Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in ℝ^𝑁

Abstract

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: −Δ𝑢+𝑢=𝑎(𝑥)|𝑢|^𝑝−2𝑢+𝜆𝑏(𝑥)|𝑢|^𝑞−2𝑢 in ℝ^𝑁, 𝑢∈𝐻¹(ℝ^𝑁), are established, where 𝜆>0,1<𝑞<2<𝑝<2^∗(2^∗=2𝑁/(𝑁−2) if 𝑁≥3,2^∗=∞ if 𝑁=1,2), 𝑎, 𝑏 satisfy suitable conditions, and 𝑏 maybe changes sign in ℝ^𝑁. The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

1. Introduction

In this paper, we deal with the multiplicity of positive solutions for the following semilinear elliptic equation:−Δ𝑢+𝑢=𝑎(𝑥𝑢)>𝑢0^𝑝−1+𝜆𝑏(𝑥)𝑢^𝑞−1inℝ^𝑁,𝑢∈𝐻₁inℝ^𝑁,ℝ^𝑁,(𝐸_{𝑎,𝜆𝑏}) where 𝜆>0, 1<𝑞<2<𝑝<2^∗ (2^∗=2𝑁/(𝑁−2) if 𝑁≥3, 2^∗=∞ if 𝑁=1,2) and 𝑎,𝑏 are measurable functions and satisfy the following conditions: (𝑎1)0<𝑎∈𝐿^∞(ℝ^𝑁), where lim_|𝑥|→∞𝑎(𝑥)=1, and there exist 𝐶₀>0 and 𝛿₀>0 such that 𝑎(𝑥)≥1−𝐶₀𝑒^{−𝛿0|𝑥|}∀𝑥∈ℝ^𝑁.(1.1)(𝑏1)𝑏∈𝐿^𝑞∗(ℝ^𝑁)(𝑞^∗=𝑝/(𝑝−𝑞)), 𝑏⁺=max{𝑏,0}≢0, 𝑏⁻=max{−𝑏,0} is bounded and 𝑏⁻ has a compact support 𝐾 in ℝ^𝑁. (𝑏2)There exist 𝐶₁>0, 0<𝛿₁<min{𝛿₀,𝑞} and 𝑅₀>0 such that 𝑏⁺(𝑥)−𝑏(𝑥)≥𝐶₁𝑒^{−𝛿1|𝑥|}∀|𝑥|≥𝑅₀.(1.2)

Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied. For example, Ambrosetti et al. [1] considered the following equation:−Δ𝑢=𝑢𝑢>^𝑝−10+𝜆𝑢^𝑞−1inΩ,𝑢=0oinnΩ𝜕Ω,,(𝐸_𝜆) where 𝜆>0, 1<𝑞<2<𝑝<2^∗. They proved that there exists 𝜆₀>0 such that (𝐸_𝜆) admits at least two positive solutions for all 𝜆∈(0,𝜆₀), has one positive solution for 𝜆=𝜆₀ and no positive solution for 𝜆>𝜆₀. Actually, Adimurthi et al. [2], Damascelli et al. [3], Korman [4], Ouyang and Shi [5], and Tang [6] proved that there exists 𝜆₀>0 such that (𝐸_𝜆) in the unit ball 𝐵^𝑁(0;1) has exactly two positive solutions for 𝜆∈(0,𝜆₀), has exactly one positive solution for 𝜆=𝜆₀ and no positive solution exists for 𝜆>𝜆₀. For more general results of (𝐸_𝜆) (involving sign-changing weights) in bounded domains; see, the work of Ambrosetti et al. in [7], of Garcia Azorero et al. in [8], of Brown and Wu in [9], of Brown and Zhang in [10], of Cao and Zhong in [11], of de Figueiredo et al. in [12], and their references.

However, little has been done for this type of problem in ℝ^𝑁. We are only aware of the works [13–17] which studied the existence of solutions for some related concave-convex elliptic problems (not involving sign-changing weights). Furthermore, we do not know of any results for concave-convex elliptic problems involving sign-changing weight functions except [18, 19]. Wu in [18] have studied the multiplicity of positive solutions for the following equation involving sign-changing weights:−Δ𝑢+𝑢=𝑓_𝜆(𝑢𝑥)>𝑢0^𝑞−1+𝑔_𝜇(𝑥)𝑢^𝑝−1inℝ^𝑁,𝑢∈𝐻₁inℝ^𝑁,ℝ^𝑁,(𝐸_{𝑓𝜆,𝑔𝜇}) where 1<𝑞<2<𝑝<2^∗ the parameters 𝜆,𝜇≥0. He also assumed that 𝑓_𝜆(𝑥)=𝜆𝑓₊(𝑥)+𝑓₋(𝑥) is sign chaning and 𝑔_𝜇(𝑥)=𝑎(𝑥)+𝜇𝑏(𝑥), where 𝑎 and 𝑏 satisfy suitable conditions and proved that (𝐸_{𝑓𝜆,𝑔𝜇}) has at least four positive solutions.

In a recent work [19], Hsu and Lin have studied (𝐸_{𝑎,𝜆𝑏}) in ℝ^𝑁 with a sign-changing weight function. They proved there exists 𝜆₀>0 such that (𝐸_{𝑎,𝜆𝑏}) has at least two positive solutions for all 𝜆∈(0,𝜆₀) provided that 𝑎, 𝑏 satisfy suitable conditions and 𝑏 maybe changes sign in ℝ^𝑁.

Continuing our previous work [19], we consider (𝐸_{𝑎,𝜆𝑏}) in ℝ^𝑁 involving a sign-changing weight function with suitable assumptions which are different from the assumptions in [19].

In order to describe our main result, we need to defineΛ₀=2−𝑞(𝑝−𝑞)‖𝑎‖_𝐿^∞^{(2−𝑞)/(𝑝−2)}(𝑝−𝑞𝑝)−‖‖𝑏2⁺‖‖_𝐿𝑞∗𝑆^{𝑝𝑝(2−𝑞)/2(𝑝−2)+𝑞/2}>0,(1.3) where ‖𝑎‖_𝐿^∞=sup_𝑥∈ℝ^𝑁𝑎(𝑥), ‖𝑏⁺‖_𝐿𝑞∗=(∫_ℝ𝑁|𝑏⁺(𝑥)|^𝑞∗𝑑𝑥)^1/𝑞∗ and 𝑆_𝑝 is the best Sobolev constant for the imbedding of 𝐻¹(ℝ^𝑁) into 𝐿^𝑝(ℝ^𝑁).

Theorem 1.1. Assume that (𝑎1), (𝑏1)-(𝑏2) hold. If 𝜆∈(0,(𝑞/2)Λ₀), (𝐸_{𝑎,𝜆𝑏}) admits at least two positive solutions in 𝐻¹(ℝ^𝑁).

This paper is organized as follows. In Section 2, we give some notations and preliminary results. In Section 3, we establish the existence of a local minimum. In Section 4, we prove the existence of a second solution of (𝐸_{𝑎,𝜆𝑏}).

At the end of this section, we explain some notations employed. In the following discussions, we will consider 𝐻=𝐻¹(ℝ^𝑁) with the norm ‖𝑢‖=(∫_ℝ𝑁(|∇𝑢|²+𝑢²)𝑑𝑥)^1/2. We denote by 𝑆_𝑝 the best constant which is given by𝑆_𝑝=inf_{𝑢∈𝐻⧵{0}}‖𝑢‖²∫_ℝ^𝑁|𝑢|^𝑝𝑑𝑥^2/𝑝.(1.4) The dual space of 𝐻 will be denoted by 𝐻^∗. ⟨⋅,⋅⟩ denote the dual pair between 𝐻^∗ and 𝐻. We denote the norm in 𝐿^𝑠(ℝ^𝑁) by ‖⋅‖_𝐿^𝑠 for 1≤𝑠≤∞. 𝐵^𝑁(𝑥;𝑟) is a ball in ℝ^𝑁 centered at 𝑥 with radius 𝑟. 𝑜_𝑛(1) denotes 𝑜_𝑛(1)→0 as 𝑛→∞. 𝐶, 𝐶_𝑖 will denote various positive constants, the exact values of which are not important.

2. Preliminary Results

Associated with (1.3), the energy functional 𝐽_𝜆∶𝐻→ℝ^𝑁 defined by𝐽_𝜆(𝑢)=12‖𝑢‖²−1𝑝_ℝ^𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥−𝜆𝑞_ℝ^𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥,(2.1) for all 𝑢∈𝐻 is considered. It is well-known that 𝐽_𝜆∈𝐶¹(𝐻,ℝ) and the solutions of (𝐸_{𝑎,𝜆𝑏}) are the critical points of 𝐽_𝜆.

Since 𝐽_𝜆 is not bounded from below on 𝐻, we will work on the Nehari manifold. For 𝜆>0 we define𝒩_𝜆=𝑢∈𝐻⧵{0}∶𝐽^𝜆(𝑢),𝑢=0.(2.2) Note that 𝒩_𝜆 contains all nonzero solutions of (𝐸_{𝑎,𝜆𝑏}) and 𝑢∈𝒩_𝜆 if and only if𝐽^𝜆(𝑢),𝑢=‖𝑢‖²−_ℝ𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥−𝜆_ℝ𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥=0.(2.3)

Lemma 2.1. 𝐽_𝜆 is coercive and bounded from below on 𝒩_𝜆.

Proof. If 𝑢∈𝒩_𝜆, then by (𝑏1), (2.3), and the Hölder and Sobolev inequalities, one has 𝐽_𝜆(𝑢)=𝑝−22𝑝‖𝑢‖²−𝜆𝑝−𝑞𝑝𝑞_ℝ^𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥(2.4)≥𝑝−22𝑝‖𝑢‖²−𝜆𝑝−𝑞𝑝𝑞𝑆^{𝑝−𝑞/2}‖‖𝑏⁺‖‖_𝐿𝑞∗‖𝑢‖^𝑞.(2.5) Since 𝑞<2<𝑝, it follows that 𝐽_𝜆 is coercive and bounded from below on 𝒩_𝜆.

The Nehari manifold is closely linked to the behavior of the function of the form 𝜑_𝑢∶𝑡→𝐽_𝜆(𝑡𝑢) for 𝑡>0. Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in [20] and are also discussed by Brown and Zhang in [10]. If 𝑢∈𝐻, we have 𝜑_𝑢(𝑡)=𝑡²2‖𝑢‖²−𝑡^𝑝𝑝_ℝ^𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥−𝑡^𝑞𝑞𝜆_ℝ^𝑁𝜑𝑏(𝑥)|𝑢|^𝑞𝑑𝑥,_𝑢(𝑡)=𝑡‖𝑢‖²−𝑡^𝑝−1_ℝ𝑁𝜑𝑎(𝑥)|𝑢|^𝑝𝑑𝑥−𝑡^𝑞−1𝜆_ℝ𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥,_𝑢(𝑡)=‖𝑢‖²−(𝑝−1)𝑡^𝑝−2_ℝ𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥−(𝑞−1)𝑡^𝑞−2𝜆_ℝ𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥.(2.6) It is easy to see that 𝑡𝜑^𝑢(𝑡)=‖𝑡𝑢‖²−_ℝ𝑁𝑎(𝑥)|𝑡𝑢|^𝑝𝑑𝑥−𝜆_ℝ𝑁𝑏(𝑥)|𝑡𝑢|^𝑞𝑑𝑥,(2.7) and so, for 𝑢∈𝐻⧵{0} and 𝑡>0, 𝜑^𝑢(𝑡)=0 if and only if 𝑡𝑢∈𝒩_𝜆 that is, the critical points of 𝜑_𝑢 correspond to the points on the Nehari manifold. In particular, 𝜑^𝑢(1)=0 if and only if 𝑢∈𝒩_𝜆. Thus, it is natural to split 𝒩_𝜆 into three parts corresponding to local minima, local maxima, and points of inflection. Accordingly, we define 𝒩𝒩^+𝜆=𝑢∈𝒩_𝜆∶𝜑^𝑢(1)>0_0𝜆,𝒩=𝑢∈𝒩_𝜆∶𝜑^𝑢(1)=0_−𝜆,=𝑢∈𝒩_𝜆∶𝜑^𝑢(1)<0,(2.8) and note that if 𝑢∈𝒩_𝜆, that is, 𝜑^𝑢(1)=0, then𝜑^𝑢(1)=(2−𝑞)‖𝑢‖²−(𝑝−𝑞)_ℝ𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥,(2.9)=(2−𝑝)‖𝑢‖²−(𝑞−𝑝)𝜆_ℝ𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥.(2.10)

We now derive some basic properties of 𝒩^+𝜆, 𝒩^0𝜆, and 𝒩^−𝜆.

Lemma 2.2. Suppose that 𝑢₀ is a local minimizer for 𝐽_𝜆 on 𝒩_𝜆 and 𝑢₀∉𝒩^0𝜆, then 𝐽^𝜆(𝑢₀)=0 in 𝐻^∗.

Proof. See the work of Brown and Zhang in [10, Theorem 2.3].

Lemma 2.3. If 𝜆∈(0,Λ₀), then 𝒩^0𝜆=∅.

Proof. We argue by contradiction. Suppose that there exists 𝜆∈(0,Λ₀) such that 𝒩^0𝜆≠∅. Then for 𝑢∈𝒩^0𝜆 by (2.9) and the Sobolev inequality, we have 2−𝑞𝑝−𝑞‖𝑢‖²=_ℝ𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥≤‖𝑎‖_𝐿^∞𝑆^{𝑝−𝑝/2}‖𝑢‖^𝑝,(2.11) and so ‖𝑢‖≥2−𝑞(𝑝−𝑞)‖𝑎‖_𝐿^∞^1/(𝑝−2)𝑆^{𝑝𝑝/2(𝑝−2)}.(2.12) Similarly, using (2.10), Hölder and Sobolev inequalities, we have ‖𝑢‖²=𝜆𝑝−𝑞𝑝−2_ℝ^𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥≤𝜆𝑝−𝑞𝑝−2‖‖𝑏⁺‖‖_𝐿𝑞∗𝑆^{𝑝−𝑞/2}‖𝑢‖^𝑞(2.13) which implies ‖𝑢‖≤𝜆𝑝−𝑞𝑝−2‖‖𝑏⁺‖‖_𝐿𝑞∗^1/(2−𝑞)𝑆^{𝑝−𝑞/2(2−𝑞)}.(2.14) Hence, we must have 𝜆≥2−𝑞(𝑝−𝑞)‖𝑎‖_𝐿^∞^{(2−𝑞)/(𝑝−2)}(𝑝−𝑞𝑝)−‖‖𝑏2⁺‖‖_𝐿𝑞∗𝑆^{𝑝𝑝(2−𝑞)/2(𝑝−2)+𝑞/2}=Λ₀(2.15) which is a contradiction.

In order to get a better understanding of the Nehari manifold and fibering maps, we consider the function 𝜓_𝑢∶ℝ⁺→ℝ defined by 𝜓_𝑢(𝑡)=𝑡^2−𝑞‖𝑢‖²−𝑡^𝑝−𝑞_ℝ𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥for𝑡>0.(2.16) Clearly, 𝑡𝑢∈𝒩_𝜆 if and only if 𝜓_𝑢(𝑡)=𝜆∫_ℝ𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥. Moreover,𝜓^𝑢(𝑡)=(2−𝑞)𝑡^1−𝑞‖𝑢‖²−(𝑝−𝑞)𝑡^{𝑝−𝑞−1}_ℝ𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥for𝑡>0,(2.17) and so it is easy to see that if 𝑡𝑢∈𝒩_𝜆, then 𝑡^𝑞−1𝜓^𝑢(𝑡)=𝜑^𝑢(𝑡). Hence, 𝑡𝑢∈𝒩^+𝜆 (or 𝑡𝑢∈𝒩^−𝜆) if and only if 𝜓^𝑢(𝑡)>0 (or 𝜓^𝑢(𝑡)<0).

Let 𝑢∈𝐻⧵{0}. Then, by (2.17), 𝜓_𝑢 has a unique critical point at 𝑡=𝑡_max(𝑢), where𝑡_max(𝑢)=(2−𝑞)‖𝑢‖²(𝑝−𝑞)∫_ℝ𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥^1/(𝑝−2)>0,(2.18) and clearly 𝜓_𝑢 is strictly increasing on (0,𝑡_max(𝑢)) and strictly decreasing on (𝑡_max(𝑢),∞) with lim_𝑡→∞𝜓_𝑢(𝑡)=−∞. Moreover, if 𝜆∈(0,Λ₀), then 𝜓_𝑢𝑡_max(𝑢)=2−𝑞𝑝−𝑞^{(2−𝑞)/(𝑝−2)}−2−𝑞𝑝−𝑞^{(𝑝−𝑞)/(𝑝−2)}‖𝑢‖^{2(𝑝−𝑞)/(𝑝−2)}∫_ℝ^𝑁𝑎(𝑥)|𝑢|^𝑝𝑑𝑥^{(2−𝑞)/(𝑝−2)}=‖𝑢‖^𝑞𝑝𝑝−−2𝑞2𝑝−−𝑞𝑞^{2−𝑞/𝑝−2}∫_ℝ𝑁𝑎‖(𝑢𝑥‖)|^𝑝𝑢|^𝑝𝑑𝑥^{(2−𝑞)/(𝑝−2)}≥‖𝑢‖^𝑞𝑝𝑝−−2𝑞2𝑝−−𝑞𝑞^{(2−𝑞)/(𝑝−2)}𝑆^{𝑝𝑝(2−𝑞)/2(𝑝−2)}>𝜆‖‖𝑏⁺‖‖_𝐿𝑞∗𝑆^{𝑝−𝑞/2}‖𝑢‖^𝑞≥≥𝜆𝜆_{ℝ𝑁ℝ𝑁}𝑏⁺(𝑥)|𝑢|^𝑞𝑑𝑥𝑏(𝑥)|𝑢|^𝑞𝑑𝑥.(2.19)

Therefore, we have the following lemma.

Lemma 2.4. Let 𝜆∈(0,Λ₀) and 𝑢∈𝐻⧵{0}. (i)If 𝜆∫_ℝ𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥≤0, then there exists a unique 𝑡⁻=𝑡⁻(𝑢)>𝑡_max(𝑢) such that 𝑡⁻𝑢∈𝒩^−𝜆, 𝜑_𝑢 is inceasing on (0,𝑡⁻) and decreasing on (𝑡⁻,∞). Moreover, 𝐽_𝜆(𝑡⁻𝑢)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢).(2.20)(ii)If 𝜆∫_ℝ𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥>0, then there exist unique 0<𝑡⁺=𝑡⁺(𝑢)<𝑡_max(𝑢)<𝑡⁻=𝑡⁻(𝑢) such that 𝑡⁺𝑢∈𝒩^+𝜆, 𝑡⁻𝑢∈𝒩^−𝜆, 𝜑_𝑢 is decreasing on (0,𝑡⁺), inceasing on (𝑡⁺,𝑡⁻) and decreasing on (𝑡⁻,∞)𝐽_𝜆𝑡⁺𝑢=inf_{0≤𝑡≤𝑡max(𝑢)}𝐽_𝜆(𝑡𝑢),𝐽_𝜆(𝑡⁻𝑢)=sup_𝑡≥𝑡+𝐽_𝜆(𝑡𝑢).(2.21)(iii)𝒩^−𝜆={𝑢∈𝐻⧵{0}∶𝑡⁻(𝑢)=(1/‖𝑢‖)𝑡⁻(𝑢/‖𝑢‖)=1}.(iv)There exists a continuous bijection between 𝑈={𝑢∈𝐻⧵{0}∶‖𝑢‖=1} and 𝒩^−𝜆. In particular, 𝑡⁻ is a continuous function for 𝑢∈𝐻⧵{0}.

Proof. See the work of Hsu and Lin in [19, Lemma 2.5].

We remark that it follows Lemma 2.4, 𝒩_𝜆=𝒩^+𝜆∪𝒩^−𝜆 for all 𝜆∈(0,Λ₀). Furthermore, by Lemma 2.4 it follows that 𝒩^+𝜆 and 𝒩^−𝜆 are non-empty and by Lemma 2.1 we may define𝛼_𝜆=inf_{𝑢∈𝒩𝜆}𝐽_𝜆(𝑢),𝛼^+𝜆=inf_{𝑢∈𝒩+𝜆}𝐽_𝜆(𝑢),𝛼^−𝜆=inf_{𝑢∈𝒩−𝜆}𝐽_𝜆(𝑢).(2.22)

Theorem 2.5. (i) If 𝜆∈(0,Λ₀), then we have 𝛼_𝜆≤𝛼^+𝜆<0.
(ii) If 𝜆∈(0,(𝑞/2)Λ₀), then 𝛼^−𝜆>𝑑₀ for some 𝑑₀>0.
In particular, for each 𝜆∈(0,(𝑞/2)Λ₀), we have 𝛼^+𝜆=𝛼_𝜆<0<𝛼^−𝜆.

Proof. See the work of Hsu and Lin in [19, Theorem 3.1].

Remark 2.6. (i) If 𝜆∈(0,Λ₀), then by (2.9), Hölder and Sobolev inequalities, for each 𝑢∈𝒩^+𝜆 we have ‖𝑢‖²<𝑝−𝑞𝑝−2𝜆_ℝ^𝑁≤𝑝−𝑞𝑏(𝑥)|𝑢|^𝑞𝑑𝑥𝑝−2𝜆‖𝑏‖^𝐿𝑞∗𝑆^{𝑝−𝑞/2}‖𝑢‖^𝑞≤𝑝−𝑞𝑝−2Λ⁰‖𝑏‖_𝐿𝑞∗𝑆^{𝑝−𝑞/2}‖𝑢‖^𝑞,(2.23) and so ‖𝑢‖≤𝑝−𝑞𝑝−2Λ⁰‖𝑏‖_𝐿𝑞∗𝑆^{𝑝−𝑞/2}^1/(2−𝑞)∀𝑢∈𝒩^+𝜆.(2.24)
 (ii) If 𝜆∈(0,(𝑞/2)Λ₀), then by Lemma 2.4(i), (ii) and Theorem 2.5(ii), for each 𝑢∈𝒩^−𝜆 we have𝐽_𝜆(𝑢)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢)≥𝛼^−𝜆>0.(2.25)

3. Existence of a Positive Solution

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values, and (PS)-conditions in 𝐻 for 𝐽_𝜆 as follows.

Definition 3.1. (i) For 𝑐∈ℝ, a sequence {𝑢_𝑛} is a (PS)_𝑐-sequence in 𝐻 for 𝐽_𝜆 if 𝐽_𝜆(𝑢_𝑛)=𝑐+𝑜_𝑛(1) and 𝐽^𝜆(𝑢_𝑛)=𝑜_𝑛(1) strongly in 𝐻^∗ as 𝑛→∞.
(ii) 𝑐∈ℝ is a (PS)-value in 𝐻 for 𝐽_𝜆 if there exists a (PS)_𝑐-sequence in 𝐻 for 𝐽_𝜆.
(iii) 𝐽_𝜆 satisfies the (PS)_𝑐-condition in 𝐻 if any (PS)_𝑐-sequence {𝑢_𝑛} in 𝐻 for 𝐽_𝜆 contains a convergent subsequence.

Now we will ensure that there are (PS)_𝛼^+𝜆-sequence and (PS)_𝛼^−𝜆-sequencein on 𝒩_𝜆 and 𝒩^−𝜆, respectively, for the functional 𝐽_𝜆.

Proposition 3.2. If 𝜆∈(0,(𝑞/2)Λ₀), then (i)there exists a (PS)_𝛼𝜆-sequence {𝑢_𝑛}⊂𝒩_𝜆 in 𝐻 for 𝐽_𝜆.(ii)there exists a (PS)_𝛼^−𝜆-sequence {𝑢_𝑛}⊂𝒩^−𝜆 in 𝐻 for 𝐽_𝜆.

Proof. See Wu [21, Proposition 9].

Now, we establish the existence of a local minimum for 𝐽_𝜆 on 𝒩^+𝜆.

Theorem 3.3. Assume (𝑎1) and (𝑏1) hold. If 𝜆∈(0,(𝑞/2)Λ₀), then there exists 𝑢_𝜆∈𝒩^+𝜆 such that (i)𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆=𝛼^+𝜆<0, (ii)𝑢_𝜆 is a positive solution of (𝐸_{𝑎,𝜆𝑏}), (iii)‖𝑢_𝜆‖→0 as 𝜆→0⁺.

Proof. From Proposition 3.2(i) it follows that there exists {𝑢_𝑛}⊂𝒩_𝜆 satisfying 𝐽_𝜆𝑢_𝑛=𝛼_𝜆+𝑜_𝑛(1)=𝛼^+𝜆+𝑜_𝑛(1),𝐽^𝜆𝑢_𝑛=𝑜_𝑛(1)in𝐻^∗.(3.1) By Lemma 2.1 we infer that {𝑢_𝑛} is bounded on 𝐻. Passing to a subsequence (Still denoted by {𝑢_𝑛}), there exists 𝑢_𝜆∈𝐻 such that as 𝑛→∞𝑢𝑢_𝑛⇀𝑢_𝜆weaklyin𝐻,_𝑛𝑢_𝑛⟶𝑢_𝜆almosteverywhereinℝ^𝑁,⟶𝑢_𝜆stronglyin𝐿^𝑠locℝ^𝑁∀1≤𝑠<2^∗.(3.2) By (𝑏1), Egorov theorem and Hölder inequality, we have 𝜆_ℝ𝑁𝑏(𝑥)||𝑢_𝑛||^𝑞𝑑𝑥=𝜆_ℝ𝑁𝑏(𝑥)||𝑢_𝜆||^𝑞𝑑𝑥+𝑜_𝑛(1)as𝑛⟶∞.(3.3) By (3.1) and (3.2), it is easy to see that 𝑢_𝜆 is a solution of (𝐸_{𝑎,𝜆𝑏}). From 𝑢_𝑛∈𝒩_𝜆 and (2.4), we deduce that 𝜆_ℝ𝑁𝑏(𝑥)||𝑢_𝑛||^𝑞𝑑𝑥=𝑞(𝑝−2)2(𝑝−𝑞)‖‖𝑢^𝑛‖‖²−𝑝𝑞𝑝−𝑞𝐽^𝜆𝑢_𝑛.(3.4) Let 𝑛→∞ in (3.4). By (3.1), (3.3) and 𝛼_𝜆<0, we get 𝜆_ℝ𝑁𝑏(𝑥)||𝑢_𝜆||^𝑞𝑑𝑥≥−𝑝𝑞𝑝−𝑞𝛼^𝜆>0.(3.5) Thus, 𝑢_𝜆∈𝒩_𝜆 is a nonzero solution of (𝐸_{𝑎,𝜆𝑏}).
Next, we prove that 𝑢_𝑛→𝑢_𝜆 strongly in 𝐻 and 𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆. From the fact 𝑢_𝑛,𝑢_𝜆∈𝒩_𝜆 and applying Fatou's lemma, we get𝛼_𝜆≤𝐽_𝜆𝑢_𝜆=𝑝−22𝑝‖‖𝑢^𝜆‖‖²−𝑝−𝑞𝑝𝑞𝜆_ℝ^𝑁≤liminf_𝑛→∞𝑏(𝑥)||𝑢_𝜆||^𝑞𝑑𝑥𝑝2−𝑝2‖‖𝑢^𝑛‖‖²−𝑝−𝑞𝑝𝑞𝜆_ℝ^𝑁≤liminf_𝑛→∞𝑏(𝑥)||𝑢_𝑛||^𝑞𝑑𝑥𝐽_𝜆𝑢_𝑛=𝛼_𝜆.(3.6) This implies that 𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆 and lim_𝑛→∞‖𝑢_𝑛‖²=‖𝑢_𝜆‖². Standard argument shows that 𝑢_𝑛→𝑢_𝜆 strongly in 𝐻. By Theorem 2.5, for all 𝜆∈(0,(𝑞/2)Λ₀) we have that 𝑢_𝜆∈𝒩_𝜆 and 𝐽_𝜆(𝑢_𝜆)=𝛼^+𝜆<𝛼^−𝜆 which implies 𝑢_𝜆∈𝒩^+𝜆. Since 𝐽_𝜆(𝑢_𝜆)=𝐽_𝜆(|𝑢_𝜆|) and |𝑢_𝜆|∈𝒩^+𝜆, by Lemma 2.2 we may assume that 𝑢_𝜆 is a nonzero nonnegative solution of (𝐸_{𝑎,𝜆𝑏}). By Harnack inequality [22] we deduce that 𝑢_𝜆>0 in ℝ^𝑁. Finally, by (2.10), Hölder and Sobolev inequlities, ‖‖𝑢_𝜆‖‖^2−𝑞<𝜆𝑝−𝑞𝑝−2‖‖𝑏⁺‖‖_𝐿𝑞∗𝑆^{𝑝−𝑞/2},(3.7) and thus we conclude the proof.

4. Second Positive Solution

In this section, we will establish the existence of the second positive solution of (𝐸_{𝑎,𝜆𝑏}) by proving that 𝐽_𝜆 satisfies the (PS)_𝛼^−𝜆-condition.

Lemma 4.1. Assume that (𝑎1) and (𝑏1) hold. If {𝑢_𝑛}⊂𝐻 is a (PS)_𝑐-sequence for 𝐽_𝜆, then {𝑢_𝑛} is bounded in 𝐻.

Proof. See the work of Hsu and Lin in [19, Lemma 4.1].

Let us introduce the problem at infinity associated with (𝐸_{𝑎,𝜆𝑏}):−Δ𝑢+𝑢=𝑢^𝑝−1inℝ^𝑁,𝑢∈𝐻,𝑢>0inℝ^𝑁.(𝐸^∞) We state some known results for problem (𝐸^∞). First of all, we recall that by Lions [23] has studied the following minimization problem closely related to problem (𝐸^∞):𝑆^∞=inf𝐽^∞(𝑢)∶𝑢∈𝐻,𝑢≢0,(𝐽^∞)^(𝑢)=0>0,(4.1) where 𝐽^∞(𝑢)=(1/2)‖𝑢‖²−(1/𝑝)∫_ℝ𝑁|𝑢|^𝑝𝑑𝑥. Note that a minimum exists and is attained by a ground state 𝑤₀>0 in ℝ^𝑁 such that 𝑆^∞=𝐽^∞𝑤₀=sup_𝑡≥0𝐽^∞𝑡𝑤₀=12−1𝑝𝑆^{𝑝𝑝/(𝑝−2)},(4.2) where 𝑆_𝑝=inf_{𝑢∈𝐻⧵{0}}‖𝑢‖²/(∫_ℝ𝑁|𝑢|^𝑝𝑑𝑥)^2/𝑝. Gidas et al. [24] showed that for every 𝜀>0, there exist positive constants 𝐶_𝜀, 𝐶₂ such that for all 𝑥∈ℝ^𝑁,𝐶_𝜀exp(−(1+𝜀)|𝑥|)≤𝑤₀(𝑥)≤𝐶₂exp(−|𝑥|).(4.3) We define𝑤_𝑛(𝑥)=𝑤₀(𝑥−𝑛𝑒),where𝑒=(0,0,…,0,1)isaunitvectorinℝ^𝑁.(4.4) Clearly, 𝑤_𝑛(𝑥)∈𝐻.

Lemma 4.2. Let Ω be a domain in ℝ^𝑁. If 𝑓∶Ω→ℝ satisfies _Ω||𝑓(𝑥)𝑒^𝜎|𝑥|||𝑑𝑥<∞forsome𝜎>0,(4.5) then _Ω𝑓(𝑥)𝑒^{−𝜎|𝑥−}𝑥̃_|𝑑𝑥𝑒^𝜎|𝑥̃_|=_Ω𝑓(𝑥)𝑒^𝜎⟨𝑥,𝑥̃_⟩/|𝑥̃_|𝑑𝑥+𝑜(1)as||𝑥̃||⟶∞.(4.6)

Proof. We know 𝜎|𝑥̃|≤𝜎|𝑥|+𝜎|𝑥−𝑥̃|. Then, |||𝑓(𝑥)𝑒_{−𝜎|𝑥−}𝑥̃_|𝑒^𝜎|𝑥̃_||||≤||𝑓(𝑥)𝑒_𝜎|𝑥|||.(4.7) Since −𝜎|𝑥−𝑥̃|+𝜎|𝑥̃|=𝜎⟨𝑥,𝑥̃⟩/|𝑥̃|+𝑜(1) as |𝑥̃|→∞, then the lemma follows from the Lebesgue dominated convergence theorem.

Lemma 4.3. Under the assumptions (𝑎1), (𝑏1)-(𝑏2) and 𝜆∈(0,Λ₀). Then there exists a number 𝑛₀∈ℕ such that for 𝑛≥𝑛₀sup_𝑡≥0𝐽_𝜆𝑡𝑤_𝑛<𝑆^∞.(4.8) In particular, 𝛼^−𝜆<𝑆^∞ for all 𝜆∈(0,Λ₀).

Proof. (i) First, since ‖𝑤_𝑛‖=‖𝑤₀‖ for all 𝑛∈ℕ and 𝐽_𝜆 is continuous in 𝐻 and 𝐽_𝜆(0)=0, we infer that there exists 𝑡₁>0 such that 𝐽_𝜆𝑡𝑤_𝑛<𝑆^∞∀𝑛∈ℕ,𝑡∈0,𝑡₁.(4.9)
 (ii) Since lim_|𝑥|→∞𝑎(𝑥)=1, there exists 𝑛₁∈ℕ such that if 𝑛≥𝑛₁, we get 𝑎(𝑥)≥1/2 for 𝑥∈𝐵^𝑁(𝑛𝑒;1). Then, for 𝑛≥𝑛₁𝐽_𝜆𝑡𝑤_𝑛=𝑡²2‖‖𝑤^𝑛‖‖²−𝑡^𝑝𝑝_ℝ^𝑁𝑎(𝑥)||𝑤_𝑛||^𝑝𝑑𝑥−𝑡^𝑞𝑞_ℝ^𝑁≤𝑡𝜆𝑏(𝑥)||𝑤_𝑛||^𝑞𝑑𝑥₂2‖‖𝑤⁰‖‖²−𝑡^𝑝𝑝_𝐵^𝑁_(0;1)𝑎(𝑥+𝑛𝑒)||𝑤₀||^𝑝𝑑𝑥+𝑡^𝑞𝑞𝜆‖𝑏⁻‖_𝐿^∞_ℝ𝑁≤𝑡||𝑤_𝑛||^𝑞𝑑𝑥₂2‖‖𝑤⁰‖‖²−𝑡^𝑝2𝑝_𝐵^𝑁_(0;1)||𝑤₀||^𝑝𝑑𝑥+𝑡^𝑞𝑞𝜆‖𝑏⁻‖_𝐿^∞_ℝ𝑁⟶−∞||𝑤₀||^𝑞𝑑𝑥as𝑡⟶∞.(4.10) Thus, there exists 𝑡₂>0 such that for any 𝑡>𝑡₂ and 𝑛>𝑛₁ we get 𝐽_𝜆𝑡𝑤_𝑛<0.(4.11)
 (iii) By (i) and (ii), we need to show that there exists 𝑛₀ such that for 𝑛≥𝑛₀sup_{𝑡1≤𝑡≤𝑡2}𝐽_𝜆𝑡𝑤_𝑛<𝑆^∞.(4.12) We know that sup_𝑡≥0𝐽^∞(𝑡𝑤₀)=𝑆^∞. Then, 𝑡₁≤𝑡≤𝑡₂, we have 𝐽_𝜆𝑡𝑤_𝑛=12‖‖𝑡𝑤^𝑛‖‖²−1𝑝_ℝ^𝑁𝑎(𝑥)𝑡𝑤_𝑛^𝑝𝑑𝑥−1𝑞_ℝ^𝑁≤𝑡𝜆𝑏(𝑥)𝑡𝑤_𝑛^𝑞𝑑𝑥₂2‖‖𝑤⁰‖‖²−𝑡^𝑝𝑝_ℝ^𝑁𝑤^𝑝0𝑑𝑥+𝑡^𝑝𝑝_ℝ^𝑁(1−𝑎(𝑥))𝑤^𝑝𝑛𝑑𝑥−𝑡^𝑞𝑞_ℝ^𝑁≤𝑆_∞𝜆𝑏(𝑥)𝑤^𝑞𝑛𝑑𝑥+𝑡^𝑝2𝑝_ℝ^𝑁(1−𝑎)⁺(𝑥)𝑤^𝑝𝑛𝑑𝑥−𝑡^𝑞1𝑞_ℝ^𝑁𝜆𝑏⁺(𝑥)𝑤^𝑞𝑛𝑑𝑥+𝑡^𝑞2𝑞_ℝ^𝑁𝜆𝑏⁻(𝑥)𝑤^𝑞𝑛𝑑𝑥.(4.13) Suppose 𝑎 satisfies (𝑎1), we get (1−𝑎)⁺(𝑥)≤𝐶₀𝑒^{−𝛿0|𝑥|} for all 𝑥∈ℝ^𝑁 and some positive constant 𝛿₀. By (4.3) and Lemma 4.3, there exists 𝑛₂>𝑛₁ such that for any 𝑛≥𝑛₂_ℝ𝑁(1−𝑎)⁺(𝑥)𝑤^𝑝𝑛𝑑𝑥≤𝐶₃𝑒^{−min{𝛿0,𝑝}𝑛}.(4.14) By (𝑏1) and (4.3), we get _ℝ𝑁𝜆𝑏⁻(𝑥)𝑤^𝑞𝑛𝑑𝑥≤𝜆‖𝑏⁻‖_𝐿^∞𝐶₂_𝐾≤𝜆𝐶₃𝑒^{−𝑞|𝑥−𝑛𝑒|}𝑑𝑥𝑒^−𝑞𝑛.(4.15) By (𝑏2), (4.3) and Lemma 4.3, we have _ℝ𝑁𝜆𝑏⁺(𝑥)𝑤^𝑞𝑛𝑑𝑥≥𝜆𝐶₁𝐶_𝜀_{|𝑥|≥𝑅0}≥𝜆𝑒^{−𝛿1|𝑥|}𝑒^{−𝑞(1+𝜀)|𝑥−𝑛𝑒|}𝑑𝑥𝐶_𝜀𝑒^−𝛿1𝑛.(4.16) Since 0<𝛿₁<min{𝛿₀,𝑞}≤min{𝛿₀,𝑝} and 𝜆∈(0,Λ₀) and using (4.13)–(4.16), we have there exists 𝑛₀>𝑛₂ such that for all 𝑛≥𝑛₀, then sup_{𝑡1≤𝑡≤𝑡2}𝐽_𝜆𝑡𝑤_𝑛<𝑆^∞,𝜆_ℝ𝑁𝑏(𝑥)||𝑤_𝑛||^𝑞𝑑𝑥>0.(4.17) This implies that if 𝜆∈(0,Λ₀), then for all 𝑛≥𝑛₀ we get sup_𝑡≥0𝐽_𝜆𝑡𝑤_𝑛<𝑆^∞.(4.18) From 𝑎(𝑥)>0 for all 𝑥∈ℝ^𝑁 and (4.17), we have _ℝ𝑁𝑎(𝑥)||𝑤_𝑛0||^𝑝𝑑𝑥>0,_ℝ𝑁𝑏(𝑥)||𝑤_𝑛0||^𝑞𝑑𝑥>0.(4.19) Combining this with Lemma 2.4(ii), from the definition of 𝛼^−𝜆 and sup_𝑡≥0𝐽_𝜆(𝑡𝑤_𝑛0)<𝑆^∞, for all 𝜆∈(0,Λ₀), we obtain that there exists 𝑡₀>0 such that 𝑡₀𝑤_𝑛0∈𝒩^−𝜆 and 𝛼^−𝜆≤𝐽_𝜆𝑡₀𝑤_𝑛0≤sup_𝑡≥0𝐽_𝜆𝑡𝑤_𝑛0<𝑆^∞.(4.20)

Lemma 4.4. Assume that (𝑎1) and (𝑏1) hold. If {𝑢_𝑛}⊂𝐻 is a (PS)_𝑐-sequence for 𝐽_𝜆 with 𝑐∈(0,𝑆^∞), then there exists a subsequence of {𝑢_𝑛} converging weakly to a nonzero solution of (𝐸_{𝑎,𝜆𝑏}) in ℝ^𝑁.

Proof. Let {𝑢_𝑛}⊂𝐻 be a (PS)_𝑐-sequence for 𝐽_𝜆 with 𝑐∈(0,𝑆^∞). We know from Lemma 4.1 that {𝑢_𝑛} is bounded in 𝐻, and then there exist a subsequence of {𝑢_𝑛} (still denoted by {𝑢_𝑛}) and 𝑢₀∈𝐻 such that 𝑢𝑢_𝑛⇀𝑢₀weaklyin𝐻,_𝑛𝑢_𝑛→𝑢₀almosteverywhereinℝ^𝑁,→𝑢₀stronglyin𝐿^𝑠locℝ^𝑁∀1≤𝑠<2^∗.(4.21) It is easy to see that 𝐽^𝜆(𝑢₀)=0 and by (𝑏1), Egorov theorem and Hölder inequality, we have 𝜆_ℝ𝑁𝑏(𝑥)||𝑢_𝑛||^𝑞𝑑𝑥=𝜆_ℝ𝑁𝑏(𝑥)||𝑢₀||^𝑞𝑑𝑥+𝑜_𝑛(1).(4.22)
Next we verify that 𝑢₀≢0. Arguing by contradiction, we assume 𝑢₀≡0. By (𝑎1), for any 𝜀>0, there exists 𝑅₀>0 such that |𝑎(𝑥)−1|<𝜀 for all 𝑥∈[𝐵^𝑁(0;𝑅₀)]^𝐶. Since 𝑢_𝑛→0 strongly in 𝐿^𝑠loc(ℝ^𝑁) for 1≤𝑠<2^∗, {𝑢_𝑛} is a bounded sequence in 𝐻, therefore ∫_ℝ𝑁(𝑎(𝑥)−1)|𝑢_𝑛|^𝑝≤𝐶∫_{𝐵𝑁(0;𝑅0)}|𝑢_𝑛|^𝑝+𝜀𝐶. Setting 𝑛→∞, then 𝜀→0, we havelim_𝑛→∞_ℝ𝑁𝑎(𝑥)||𝑢_𝑛||^𝑝𝑑𝑥=lim_𝑛→∞_ℝ𝑁||𝑢_𝑛||^𝑝𝑑𝑥.(4.23) We set 𝑙=lim_𝑛→∞_ℝ𝑁𝑎(𝑥)||𝑢_𝑛||^𝑝𝑑𝑥=lim_𝑛→∞_ℝ𝑁||𝑢_𝑛||^𝑝𝑑𝑥.(4.24) Since 𝐽^𝜆(𝑢_𝑛)=𝑜_𝑛(1) and {𝑢_𝑛} is bounded, then by (4.22), we can deduce that 0=lim_𝑛→∞𝐽^𝜆𝑢_𝑛,𝑢_𝑛=lim_𝑛→∞‖‖𝑢^𝑛‖‖²−_ℝ𝑁𝑎(𝑥)||𝑢_𝑛||^𝑝𝑑𝑥=lim_𝑛→∞‖‖𝑢_𝑛‖‖²−𝑙,(4.25) that is, lim_𝑛→∞‖‖𝑢_𝑛‖‖²=𝑙.(4.26)
If 𝑙=0, then we get 𝑐=lim_𝑛→∞𝐽_𝜆(𝑢_𝑛)=0, which contradicts to 𝑐>0. Thus we conclude that 𝑙>0. Furthermore, by the definition of 𝑆_𝑝 we obtain‖‖𝑢_𝑛‖‖²≥𝑆_𝑝_ℝ𝑁||𝑢_𝑛||^𝑝𝑑𝑥^2/𝑝.(4.27) Then, as 𝑛→∞, we have 𝑙=lim_𝑛→∞‖‖𝑢_𝑛‖‖²≥𝑆_𝑝𝑙^2/𝑝,(4.28) which implies that 𝑙≥𝑆^{𝑝𝑝/(𝑝−2)}.(4.29) Hence, from (4.2) and (4.22)–(4.29), we get 𝑐==1lim_𝑛→∞𝐽_𝜆𝑢_𝑛2lim^𝑛→∞‖‖𝑢_𝑛‖‖²−1𝑝lim^𝑛→∞_ℝ𝑁𝑎(𝑥)||𝑢_𝑛||^𝑝𝑑𝑥−𝜆𝑞lim^𝑛→∞_ℝ𝑁=1𝑏(𝑥)||𝑢_𝑛||^𝑞𝑑𝑥≥𝑝−2−21𝑝𝑙2𝑝𝑆^{𝑝𝑝/(𝑝−2)}=𝑆^∞.(4.30) This is a contradiction to 𝑐<𝑆^∞. Therefore, 𝑢₀ is a nonzero solution of (𝐸_{𝑎,𝜆𝑏}).

Now, we establish the existence of a local minimum of 𝐽_𝜆 on 𝒩^−𝜆.

Theorem 4.5. Assume that (𝑎1) and (𝑏1)-(𝑏2) hold. If 𝜆∈(0,(𝑞/2)Λ₀), then there exists 𝑈_𝜆∈𝒩^−𝜆 such that (i)𝐽_𝜆(𝑈_𝜆)=𝛼^−𝜆,(ii)𝑈_𝜆 is a positive solution of (𝐸_{𝑎,𝜆𝑏}).

Proof. If 𝜆∈(0,(𝑞/2)Λ₀), then by Theorem 2.5(ii), Proposition 3.2(ii) and Lemma 4.3(ii), there exists a (PS)_𝛼^−𝜆-sequence {𝑢_𝑛}⊂𝒩^−𝜆 in 𝐻 for 𝐽_𝜆 with 𝛼^−𝜆∈(0,𝑆^∞). From Lemma 4.4, there exist a subsequence still denoted by {𝑢_𝑛} and a nonzero solution 𝑈_𝜆∈𝐻 of (𝐸_{𝑎,𝜆𝑏}) such that 𝑢_𝑛⇀𝑈_𝜆 weakly in 𝐻.
First, we prove that 𝑈_𝜆∈𝒩^−𝜆. On the contrary, if 𝑈_𝜆∈𝒩^+𝜆, then by 𝒩^−𝜆 is closed in 𝐻, we have ‖𝑈_𝜆‖²<liminf_𝑛→∞‖𝑢_𝑛‖². From (2.9) and 𝑎(𝑥)>0 for all 𝑥∈ℝ^𝑁, we get_ℝ𝑁𝑏(𝑥)||𝑈_𝜆||^𝑞𝑑𝑥>0,_ℝ𝑁𝑎(𝑥)||𝑈_𝜆||^𝑝𝑑𝑥>0.(4.31) By Lemma 2.4(ii), there exists a unique 𝑡^−𝜆 such that 𝑡^−𝜆𝑈_𝜆∈𝒩^−𝜆. If 𝑢∈𝒩_𝜆, then it is easy to see that 𝐽_𝜆(𝑢)=𝑝−22𝑝‖𝑢‖²−𝑝−𝑞𝑝𝑞𝜆_ℝ^𝑁𝑏(𝑥)|𝑢|^𝑞𝑑𝑥.(4.32) From (3.1), 𝑢_𝑛∈𝒩^−𝜆 and (4.32), we can deduce that 𝛼^−𝜆≤𝐽_𝜆𝑡^−𝜆𝑈_𝜆<lim_𝑛→∞𝐽_𝜆𝑡^−𝜆𝑢_𝑛≤lim_𝑛→∞𝐽_𝜆𝑢_𝑛=𝛼^−𝜆(4.33) which is a contradiction. Thus, 𝑈_𝜆∈𝒩^−𝜆.
Next, by the same argument as that in Theorem 3.3, we get that 𝑢_𝑛→𝑈_𝜆 strongly in 𝐻 and 𝐽_𝜆(𝑈_𝜆)=𝛼^−𝜆>0 for all 𝜆∈(0,(𝑞/2)Λ₀). Since 𝐽_𝜆(𝑈_𝜆)=𝐽_𝜆(|𝑈_𝜆|) and |𝑈_𝜆|∈𝒩^−𝜆, by Lemma 2.2 we may assume that 𝑈_𝜆 is a nonzero nonnegative solution of (𝐸_{𝑎,𝜆𝑏}). Finally, by the Harnack inequality [22] we deduce that 𝑈_𝜆>0 in ℝ^𝑁.

Now, we complete the proof of Theorem 1.1. By Theorems 3.3, 4.5, we obtain (𝐸_{𝑎,𝜆𝑏}) has two positive solutions 𝑢_𝜆 and 𝑈_𝜆 such that 𝑢_𝜆∈𝒩^+𝜆, 𝑈_𝜆∈𝒩^−𝜆. Since 𝒩^+𝜆∩𝒩^−𝜆=∅, this implies that 𝑢_𝜆 and 𝑈_𝜆 are distinct. It completes the proof of Theorem 1.1.