜆) and 𝐼_𝜆(𝑢)≤Φ_𝜆(𝑢) for all 𝑢∈𝐸_𝜆.

Note that inf_ℝ𝑁∇𝜙^𝑝∶𝜙∈𝐶^∞₀ℝ^𝑁,ℝ,𝜙_{𝐿𝛼(ℝ𝑁)}=1=0. (43) For any 𝛿>0, there is 𝜙_𝛿∈𝐶^∞₀(ℝ^𝑁,ℝ) with ‖𝜙_𝛿‖_{𝐿𝛼(ℝ𝑁)}=1 and supp𝜙_𝛿⊂𝐵_𝑟𝛿(0) such that ‖∇𝜙_𝛿‖^𝑝_𝑝<𝛿. Let 𝑒_𝜆(𝑥)=𝜙_𝛿(√^𝑝𝜆𝑥), then supp𝑒_𝜆⊂𝐵_𝜆^−1/𝑝_𝑟𝛿(0). For any 𝑡≥0, we have Φ_𝜆𝑡𝑒_𝜆=𝑡^𝑝𝑝𝑒_𝜆^𝑝_𝐸𝜆−𝑏₀𝜆𝑡^𝛼_ℝ𝑁𝜙√_𝛿^𝑝𝜆𝑥^𝛼=𝜆^{1−𝑁/𝑝}𝐽_𝜆𝑡𝜙_𝛿, (44) where 𝐽_𝜆(𝑢)=1𝑝_ℝ^𝑁|∇𝑢|^𝑝+𝑉𝜆^−(1/𝑝)𝑥|𝑢|^𝑝−𝑏_0ℝ𝑁|𝑢|^𝛼. (45) By direct computation, we easily get max_𝑡≥0𝐽_𝜆𝑡𝜙_𝛿≤𝛼−𝑝𝑝𝛼𝛼𝑏₀^{𝑝/(𝛼−𝑝)ℝ𝑁}∇𝜙_𝛿^𝑝+𝑉𝜆^−1/𝑝𝑥𝜙_𝛿^{𝑝𝛼/(𝛼−𝑝)}. (46) In connection with 𝑉(0)=0 and ‖∇𝜙_𝛿‖^𝑝_𝑝<𝛿, it shows that there exists Λ_𝛿>0 such that for all 𝜆≥Λ_𝛿, we have max_𝑡≥0𝐼_𝜆𝑡𝜙_𝛿≤𝛼−𝑝𝑝𝛼𝛼𝑏₀^{𝑝/(𝛼−𝑝)}(2𝛿)^{𝛼/(𝛼−𝑝)}𝜆^{1−𝑁/𝑝}. (47) It follows from (47) that

Lemma 11. Assume that (𝐻₁)–(𝐻₅) is satisfied. For any 𝜎>0, there is Λ_𝜎>0 such that 𝜆≥Λ_𝜎; there exists 𝑒_𝜆∈𝐸_𝜆 with ‖𝑒_𝜆‖_𝐸𝜆>𝜌_𝜆; we have 𝐼_𝜆(𝑒_𝜆)≤0 and max_𝑡≥0𝐼_𝜆𝑡𝑒_𝜆≤𝜎𝜆^{1−𝑁/𝑝}, (48) where 𝜌_𝜆 is defined in Lemma 9.

Proof. This proof is similar to the one of Lemma 4.3 in [8], so we omit it.

4. Proof of Theorem 2

In the following, we will give the proof of Theorem 2.

Proof. By Lemma 11, for any 𝜎>0 with 0<𝜎<𝛼₀, there is Λ_𝜎>0 such that for 𝜆≥Λ_𝜎, we obtain 𝑐_𝜆=inf_{𝛾∈Γ𝜆}max_𝑡∈[0,1]𝐼_𝜆𝛾(𝑡)≤𝜎𝜆^{1−𝑁/𝑝}, (49) where Γ_𝜆={𝛾∈𝐶([0,1],𝐸_𝜆)∶𝛾(0)=0,𝛾(1)=𝑒_𝜆}.
It follows from Lemma 8 that 𝐼_𝜆 satisfies (𝑃𝑆)_𝑐𝜆 condition. Hence, by the mountain-pass theorem, there exists 𝑢_𝜆∈𝐸_𝜆 which satisfies 𝐼_𝜆(𝑢_𝜆)=𝑐_𝜆 and 𝐼^′_𝜆(𝑢_𝜆)=0. Actually, 𝑢_𝜆 is a weak solution of (4). Similar to the argument in [8], we also get that 𝑢_𝜆 is a positive least energy solution.
In the end, we show that the solution 𝑢_𝜆 satisfies the estimate (5). We easily get 𝐼_𝜆𝑢_𝜆=𝐼_𝜆𝑢_𝜆−1𝜇𝐼^′_𝜆𝑢_𝜆𝑢_𝜆=1𝑝−1𝜇𝑢_𝜆^𝑝_𝐸𝜆+1𝜇−1𝑝^∗𝜆_ℝ𝑁𝐾(𝑥)𝑢_𝜆^𝑝∗+𝜆_ℝ𝑁1𝜇𝑢_𝜆𝑓𝑥,𝑢_𝜆−𝐹𝑥,𝑢_𝜆≥1𝑝−1𝜇𝑢_𝜆^𝑝_𝐸𝜆. (50) Note that 𝐼_𝜆(𝑢_𝜆)=𝑐_𝜆 and 𝑐_𝜆≤𝜎𝜆^{1−𝑁/𝑝} and it implies the required conclusion. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2013XK03) and the National Training Programs of Innovation and Entrepreneurship for Undergraduates (201310290049).